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207 KiB
Plaintext
4909 lines
207 KiB
Plaintext
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College Physics 2e
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SENIOR CONTRIBUTING AUTHORS PAUL PETER URONE, CALIFORNIA STATE UNIVERSITY, SACRAMENTO ROGER HINRICHS, STATE UNIVERSITY OF NEW YORK, COLLEGE AT OSWEGO
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OpenStax Rice University 6100 Main Street MS-375 Houston, Texas 77005
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To learn more about OpenStax, visit https://openstax.org. Individual print copies and bulk orders can be purchased through our website.
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©2022 Rice University. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). Under this license, any user of this textbook or the textbook contents herein must provide proper attribution as follows:
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- If you redistribute this textbook in a digital format (including but not limited to PDF and HTML), then you must retain on every page the following attribution: “Access for free at openstax.org.”
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For questions regarding this licensing, please contact support@openstax.org.
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Trademarks The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, OpenStax CNX logo, OpenStax Tutor name, Openstax Tutor logo, Connexions name, Connexions logo, Rice University name, and Rice University logo are not subject to the license and may not be reproduced without the prior and express written consent of Rice University.
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HARDCOVER BOOK ISBN-13 B&W PAPERBACK BOOK ISBN-13 DIGITAL VERSION ISBN-13 ORIGINAL PUBLICATION YEAR 1 2 3 4 5 6 7 8 9 10 RS 22
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978-1-711470-83-2 978-1-711470-82-5 978-1-951693-60-2
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2022
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OPENSTAX
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OpenStax provides free, peer-reviewed, openly licensed textbooks for introductory college and Advanced Placement® courses and low-cost, personalized courseware that helps students learn. A nonprofit ed tech initiative based at Rice University, we’re committed to helping students access the tools they need to complete their courses and meet their educational goals.
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RICE UNIVERSITY
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OpenStax, OpenStax CNX, and OpenStax Tutor are initiatives of Rice University. As a leading research university with a distinctive commitment to undergraduate education, Rice University aspires to path-breaking research, unsurpassed teaching, and contributions to the betterment of our world. It seeks to fulfill this mission by cultivating a diverse community of learning and discovery that produces leaders across the spectrum of human endeavor.
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PHILANTHROPIC SUPPORT
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OpenStax is grateful for the generous philanthropic partners who advance our mission to improve educational access and learning for everyone. To see the impact of our supporter community and our most updated list of partners, please visit openstax.org/impact.
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Arnold Ventures Chan Zuckerberg Initiative Chegg, Inc. Arthur and Carlyse Ciocca Charitable Foundation Digital Promise Ann and John Doerr Bill & Melinda Gates Foundation Girard Foundation Google Inc. The William and Flora Hewlett Foundation The Hewlett-Packard Company Intel Inc. Rusty and John Jaggers The Calvin K. Kazanjian Economics Foundation Charles Koch Foundation Leon Lowenstein Foundation, Inc. The Maxfield Foundation
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Burt and Deedee McMurtry Michelson 20MM Foundation National Science Foundation The Open Society Foundations Jumee Yhu and David E. Park III Brian D. Patterson USA-International Foundation The Bill and Stephanie Sick Fund Steven L. Smith & Diana T. Go Stand Together Robin and Sandy Stuart Foundation The Stuart Family Foundation Tammy and Guillermo Treviño Valhalla Charitable Foundation White Star Education Foundation Schmidt Futures William Marsh Rice University
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Our books are free and flexible, forever. Get started at openstax.org/details/books/college-physics-2e
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Access. The future of education. openstax.org
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Contents
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Preface
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1
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CHAPTER 1
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Introduction: The Nature of Science and Physics 5
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Introduction to Science and the Realm of Physics, Physical Quantities, and Units
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5
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1.1 Physics: An Introduction
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6
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1.2 Physical Quantities and Units
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15
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1.3 Accuracy, Precision, and Significant Figures
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23
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1.4 Approximation
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29
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Glossary
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32
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Section Summary
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32
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Conceptual Questions
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33
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Problems & Exercises
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33
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CHAPTER 2
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Kinematics 37
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Introduction to One-Dimensional Kinematics
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37
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2.1 Displacement
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38
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2.2 Vectors, Scalars, and Coordinate Systems
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40
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2.3 Time, Velocity, and Speed
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42
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2.4 Acceleration
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47
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2.5 Motion Equations for Constant Acceleration in One Dimension
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58
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2.6 Problem-Solving Basics for One-Dimensional Kinematics
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69
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2.7 Falling Objects
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71
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2.8 Graphical Analysis of One-Dimensional Motion
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80
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Glossary
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88
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Section Summary
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88
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Conceptual Questions
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90
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Problems & Exercises
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92
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CHAPTER 3
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Two-Dimensional Kinematics
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Introduction to Two-Dimensional Kinematics
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99
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3.1 Kinematics in Two Dimensions: An Introduction
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3.2 Vector Addition and Subtraction: Graphical Methods
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3.3 Vector Addition and Subtraction: Analytical Methods
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3.4 Projectile Motion
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117
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3.5 Addition of Velocities
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126
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Glossary
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135
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Section Summary
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135
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Conceptual Questions
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137
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Problems & Exercises
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138
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99
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CHAPTER 4
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Dynamics: Force and Newton's Laws of Motion 147
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Introduction to Dynamics: Newton’s Laws of Motion
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147
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4.1 Development of Force Concept
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149
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4.2 Newton’s First Law of Motion: Inertia
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150
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4.3 Newton’s Second Law of Motion: Concept of a System
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151
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4.4 Newton’s Third Law of Motion: Symmetry in Forces
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157
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4.5 Normal, Tension, and Other Examples of Forces
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161
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4.6 Problem-Solving Strategies
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170
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4.7 Further Applications of Newton’s Laws of Motion
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172
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4.8 Extended Topic: The Four Basic Forces—An Introduction
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179
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Glossary
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185
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Section Summary
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185
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Conceptual Questions
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187
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Problems & Exercises
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189
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CHAPTER 5
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Further Applications of Newton's Laws: Friction, Drag, and Elasticity 197
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Introduction: Further Applications of Newton’s Laws
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197
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5.1 Friction
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198
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5.2 Drag Forces
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204
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5.3 Elasticity: Stress and Strain
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209
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Glossary
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220
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Section Summary
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220
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Conceptual Questions
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221
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Problems & Exercises
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221
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CHAPTER 6
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Uniform Circular Motion and Gravitation 227
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Introduction to Uniform Circular Motion and Gravitation
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227
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6.1 Rotation Angle and Angular Velocity
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228
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6.2 Centripetal Acceleration
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232
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6.3 Centripetal Force
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235
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6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force
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240
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6.5 Newton’s Universal Law of Gravitation
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244
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6.6 Satellites and Kepler’s Laws: An Argument for Simplicity
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252
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Glossary
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258
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Section Summary
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258
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Conceptual Questions
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259
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Problems & Exercises
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262
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CHAPTER 7
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Work, Energy, and Energy Resources 269
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Introduction to Work, Energy, and Energy Resources
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269
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7.1 Work: The Scientific Definition
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270
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7.2 Kinetic Energy and the Work-Energy Theorem
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273
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7.3 Gravitational Potential Energy
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278
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7.4 Conservative Forces and Potential Energy
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284
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Access for free at openstax.org
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7.5 Nonconservative Forces
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288
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7.6 Conservation of Energy
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293
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7.7 Power
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297
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7.8 Work, Energy, and Power in Humans
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302
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7.9 World Energy Use
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306
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Glossary
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310
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Section Summary
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310
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Conceptual Questions
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312
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Problems & Exercises
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313
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CHAPTER 8
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Linear Momentum and Collisions 323
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Introduction to Linear Momentum and Collisions
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323
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8.1 Linear Momentum and Force
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324
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8.2 Impulse
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326
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8.3 Conservation of Momentum
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329
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8.4 Elastic Collisions in One Dimension
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333
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8.5 Inelastic Collisions in One Dimension
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336
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8.6 Collisions of Point Masses in Two Dimensions
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340
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8.7 Introduction to Rocket Propulsion
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344
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Glossary
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348
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Section Summary
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348
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Conceptual Questions
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349
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Problems & Exercises
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350
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CHAPTER 9
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Statics and Torque 357
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Introduction to Statics and Torque
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357
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9.1 The First Condition for Equilibrium
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358
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9.2 The Second Condition for Equilibrium
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359
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|
|
|||
|
|
9.3 Stability
|
|||
|
|
|
|||
|
|
364
|
|||
|
|
|
|||
|
|
9.4 Applications of Statics, Including Problem-Solving Strategies
|
|||
|
|
|
|||
|
|
368
|
|||
|
|
|
|||
|
|
9.5 Simple Machines
|
|||
|
|
|
|||
|
|
372
|
|||
|
|
|
|||
|
|
9.6 Forces and Torques in Muscles and Joints
|
|||
|
|
|
|||
|
|
376
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
382
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
382
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
383
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
384
|
|||
|
|
|
|||
|
|
CHAPTER 10
|
|||
|
|
Rotational Motion and Angular Momentum 391
|
|||
|
|
|
|||
|
|
Introduction to Rotational Motion and Angular Momentum
|
|||
|
|
|
|||
|
|
391
|
|||
|
|
|
|||
|
|
10.1 Angular Acceleration
|
|||
|
|
|
|||
|
|
392
|
|||
|
|
|
|||
|
|
10.2 Kinematics of Rotational Motion
|
|||
|
|
|
|||
|
|
397
|
|||
|
|
|
|||
|
|
10.3 Dynamics of Rotational Motion: Rotational Inertia
|
|||
|
|
|
|||
|
|
402
|
|||
|
|
|
|||
|
|
10.4 Rotational Kinetic Energy: Work and Energy Revisited
|
|||
|
|
|
|||
|
|
407
|
|||
|
|
|
|||
|
|
10.5 Angular Momentum and Its Conservation
|
|||
|
|
|
|||
|
|
415
|
|||
|
|
|
|||
|
|
10.6 Collisions of Extended Bodies in Two Dimensions
|
|||
|
|
|
|||
|
|
421
|
|||
|
|
|
|||
|
|
10.7 Gyroscopic Effects: Vector Aspects of Angular Momentum
|
|||
|
|
|
|||
|
|
426
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
429
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
429 430 433
|
|||
|
|
|
|||
|
|
CHAPTER 11
|
|||
|
|
Fluid Statics 439
|
|||
|
|
|
|||
|
|
Introduction to Fluid Statics
|
|||
|
|
|
|||
|
|
439
|
|||
|
|
|
|||
|
|
11.1 What Is a Fluid?
|
|||
|
|
|
|||
|
|
439
|
|||
|
|
|
|||
|
|
11.2 Density
|
|||
|
|
|
|||
|
|
441
|
|||
|
|
|
|||
|
|
11.3 Pressure
|
|||
|
|
|
|||
|
|
444
|
|||
|
|
|
|||
|
|
11.4 Variation of Pressure with Depth in a Fluid
|
|||
|
|
|
|||
|
|
446
|
|||
|
|
|
|||
|
|
11.5 Pascal’s Principle
|
|||
|
|
|
|||
|
|
450
|
|||
|
|
|
|||
|
|
11.6 Gauge Pressure, Absolute Pressure, and Pressure Measurement
|
|||
|
|
|
|||
|
|
453
|
|||
|
|
|
|||
|
|
11.7 Archimedes’ Principle
|
|||
|
|
|
|||
|
|
457
|
|||
|
|
|
|||
|
|
11.8 Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
|
|||
|
|
|
|||
|
|
464
|
|||
|
|
|
|||
|
|
11.9 Pressures in the Body
|
|||
|
|
|
|||
|
|
473
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
479
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
479
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
480
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
482
|
|||
|
|
|
|||
|
|
CHAPTER 12
|
|||
|
|
Fluid Dynamics and Its Biological and Medical Applications
|
|||
|
|
|
|||
|
|
489
|
|||
|
|
|
|||
|
|
Introduction to Fluid Dynamics and Its Biological and Medical Applications
|
|||
|
|
|
|||
|
|
489
|
|||
|
|
|
|||
|
|
12.1 Flow Rate and Its Relation to Velocity
|
|||
|
|
|
|||
|
|
490
|
|||
|
|
|
|||
|
|
12.2 Bernoulli’s Equation
|
|||
|
|
|
|||
|
|
493
|
|||
|
|
|
|||
|
|
12.3 The Most General Applications of Bernoulli’s Equation
|
|||
|
|
|
|||
|
|
498
|
|||
|
|
|
|||
|
|
12.4 Viscosity and Laminar Flow; Poiseuille’s Law
|
|||
|
|
|
|||
|
|
501
|
|||
|
|
|
|||
|
|
12.5 The Onset of Turbulence
|
|||
|
|
|
|||
|
|
509
|
|||
|
|
|
|||
|
|
12.6 Motion of an Object in a Viscous Fluid
|
|||
|
|
|
|||
|
|
511
|
|||
|
|
|
|||
|
|
12.7 Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes
|
|||
|
|
|
|||
|
|
513
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
518
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
518
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
519
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
522
|
|||
|
|
|
|||
|
|
CHAPTER 13
|
|||
|
|
Temperature, Kinetic Theory, and the Gas Laws
|
|||
|
|
|
|||
|
|
Introduction to Temperature, Kinetic Theory, and the Gas Laws
|
|||
|
|
|
|||
|
|
529
|
|||
|
|
|
|||
|
|
13.1 Temperature
|
|||
|
|
|
|||
|
|
530
|
|||
|
|
|
|||
|
|
13.2 Thermal Expansion of Solids and Liquids
|
|||
|
|
|
|||
|
|
537
|
|||
|
|
|
|||
|
|
13.3 The Ideal Gas Law
|
|||
|
|
|
|||
|
|
544
|
|||
|
|
|
|||
|
|
13.4 Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature
|
|||
|
|
|
|||
|
|
13.5 Phase Changes
|
|||
|
|
|
|||
|
|
559
|
|||
|
|
|
|||
|
|
13.6 Humidity, Evaporation, and Boiling
|
|||
|
|
|
|||
|
|
563
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
569
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
569
|
|||
|
|
|
|||
|
|
529 551
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
571
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
572
|
|||
|
|
|
|||
|
|
CHAPTER 14
|
|||
|
|
Heat and Heat Transfer Methods 577
|
|||
|
|
|
|||
|
|
Introduction to Heat and Heat Transfer Methods
|
|||
|
|
|
|||
|
|
577
|
|||
|
|
|
|||
|
|
14.1 Heat
|
|||
|
|
|
|||
|
|
577
|
|||
|
|
|
|||
|
|
14.2 Temperature Change and Heat Capacity
|
|||
|
|
|
|||
|
|
579
|
|||
|
|
|
|||
|
|
14.3 Phase Change and Latent Heat
|
|||
|
|
|
|||
|
|
585
|
|||
|
|
|
|||
|
|
14.4 Heat Transfer Methods
|
|||
|
|
|
|||
|
|
592
|
|||
|
|
|
|||
|
|
14.5 Conduction
|
|||
|
|
|
|||
|
|
593
|
|||
|
|
|
|||
|
|
14.6 Convection
|
|||
|
|
|
|||
|
|
599
|
|||
|
|
|
|||
|
|
14.7 Radiation
|
|||
|
|
|
|||
|
|
604
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
611
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
611
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
612
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
614
|
|||
|
|
|
|||
|
|
CHAPTER 15
|
|||
|
|
Thermodynamics 623
|
|||
|
|
|
|||
|
|
Introduction to Thermodynamics
|
|||
|
|
|
|||
|
|
623
|
|||
|
|
|
|||
|
|
15.1 The First Law of Thermodynamics
|
|||
|
|
|
|||
|
|
624
|
|||
|
|
|
|||
|
|
15.2 The First Law of Thermodynamics and Some Simple Processes
|
|||
|
|
|
|||
|
|
629
|
|||
|
|
|
|||
|
|
15.3 Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
|
|||
|
|
|
|||
|
|
637
|
|||
|
|
|
|||
|
|
15.4 Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
|
|||
|
|
|
|||
|
|
644
|
|||
|
|
|
|||
|
|
15.5 Applications of Thermodynamics: Heat Pumps and Refrigerators
|
|||
|
|
|
|||
|
|
648
|
|||
|
|
|
|||
|
|
15.6 Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
|
|||
|
|
|
|||
|
|
653
|
|||
|
|
|
|||
|
|
15.7 Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation
|
|||
|
|
|
|||
|
|
661
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
667
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
667
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
669
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
671
|
|||
|
|
|
|||
|
|
CHAPTER 16
|
|||
|
|
Oscillatory Motion and Waves 677
|
|||
|
|
|
|||
|
|
Introduction to Oscillatory Motion and Waves
|
|||
|
|
|
|||
|
|
677
|
|||
|
|
|
|||
|
|
16.1 Hooke’s Law: Stress and Strain Revisited
|
|||
|
|
|
|||
|
|
678
|
|||
|
|
|
|||
|
|
16.2 Period and Frequency in Oscillations
|
|||
|
|
|
|||
|
|
682
|
|||
|
|
|
|||
|
|
16.3 Simple Harmonic Motion: A Special Periodic Motion
|
|||
|
|
|
|||
|
|
684
|
|||
|
|
|
|||
|
|
16.4 The Simple Pendulum
|
|||
|
|
|
|||
|
|
689
|
|||
|
|
|
|||
|
|
16.5 Energy and the Simple Harmonic Oscillator
|
|||
|
|
|
|||
|
|
691
|
|||
|
|
|
|||
|
|
16.6 Uniform Circular Motion and Simple Harmonic Motion
|
|||
|
|
|
|||
|
|
694
|
|||
|
|
|
|||
|
|
16.7 Damped Harmonic Motion
|
|||
|
|
|
|||
|
|
697
|
|||
|
|
|
|||
|
|
16.8 Forced Oscillations and Resonance
|
|||
|
|
|
|||
|
|
701
|
|||
|
|
|
|||
|
|
16.9 Waves
|
|||
|
|
|
|||
|
|
704
|
|||
|
|
|
|||
|
|
16.10 Superposition and Interference
|
|||
|
|
|
|||
|
|
707
|
|||
|
|
|
|||
|
|
16.11 Energy in Waves: Intensity
|
|||
|
|
|
|||
|
|
713
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
717
|
|||
|
|
|
|||
|
|
Section Summary Conceptual Questions Problems & Exercises
|
|||
|
|
|
|||
|
|
717 719 720
|
|||
|
|
|
|||
|
|
CHAPTER 17
|
|||
|
|
Physics of Hearing 725
|
|||
|
|
|
|||
|
|
Introduction to the Physics of Hearing
|
|||
|
|
|
|||
|
|
725
|
|||
|
|
|
|||
|
|
17.1 Sound
|
|||
|
|
|
|||
|
|
726
|
|||
|
|
|
|||
|
|
17.2 Speed of Sound, Frequency, and Wavelength
|
|||
|
|
|
|||
|
|
728
|
|||
|
|
|
|||
|
|
17.3 Sound Intensity and Sound Level
|
|||
|
|
|
|||
|
|
732
|
|||
|
|
|
|||
|
|
17.4 Doppler Effect and Sonic Booms
|
|||
|
|
|
|||
|
|
738
|
|||
|
|
|
|||
|
|
17.5 Sound Interference and Resonance: Standing Waves in Air Columns
|
|||
|
|
|
|||
|
|
743
|
|||
|
|
|
|||
|
|
17.6 Hearing
|
|||
|
|
|
|||
|
|
750
|
|||
|
|
|
|||
|
|
17.7 Ultrasound
|
|||
|
|
|
|||
|
|
757
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
766
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
766
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
767
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
768
|
|||
|
|
|
|||
|
|
CHAPTER 18
|
|||
|
|
Electric Charge and Electric Field
|
|||
|
|
|
|||
|
|
Introduction to Electric Charge and Electric Field
|
|||
|
|
|
|||
|
|
773
|
|||
|
|
|
|||
|
|
18.1 Static Electricity and Charge: Conservation of Charge
|
|||
|
|
|
|||
|
|
18.2 Conductors and Insulators
|
|||
|
|
|
|||
|
|
780
|
|||
|
|
|
|||
|
|
18.3 Coulomb’s Law
|
|||
|
|
|
|||
|
|
783
|
|||
|
|
|
|||
|
|
18.4 Electric Field: Concept of a Field Revisited
|
|||
|
|
|
|||
|
|
786
|
|||
|
|
|
|||
|
|
18.5 Electric Field Lines: Multiple Charges
|
|||
|
|
|
|||
|
|
788
|
|||
|
|
|
|||
|
|
18.6 Electric Forces in Biology
|
|||
|
|
|
|||
|
|
792
|
|||
|
|
|
|||
|
|
18.7 Conductors and Electric Fields in Static Equilibrium
|
|||
|
|
|
|||
|
|
18.8 Applications of Electrostatics
|
|||
|
|
|
|||
|
|
797
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
804
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
805
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
806
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
809
|
|||
|
|
|
|||
|
|
773 775
|
|||
|
|
793
|
|||
|
|
|
|||
|
|
CHAPTER 19
|
|||
|
|
Electric Potential and Electric Field 817
|
|||
|
|
|
|||
|
|
Introduction to Electric Potential and Electric Energy
|
|||
|
|
|
|||
|
|
817
|
|||
|
|
|
|||
|
|
19.1 Electric Potential Energy: Potential Difference
|
|||
|
|
|
|||
|
|
818
|
|||
|
|
|
|||
|
|
19.2 Electric Potential in a Uniform Electric Field
|
|||
|
|
|
|||
|
|
824
|
|||
|
|
|
|||
|
|
19.3 Electrical Potential Due to a Point Charge
|
|||
|
|
|
|||
|
|
828
|
|||
|
|
|
|||
|
|
19.4 Equipotential Lines
|
|||
|
|
|
|||
|
|
830
|
|||
|
|
|
|||
|
|
19.5 Capacitors and Dielectrics
|
|||
|
|
|
|||
|
|
833
|
|||
|
|
|
|||
|
|
19.6 Capacitors in Series and Parallel
|
|||
|
|
|
|||
|
|
841
|
|||
|
|
|
|||
|
|
19.7 Energy Stored in Capacitors
|
|||
|
|
|
|||
|
|
845
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
848
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
848
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
849
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
850
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
CHAPTER 20
|
|||
|
|
Electric Current, Resistance, and Ohm's Law 857
|
|||
|
|
|
|||
|
|
Introduction to Electric Current, Resistance, and Ohm's Law
|
|||
|
|
|
|||
|
|
857
|
|||
|
|
|
|||
|
|
20.1 Current
|
|||
|
|
|
|||
|
|
858
|
|||
|
|
|
|||
|
|
20.2 Ohm’s Law: Resistance and Simple Circuits
|
|||
|
|
|
|||
|
|
864
|
|||
|
|
|
|||
|
|
20.3 Resistance and Resistivity
|
|||
|
|
|
|||
|
|
866
|
|||
|
|
|
|||
|
|
20.4 Electric Power and Energy
|
|||
|
|
|
|||
|
|
872
|
|||
|
|
|
|||
|
|
20.5 Alternating Current versus Direct Current
|
|||
|
|
|
|||
|
|
875
|
|||
|
|
|
|||
|
|
20.6 Electric Hazards and the Human Body
|
|||
|
|
|
|||
|
|
880
|
|||
|
|
|
|||
|
|
20.7 Nerve Conduction–Electrocardiograms
|
|||
|
|
|
|||
|
|
885
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
893
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
893
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
894
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
896
|
|||
|
|
|
|||
|
|
CHAPTER 21
|
|||
|
|
Circuits and DC Instruments
|
|||
|
|
|
|||
|
|
Introduction to Circuits and DC Instruments
|
|||
|
|
|
|||
|
|
903
|
|||
|
|
|
|||
|
|
21.1 Resistors in Series and Parallel
|
|||
|
|
|
|||
|
|
904
|
|||
|
|
|
|||
|
|
21.2 Electromotive Force: Terminal Voltage
|
|||
|
|
|
|||
|
|
913
|
|||
|
|
|
|||
|
|
21.3 Kirchhoff’s Rules
|
|||
|
|
|
|||
|
|
922
|
|||
|
|
|
|||
|
|
21.4 DC Voltmeters and Ammeters
|
|||
|
|
|
|||
|
|
927
|
|||
|
|
|
|||
|
|
21.5 Null Measurements
|
|||
|
|
|
|||
|
|
931
|
|||
|
|
|
|||
|
|
21.6 DC Circuits Containing Resistors and Capacitors
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
940
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
941
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
942
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
945
|
|||
|
|
|
|||
|
|
903 934
|
|||
|
|
|
|||
|
|
CHAPTER 22
|
|||
|
|
Magnetism 953
|
|||
|
|
|
|||
|
|
Introduction to Magnetism
|
|||
|
|
|
|||
|
|
953
|
|||
|
|
|
|||
|
|
22.1 Magnets
|
|||
|
|
|
|||
|
|
954
|
|||
|
|
|
|||
|
|
22.2 Ferromagnets and Electromagnets
|
|||
|
|
|
|||
|
|
956
|
|||
|
|
|
|||
|
|
22.3 Magnetic Fields and Magnetic Field Lines
|
|||
|
|
|
|||
|
|
960
|
|||
|
|
|
|||
|
|
22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
|
|||
|
|
|
|||
|
|
22.5 Force on a Moving Charge in a Magnetic Field: Examples and Applications
|
|||
|
|
|
|||
|
|
22.6 The Hall Effect
|
|||
|
|
|
|||
|
|
968
|
|||
|
|
|
|||
|
|
22.7 Magnetic Force on a Current-Carrying Conductor
|
|||
|
|
|
|||
|
|
971
|
|||
|
|
|
|||
|
|
22.8 Torque on a Current Loop: Motors and Meters
|
|||
|
|
|
|||
|
|
973
|
|||
|
|
|
|||
|
|
22.9 Magnetic Fields Produced by Currents: Ampere’s Law
|
|||
|
|
|
|||
|
|
977
|
|||
|
|
|
|||
|
|
22.10 Magnetic Force between Two Parallel Conductors
|
|||
|
|
|
|||
|
|
981
|
|||
|
|
|
|||
|
|
22.11 More Applications of Magnetism
|
|||
|
|
|
|||
|
|
983
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
988
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
989
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
990
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
993
|
|||
|
|
|
|||
|
|
961 964
|
|||
|
|
|
|||
|
|
CHAPTER 23
|
|||
|
|
Electromagnetic Induction, AC Circuits, and Electrical
|
|||
|
|
|
|||
|
|
Technologies 1003
|
|||
|
|
|
|||
|
|
Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
|
|||
|
|
|
|||
|
|
1003
|
|||
|
|
|
|||
|
|
23.1 Induced Emf and Magnetic Flux
|
|||
|
|
|
|||
|
|
1004
|
|||
|
|
|
|||
|
|
23.2 Faraday’s Law of Induction: Lenz’s Law
|
|||
|
|
|
|||
|
|
1007
|
|||
|
|
|
|||
|
|
23.3 Motional Emf
|
|||
|
|
|
|||
|
|
1010
|
|||
|
|
|
|||
|
|
23.4 Eddy Currents and Magnetic Damping
|
|||
|
|
|
|||
|
|
1013
|
|||
|
|
|
|||
|
|
23.5 Electric Generators
|
|||
|
|
|
|||
|
|
1017
|
|||
|
|
|
|||
|
|
23.6 Back Emf
|
|||
|
|
|
|||
|
|
1021
|
|||
|
|
|
|||
|
|
23.7 Transformers
|
|||
|
|
|
|||
|
|
1022
|
|||
|
|
|
|||
|
|
23.8 Electrical Safety: Systems and Devices
|
|||
|
|
|
|||
|
|
1027
|
|||
|
|
|
|||
|
|
23.9 Inductance
|
|||
|
|
|
|||
|
|
1031
|
|||
|
|
|
|||
|
|
23.10 RL Circuits
|
|||
|
|
|
|||
|
|
1036
|
|||
|
|
|
|||
|
|
23.11 Reactance, Inductive and Capacitive
|
|||
|
|
|
|||
|
|
1038
|
|||
|
|
|
|||
|
|
23.12 RLC Series AC Circuits
|
|||
|
|
|
|||
|
|
1042
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
1049
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
1049
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1052
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1054
|
|||
|
|
|
|||
|
|
CHAPTER 24
|
|||
|
|
Electromagnetic Waves 1063
|
|||
|
|
|
|||
|
|
Introduction to Electromagnetic Waves
|
|||
|
|
|
|||
|
|
1063
|
|||
|
|
|
|||
|
|
24.1 Maxwell’s Equations: Electromagnetic Waves Predicted and Observed
|
|||
|
|
|
|||
|
|
24.2 Production of Electromagnetic Waves
|
|||
|
|
|
|||
|
|
1067
|
|||
|
|
|
|||
|
|
24.3 The Electromagnetic Spectrum
|
|||
|
|
|
|||
|
|
1070
|
|||
|
|
|
|||
|
|
24.4 Energy in Electromagnetic Waves
|
|||
|
|
|
|||
|
|
1086
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
1089
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
1090
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1091
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1092
|
|||
|
|
|
|||
|
|
1064
|
|||
|
|
|
|||
|
|
CHAPTER 25
|
|||
|
|
Geometric Optics 1097
|
|||
|
|
|
|||
|
|
Introduction to Geometric Optics
|
|||
|
|
|
|||
|
|
1097
|
|||
|
|
|
|||
|
|
25.1 The Ray Aspect of Light
|
|||
|
|
|
|||
|
|
1098
|
|||
|
|
|
|||
|
|
25.2 The Law of Reflection
|
|||
|
|
|
|||
|
|
1099
|
|||
|
|
|
|||
|
|
25.3 The Law of Refraction
|
|||
|
|
|
|||
|
|
1102
|
|||
|
|
|
|||
|
|
25.4 Total Internal Reflection
|
|||
|
|
|
|||
|
|
1108
|
|||
|
|
|
|||
|
|
25.5 Dispersion: The Rainbow and Prisms
|
|||
|
|
|
|||
|
|
25.6 Image Formation by Lenses
|
|||
|
|
|
|||
|
|
1118
|
|||
|
|
|
|||
|
|
25.7 Image Formation by Mirrors
|
|||
|
|
|
|||
|
|
1132
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
1141
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
1141
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1142
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1144
|
|||
|
|
|
|||
|
|
1114
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
CHAPTER 26
|
|||
|
|
Vision and Optical Instruments
|
|||
|
|
|
|||
|
|
Introduction to Vision and Optical Instruments
|
|||
|
|
|
|||
|
|
1151
|
|||
|
|
|
|||
|
|
26.1 Physics of the Eye
|
|||
|
|
|
|||
|
|
1152
|
|||
|
|
|
|||
|
|
26.2 Vision Correction
|
|||
|
|
|
|||
|
|
1156
|
|||
|
|
|
|||
|
|
26.3 Color and Color Vision
|
|||
|
|
|
|||
|
|
1161
|
|||
|
|
|
|||
|
|
26.4 Microscopes
|
|||
|
|
|
|||
|
|
1165
|
|||
|
|
|
|||
|
|
26.5 Telescopes
|
|||
|
|
|
|||
|
|
1171
|
|||
|
|
|
|||
|
|
26.6 Aberrations
|
|||
|
|
|
|||
|
|
1175
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
1178
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
1178
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1179
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1180
|
|||
|
|
|
|||
|
|
1151
|
|||
|
|
|
|||
|
|
CHAPTER 27
|
|||
|
|
Wave Optics 1185
|
|||
|
|
|
|||
|
|
Introduction to Wave Optics
|
|||
|
|
|
|||
|
|
1185
|
|||
|
|
|
|||
|
|
27.1 The Wave Aspect of Light: Interference
|
|||
|
|
|
|||
|
|
1186
|
|||
|
|
|
|||
|
|
27.2 Huygens's Principle: Diffraction
|
|||
|
|
|
|||
|
|
1187
|
|||
|
|
|
|||
|
|
27.3 Young’s Double Slit Experiment
|
|||
|
|
|
|||
|
|
1190
|
|||
|
|
|
|||
|
|
27.4 Multiple Slit Diffraction
|
|||
|
|
|
|||
|
|
1195
|
|||
|
|
|
|||
|
|
27.5 Single Slit Diffraction
|
|||
|
|
|
|||
|
|
1198
|
|||
|
|
|
|||
|
|
27.6 Limits of Resolution: The Rayleigh Criterion
|
|||
|
|
|
|||
|
|
1202
|
|||
|
|
|
|||
|
|
27.7 Thin Film Interference
|
|||
|
|
|
|||
|
|
1207
|
|||
|
|
|
|||
|
|
27.8 Polarization
|
|||
|
|
|
|||
|
|
1212
|
|||
|
|
|
|||
|
|
27.9 *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
1225
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
1225
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1227
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1229
|
|||
|
|
|
|||
|
|
CHAPTER 28
|
|||
|
|
Special Relativity 1237
|
|||
|
|
|
|||
|
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Introduction to Special Relativity
|
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1237
|
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28.1 Einstein’s Postulates
|
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1238
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28.2 Simultaneity And Time Dilation
|
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1241
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28.3 Length Contraction
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1248
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28.4 Relativistic Addition of Velocities
|
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1252
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28.5 Relativistic Momentum
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1258
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28.6 Relativistic Energy
|
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1260
|
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|
Glossary
|
|||
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|
|||
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1269
|
|||
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|||
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Section Summary
|
|||
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|
|
|||
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1269
|
|||
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|||
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Conceptual Questions
|
|||
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|
|
|||
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1271
|
|||
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|||
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Problems & Exercises
|
|||
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1272
|
|||
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|||
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CHAPTER 29
|
|||
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Quantum Physics 1277
|
|||
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|
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|||
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Introduction to Quantum Physics
|
|||
|
|
|
|||
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1277
|
|||
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|||
|
|
29.1 Quantization of Energy
|
|||
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1278
|
|||
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|
|||
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|
1221
|
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29.2 The Photoelectric Effect
|
|||
|
|
|
|||
|
|
1281
|
|||
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|
|
|||
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|
29.3 Photon Energies and the Electromagnetic Spectrum
|
|||
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|
|
|||
|
|
29.4 Photon Momentum
|
|||
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|
|
|||
|
|
1292
|
|||
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|
|
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|
29.5 The Particle-Wave Duality
|
|||
|
|
|
|||
|
|
1296
|
|||
|
|
|
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|
29.6 The Wave Nature of Matter
|
|||
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|
|
|||
|
|
1297
|
|||
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|
|
|||
|
|
29.7 Probability: The Heisenberg Uncertainty Principle
|
|||
|
|
|
|||
|
|
29.8 The Particle-Wave Duality Reviewed
|
|||
|
|
|
|||
|
|
1306
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
1310
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
1310
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1311
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1312
|
|||
|
|
|
|||
|
|
1284 1301
|
|||
|
|
|
|||
|
|
CHAPTER 30
|
|||
|
|
Atomic Physics 1319
|
|||
|
|
|
|||
|
|
Introduction to Atomic Physics
|
|||
|
|
|
|||
|
|
1319
|
|||
|
|
|
|||
|
|
30.1 Discovery of the Atom
|
|||
|
|
|
|||
|
|
1319
|
|||
|
|
|
|||
|
|
30.2 Discovery of the Parts of the Atom: Electrons and Nuclei
|
|||
|
|
|
|||
|
|
1322
|
|||
|
|
|
|||
|
|
30.3 Bohr’s Theory of the Hydrogen Atom
|
|||
|
|
|
|||
|
|
1329
|
|||
|
|
|
|||
|
|
30.4 X Rays: Atomic Origins and Applications
|
|||
|
|
|
|||
|
|
1336
|
|||
|
|
|
|||
|
|
30.5 Applications of Atomic Excitations and De-Excitations
|
|||
|
|
|
|||
|
|
1342
|
|||
|
|
|
|||
|
|
30.6 The Wave Nature of Matter Causes Quantization
|
|||
|
|
|
|||
|
|
1352
|
|||
|
|
|
|||
|
|
30.7 Patterns in Spectra Reveal More Quantization
|
|||
|
|
|
|||
|
|
1354
|
|||
|
|
|
|||
|
|
30.8 Quantum Numbers and Rules
|
|||
|
|
|
|||
|
|
1357
|
|||
|
|
|
|||
|
|
30.9 The Pauli Exclusion Principle
|
|||
|
|
|
|||
|
|
1362
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
1370
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
1371
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1373
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1374
|
|||
|
|
|
|||
|
|
CHAPTER 31
|
|||
|
|
Radioactivity and Nuclear Physics
|
|||
|
|
|
|||
|
|
Introduction to Radioactivity and Nuclear Physics
|
|||
|
|
|
|||
|
|
1379
|
|||
|
|
|
|||
|
|
31.1 Nuclear Radioactivity
|
|||
|
|
|
|||
|
|
1380
|
|||
|
|
|
|||
|
|
31.2 Radiation Detection and Detectors
|
|||
|
|
|
|||
|
|
1384
|
|||
|
|
|
|||
|
|
31.3 Substructure of the Nucleus
|
|||
|
|
|
|||
|
|
1388
|
|||
|
|
|
|||
|
|
31.4 Nuclear Decay and Conservation Laws
|
|||
|
|
|
|||
|
|
1392
|
|||
|
|
|
|||
|
|
31.5 Half-Life and Activity
|
|||
|
|
|
|||
|
|
1399
|
|||
|
|
|
|||
|
|
31.6 Binding Energy
|
|||
|
|
|
|||
|
|
1405
|
|||
|
|
|
|||
|
|
31.7 Tunneling
|
|||
|
|
|
|||
|
|
1410
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
1413
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
1414
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1415
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1417
|
|||
|
|
|
|||
|
|
1379
|
|||
|
|
|
|||
|
|
CHAPTER 32
|
|||
|
|
Medical Applications of Nuclear Physics
|
|||
|
|
|
|||
|
|
Introduction to Applications of Nuclear Physics
|
|||
|
|
|
|||
|
|
1423
|
|||
|
|
|
|||
|
|
32.1 Diagnostics and Medical Imaging
|
|||
|
|
|
|||
|
|
1424
|
|||
|
|
|
|||
|
|
32.2 Biological Effects of Ionizing Radiation
|
|||
|
|
|
|||
|
|
1428
|
|||
|
|
|
|||
|
|
1423
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
32.3 Therapeutic Uses of Ionizing Radiation
|
|||
|
|
|
|||
|
|
32.4 Food Irradiation
|
|||
|
|
|
|||
|
|
1438
|
|||
|
|
|
|||
|
|
32.5 Fusion
|
|||
|
|
|
|||
|
|
1439
|
|||
|
|
|
|||
|
|
32.6 Fission
|
|||
|
|
|
|||
|
|
1445
|
|||
|
|
|
|||
|
|
32.7 Nuclear Weapons
|
|||
|
|
|
|||
|
|
1451
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
1457
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
1457
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1459
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1461
|
|||
|
|
|
|||
|
|
1436
|
|||
|
|
|
|||
|
|
CHAPTER 33
|
|||
|
|
Particle Physics 1467
|
|||
|
|
|
|||
|
|
Introduction to Particle Physics
|
|||
|
|
|
|||
|
|
1467
|
|||
|
|
|
|||
|
|
33.1 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited
|
|||
|
|
|
|||
|
|
33.2 The Four Basic Forces
|
|||
|
|
|
|||
|
|
1470
|
|||
|
|
|
|||
|
|
33.3 Accelerators Create Matter from Energy
|
|||
|
|
|
|||
|
|
1472
|
|||
|
|
|
|||
|
|
33.4 Particles, Patterns, and Conservation Laws
|
|||
|
|
|
|||
|
|
1476
|
|||
|
|
|
|||
|
|
33.5 Quarks: Is That All There Is?
|
|||
|
|
|
|||
|
|
1482
|
|||
|
|
|
|||
|
|
33.6 GUTs: The Unification of Forces
|
|||
|
|
|
|||
|
|
1489
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
1494
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
1495
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1496
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1497
|
|||
|
|
|
|||
|
|
CHAPTER 34
|
|||
|
|
Frontiers of Physics 1503
|
|||
|
|
|
|||
|
|
Introduction to Frontiers of Physics
|
|||
|
|
|
|||
|
|
1503
|
|||
|
|
|
|||
|
|
34.1 Cosmology and Particle Physics
|
|||
|
|
|
|||
|
|
1504
|
|||
|
|
|
|||
|
|
34.2 General Relativity and Quantum Gravity
|
|||
|
|
|
|||
|
|
1512
|
|||
|
|
|
|||
|
|
34.3 Superstrings
|
|||
|
|
|
|||
|
|
1518
|
|||
|
|
|
|||
|
|
34.4 Dark Matter and Closure
|
|||
|
|
|
|||
|
|
1519
|
|||
|
|
|
|||
|
|
34.5 Complexity and Chaos
|
|||
|
|
|
|||
|
|
1523
|
|||
|
|
|
|||
|
|
34.6 High-temperature Superconductors
|
|||
|
|
|
|||
|
|
1525
|
|||
|
|
|
|||
|
|
34.7 Some Questions We Know to Ask
|
|||
|
|
|
|||
|
|
1527
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
|
|||
|
|
1530
|
|||
|
|
|
|||
|
|
Section Summary
|
|||
|
|
|
|||
|
|
1530
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1532
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1534
|
|||
|
|
|
|||
|
|
Appendix A Atomic Masses
|
|||
|
|
|
|||
|
|
1537
|
|||
|
|
|
|||
|
|
Appendix B Selected Radioactive Isotopes
|
|||
|
|
|
|||
|
|
1549
|
|||
|
|
|
|||
|
|
Appendix C Useful Information
|
|||
|
|
|
|||
|
|
1553
|
|||
|
|
|
|||
|
|
Appendix D Glossary of Key Symbols and Notation
|
|||
|
|
|
|||
|
|
1559
|
|||
|
|
|
|||
|
|
Answer Key
|
|||
|
|
|
|||
|
|
1575
|
|||
|
|
|
|||
|
|
Index
|
|||
|
|
|
|||
|
|
1673
|
|||
|
|
|
|||
|
|
1468
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
Preface
|
|||
|
|
|
|||
|
|
1
|
|||
|
|
|
|||
|
|
PREFACE
|
|||
|
|
|
|||
|
|
About OpenStax
|
|||
|
|
OpenStax is part of Rice University, which is a 501(c)(3) nonprofit charitable corporation. As an educational initiative, it's our mission to transform learning so that education works for every student. Through our partnerships with philanthropic organizations and our alliance with other educational resource companies, we're breaking down the most common barriers to learning. Because we believe that everyone should and can have access to knowledge.
|
|||
|
|
About OpenStax Resources
|
|||
|
|
Customization College Physics 2e is licensed under a Creative Commons Attribution 4.0 International (CC BY) license, which means that you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its content contributors.
|
|||
|
|
Because our books are openly licensed, you are free to use the entire book or select only the sections that are most relevant to the needs of your course. Feel free to remix the content by assigning your students certain chapters and sections in your syllabus, in the order that you prefer. You can even provide a direct link in your syllabus to the sections in the web view of your book.
|
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Instructors also have the option of creating a customized version of their OpenStax book. The custom version can be made available to students in low-cost print or digital form through their campus bookstore. Visit the Instructor Resources section of your book page on OpenStax.org for more information.
|
|||
|
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Art Attribution In College Physics 2e, most photos and third-party illustrations contain attribution to their creator, rights holder, host platform, and/or license within the caption. Because the art is openly licensed, anyone may reuse the art as long as they provide the same attribution to its original source. To maximize readability and content flow, some art does not include attribution in the text. This art is part of the public domain or under a CC0 or similar license, and can be reused without attribution. For illustrations (e.g., graphs, charts, etc.) that are not credited, use the following attribution: Copyright Rice University, OpenStax, under CC BY 4.0 license.
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Errata All OpenStax textbooks undergo a rigorous review
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process. However, like any professional-grade textbook, errors sometimes occur. Since our books are web-based, we can make updates periodically when deemed pedagogically necessary. If you have a correction to suggest, submit it through the link on your book page on OpenStax.org. Subject matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you will also find a list of past errata changes on your book page on OpenStax.org.
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Format You can access this textbook for free in web view or PDF through OpenStax.org, and for a low cost in print.
|
|||
|
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About College Physics 2e
|
|||
|
|
College Physics 2e provides a comprehensive and welcoming introduction to the principles, concepts, and applications typically covered in introductory physics courses. The book progresses through a learning model intended to support students at various levels, and offers faculty a great deal of flexibility in the approach and organization of their course. The text is grounded in real-world examples to help students understand the importance of physics in their lives and especially their future careers. It requires knowledge of algebra and some trigonometry, but not calculus; quantitative explanations and solutions are extremely detailed in order to build a thorough and useful understanding among students. College Physics 2e orients its coverage around clear and widely accepted learning outcomes. It includes links to simulations and other multimedia, and each section contains ample practice opportunities in a wide array of question types.
|
|||
|
|
Coverage and Scope College Physics 2e introduces topics conceptually and progresses through clear explanations in the context of career-oriented, practical applications. Consistency among the various definitions, foundational concepts, worked examples, and features provides a memorable pathway for student learning and helps maximize the impacts of study and practice.
|
|||
|
|
The text aligns to the scope and sequence of most introductory physics courses and uses algebra as a basis for calculations. Extensive faculty feedback informed the sequence presented in the standard table of contents, but the open nature of the book—both in license and available formats—allows for significant
|
|||
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2
|
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Preface
|
|||
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|
|||
|
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rearrangement by faculty. Mechanics and electricity & magnetism anchor each half of the text, and optics, waves, modern physics, and other topics are arranged in units of their own for flexibility in course placement.
|
|||
|
|
Changes to the Second Edition
|
|||
|
|
College Physics 2e builds on the first edition’s guiding principle that physics is a discipline undertaken by and for people. Throughout the text, the human impact of physics understanding, phenomena, discoveries, and applications is made clear through widespread examples, scenarios, and explanations. The narrative of physics and scientific discovery has been even further expanded to focus on including more diverse contributors to the field. From Ibn al-Haytham’s 11th century foundation of the scientific method to Gladys West’s complex models enabling GPS, the second edition broadens the discussion of pioneering and current researchers in an effort to tell a more accurate and inclusive scientific and societal story.
|
|||
|
|
Relevance and Responsiveness The impact of physics on engineering, urban development, the environment, medicine, energy production, and other aspects of everyday life have been updated and expanded to reflect more student experiences and interests. Techniques and developments in related disciplines are covered in context—not only in opening vignettes—so that students encounter the deep impact of evolving knowledge relevant to their potential fields of study.
|
|||
|
|
Since many introductory physics students are focused on medicine, sections and examples related to biology have been significantly expanded. The section on electric forces in biology (18.6), for example, has been deepened to include Ernest Everett Just’s work on electronegativity in ova, as well as the emerging practice of electrical stimulation in wound healing. Additional biological application narratives include Yalow and Berson’s development of radioimmunoassay, and Strickland and Mourou’s invention of chirped lasers used in vision correction.
|
|||
|
|
Currency and Accuracy We have updated sections related to ongoing research, frontiers of physics, and emerging information. In particular, section 4.8 on the four basic forces has been revised with information about recent discoveries and ongoing research, as well as with additional context about the ongoing process of discovery—for example, the progression from Einstein’s black hole predictions to the first black hole images produced in 2019. The section on world energy use (7.9), the section on ozone depletion (24.3), and several sections discussing space
|
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|
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telescopes have been similarly updated to reflect current research and data.
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Over ten years of widespread usage, OpenStax College Physics has benefitted from suggestions, corrections, and clarifications submitted by hundreds of faculty and also from students. We have made the requisite corrections and improvements over time, but the second edition unifies those edits for more consistency and ease of use.
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Improving Problem-Solving and Deepening Understanding
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College Physics 2e employs the best practices of physics teaching, informed by education research and extensive adopter feedback. In order to unify conceptual, analytical, and calculation skills within the learning process, the authors have integrated a wide array of strategies and supports throughout the text.
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Worked Examples Worked examples have four distinct parts to promote both analytical and conceptual skills. Worked examples are introduced in words, always using some application that should be of interest. This is followed by a Strategy section that emphasizes the concepts involved and how solving the problem relates to those concepts. This is followed by the mathematical Solution and Discussion.
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Many worked examples contain multiple-part problems to help the students learn how to approach normal situations, in which problems tend to have multiple parts. Finally, worked examples employ the techniques of the problem-solving strategies so that students can see how those strategies succeed in practice as well as in theory.
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Problem-Solving Strategies Problem-solving strategies are first presented in a special section and subsequently appear at crucial points in the text where students can benefit most from them. Problem-solving strategies have a logical structure that is reinforced in the worked examples and supported in certain places by line drawings that illustrate various steps.
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Misconception Alerts Students come to physics with preconceptions from everyday experiences and from previous courses. Some of these preconceptions are misconceptions, and many are very common among students and the general public. Some are inadvertently picked up through misunderstandings of lectures and texts. The Misconception Alerts feature is designed to point these
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Access for free at openstax.org
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Preface
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3
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out and correct them explicitly.
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Take-Home Investigations Take Home Investigations provide the opportunity for students to apply or explore what they have learned with a hands-on activity.
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Things Great and Small In these special topic essays, macroscopic phenomena (such as air pressure) are explained with submicroscopic phenomena (such as atoms bouncing off walls). These essays support the modern perspective by describing aspects of modern physics before they are formally treated in later chapters. Connections are also made between apparently disparate phenomena.
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Module Summaries Module summaries are thorough and functional and present all important definitions and equations. Students are able to find the definitions of all terms and symbols as well as their physical relationships. The structure of the summary makes plain the fundamental principles of the module or collection and serves as a useful study guide.
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Engaging Students Concept Trailers Click to view content (https://openstax.org/r/ concepttrailers) Concept Trailers are twenty-four videos designed to engage and introduce students to key chapter concepts. These professionally produced videos are like a movie trailer and are approximately 90 seconds in length. These can be used independently by students or in lecture. Concept Trailers are available on YouTube.
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Simulations Where applicable, students are directed to the interactive PhET physics simulations developed by the University of Colorado Boulder. There they can further explore the physics concepts they have learned about in the module.
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Flexibility and Progressions in Practice and Assessment
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College Physics 2e provides a rich array of question types that promote faculty choice and ample opportunity for student practice and advancement.
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• Conceptual Questions challenge students' ability to explain what they have learned conceptually, independent of the mathematical details.
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• Problems & Exercises challenge students to apply both concepts and skills to solve mathematical
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physics problems. • Integrated Concept Problems ask students to
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apply what they have learned about two or more concepts to arrive at a solution to a problem. • Create Your Own Problems require students to construct the details of a problem, justify their starting assumptions, show specific steps in the problem's solution, and discuss the meaning of the result. These types of problems relate well to both conceptual and analytical aspects of physics, emphasizing that physics must describe nature. Often they involve an integration of topics from more than one chapter. Unlike other problems, solutions are not provided since there is no single correct answer. Instructors should feel free to direct students regarding the level and scope of their considerations. Whether the problem is solved and described correctly will depend on initial assumptions. • Unreasonable Results Problems drive students to both solve a problem and analyze the answer's likelihood and realism. These problems contain a premise that produces an unreasonable answer and are designed to further emphasize that properly applied physics must describe nature accurately and is not simply the process of solving equations. • Critical Thinking Questions are new additions to the text. These challenging, multi-part problems typically integrate conceptual, quantitative, and graphical response elements in order to deeply investigate student understanding. Most chapters provide one Critical Thinking Question, and we have reserved additional questions and solutions only for instructor use.
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About the Authors
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Senior Contributing Authors Paul Peter Urone, Professor Emeritus at California State University, Sacramento Roger Hinrichs, State University of New York, Oswego (retired)
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Contributing Authors Kim Dirks, University of Auckland Kenneth Podolak, State University of New York, Plattsburgh Manjula Sharma, University of Sydney Henry Smith, River Parishes Community College
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Reviewers Matthew Adams, Crafton Hills College, San Bernardino Community College District Erik Christensen, South Florida Community College
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4
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Preface
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Douglas Ingram, Texas Christian University Eric Kincanon, Gonzaga University Lee H. LaRue, Paris Junior College Chuck Pearson, Virginia Intermont College Marc Sher, College of William and Mary Ulrich Zurcher, Cleveland State University
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Additional Resources
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Student and Instructor Resources We’ve compiled additional resources for both students and instructors, including Getting Started Guides, an instructor’s manual, a test bank, and image slides. Instructor resources require a verified instructor account, which you can apply for when you log in or create your account on OpenStax.org. Take advantage of these resources to supplement your OpenStax book.
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Instructor’s solutions manual. The instructor solutions manual contains the instructor-facing answers to the problems and exercises within the textbook. Since many instructors use these questions in graded assignments, we ask that you not post these questions and the answers on any publicly available websites.
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PowerPoint lecture slides. The PowerPoint slides provide images and descriptions as a starting place for instructors to build their lectures.
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Concept Trailer instructor notes. These teaching notes support implementation of the OpenStax Physics Concept Trailers. The notes contain tips for usage, clarifications of coverage, and guidance on how to use the trailers in different educational situations.
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Academic Integrity
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Academic integrity builds trust, understanding, equity, and genuine learning. While students may encounter significant challenges in their courses and their lives, doing their own work and maintaining a high degree of authenticity will result in meaningful outcomes that will extend far beyond their college career. Faculty, administrators, resource providers, and students can work together to maintain a fair and positive experience.
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We realize that students benefit when academic integrity ground rules are established early in the course. To that end, OpenStax has created an interactive to aid with academic integrity discussions in
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your course.
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Visit our academic integrity slider (https://view.genial.ly/61e08a7af6db870d591078c1/ interactive-image-defining-academic-integrityinteractive-slider). Click and drag icons along the continuum to align these practices with your institution and course policies. You may then include the graphic on your syllabus, present it in your first course meeting, or create a handout for students.
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At OpenStax we are also developing resources supporting authentic learning experiences and assessment. Please visit this book’s page for updates. For an in-depth review of academic integrity strategies, we highly recommend visiting the International Center of Academic Integrity (ICAI) website at https://academicintegrity.org/ (https://academicintegrity.org/).
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Community Hubs OpenStax partners with the Institute for the Study of Knowledge Management in Education (ISKME) to offer Community Hubs on OER Commons—a platform for instructors to share community-created resources that support OpenStax books, free of charge. Through our Community Hubs, instructors can upload their own materials or download resources to use in their own courses, including additional ancillaries, teaching material, multimedia, and relevant course content. We encourage instructors to join the hubs for the subjects most relevant to your teaching and research as an opportunity both to enrich your courses and to engage with other faculty. To reach the Community Hubs, visit www.oercommons.org/hubs/openstax (https://www.oercommons.org/hubs/openstax).
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Technology Partners As allies in making high-quality learning materials accessible, our technology partners offer optional lowcost tools that are integrated with OpenStax books. To access the technology options for your text, visit your book page on OpenStax.org.
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Access for free at openstax.org
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CHAPTER 1
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Introduction: The Nature of Science and Physics
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FIGURE 1.1 The laws of physics describe the smallest and largest forces and structures, such as the Veil Nebula, a supernova remnant from a star that was 20 times larger than our Sun. Stretching over 110 light years across, the spectacular image is created by the stellar explosion's blast wave impacting clouds of gas. Scientists use successive observations of the still-moving wave to learn about nebula formation in ways they can apply to other systems. The laws of physics are surprisingly few in number, implying an underlying simplicity to nature’s apparent complexity. (credit: NASA, ESA, Hubble Heritage, and the Digitized Sky Survey 2)
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CHAPTER OUTLINE 1.1 Physics: An Introduction 1.2 Physical Quantities and Units 1.3 Accuracy, Precision, and Significant Figures 1.4 Approximation
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INTRODUCTION TO SCIENCE AND THE REALM OF PHYSICS, PHYSICAL QUANTITIES, AND UNITS What is your first reaction when you hear the word “physics”? Did you imagine working through difficult equations or memorizing formulas that seem to have no real use in life outside the physics classroom? Many people come to the subject of physics with a bit of fear. But as you begin your exploration of this broad-ranging subject, you may soon come to realize that physics plays a much larger role in your life than you first thought, no matter your life goals or career choice. Consider the Veil Nebula, a cloud of heated dust and gas located about 2,400 light years from Earth (a light year is the distance light travels in one year, or approximately 9.5 trillion kilometers). The unique structure is the ongoing result of a supernova that occurred 8,000 years ago. The shock wave from the explosion is colliding with a cloud of gas and dust, creating the rope-like filaments and sheet-like appearance. Scientists compare newer images, such as the one above, with detailed images taken by the Hubble Space Telescope in 1997 in order to understand nebula expansion and other properties of astronomical objects. The forces that cause the supernova remnant to act as it
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6
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1 • Introduction: The Nature of Science and Physics
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does are the same forces we contend with here on Earth, whether we are planning to send a rocket into space or simply heating a new home. Although the scale is much different, the interaction of gasses in the Veil Nebula resembles those on Earth; and the vibrant color combinations are created by the familiar glows of hydrogen, sulfur, oxygen, and similar elements that make up everything we know. Tonight, take a moment to look up at the stars. The forces out there are the same as the ones here on Earth. Through a study of physics, you may gain a greater understanding of the interconnectedness of everything we can see and know in this universe.
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Humans have created and manufactured millions of different objects over the history of our species. Successive technological periods (often referred to as the Stone Age, the Bronze Age, the Iron Age, and so on) were marked by our knowledge of the physical properties of certain materials and our ability to manipulate them. This knowledge all stems from physics, whether it's the way a rock would flake when constructing a spear point, the effect of integrating carbon with iron in South Indian and Sri Lankan furnaces to create the earliest high-quality steel, or the proper way to combine perfectly ground and polished pieces of glass to create optical instruments. Our current technological age, the Information Age, builds on all that knowledge and can be traced to critical innovations made by people from all backgrounds working together. Mohamed M. Atalla and Dawon Kahng, for example, invented the MOSFET (metal-oxide-semiconductor field-effect transistor). Although unknown to most people, this tiny device, created in 1959 by an Egyptian-born scientist and Korean-born scientist working in a lab in New Jersey, is the basis for modern electronics. More MOSFETs have been produced than any other object in human history. They are used in computers, smart phones, microwave ovens, automotive controls, medical instruments, and nearly every other electronic device.
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Next, think about the most exciting modern technologies that you have heard about in the news, such as trains that levitate above tracks, “invisibility cloaks” that bend light around them, and microscopic robots that fight cancer cells in our bodies. All of these groundbreaking advancements, commonplace or unbelievable, rely on the principles of physics. Aside from playing a significant role in technology, professionals such as engineers, pilots, physicians, physical therapists, electricians, and computer programmers apply physics concepts in their daily work. For example, a pilot must understand how wind forces affect a flight path and a physical therapist must understand how the muscles in the body experience forces as they move and bend. As you will learn in this text, physics principles are propelling new, exciting technologies, and these principles are applied in a wide range of careers.
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In this text, you will begin to explore the history of the formal study of physics, beginning with natural philosophy and the ancient thinkers from the Middle East and the Mediterranean, and leading up through a review of Sir Isaac Newton and the laws of physics that bear his name. You will also be introduced to the standards scientists use when they study physical quantities and the interrelated system of measurements most of the scientific community uses to communicate in a single mathematical language. Finally, you will study the limits of our ability to be accurate and precise, and the reasons scientists go to painstaking lengths to be as clear as possible regarding their own limitations.
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1.1 Physics: An Introduction
|
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LEARNING OBJECTIVES By the end of this section, you will be able to:
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• Explain the difference between a principle and a law. • Explain the difference between a model and a theory.
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Access for free at openstax.org
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1.1 • Physics: An Introduction
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7
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FIGURE 1.2 The flight formations of migratory birds such as Canada geese are governed by the laws of physics. (credit: David Merrett)
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The physical universe is enormously complex in its detail. Every day, each of us observes a great variety of objects and phenomena. Over the centuries, the curiosity of the human race has led us collectively to explore and catalog a tremendous wealth of information. From the flight of birds to the colors of flowers, from lightning to gravity, from quarks to clusters of galaxies, from the flow of time to the mystery of the creation of the universe, we have asked questions and assembled huge arrays of facts. In the face of all these details, we have discovered that a surprisingly small and unified set of physical laws can explain what we observe. As humans, we make generalizations and seek order. We have found that nature is remarkably cooperative—it exhibits the underlying order and simplicity we so value.
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It is the underlying order of nature that makes science in general, and physics in particular, so enjoyable to study. For example, what do a bag of chips and a car battery have in common? Both contain energy that can be converted to other forms. The law of conservation of energy (which says that energy can change form but is never lost) ties together such topics as food calories, batteries, heat, light, and watch springs. Understanding this law makes it easier to learn about the various forms energy takes and how they relate to one another. Apparently unrelated topics are connected through broadly applicable physical laws, permitting an understanding beyond just the memorization of lists of facts.
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The unifying aspect of physical laws and the basic simplicity of nature form the underlying themes of this text. In learning to apply these laws, you will, of course, study the most important topics in physics. More importantly, you will gain analytical abilities that will enable you to apply these laws far beyond the scope of what can be included in a single book. These analytical skills will help you to excel academically, and they will also help you to think critically in any professional career you choose to pursue. This module discusses the realm of physics (to define what physics is), some applications of physics (to illustrate its relevance to other disciplines), and more precisely what constitutes a physical law (to illuminate the importance of experimentation to theory).
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Science and the Realm of Physics
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Science consists of the theories and laws that are the general truths of nature as well as the body of knowledge they encompass. Scientists are continually trying to expand this body of knowledge and to perfect the expression of the laws that describe it. Physics is concerned with describing the interactions of energy, matter, space, and time, and it is especially interested in what fundamental mechanisms underlie every phenomenon. The concern for describing the basic phenomena in nature essentially defines the realm of physics.
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Physics aims to describe the function of everything around us, from the movement of tiny charged particles to the motion of people, cars, and spaceships. In fact, almost everything around you can be described quite accurately by the laws of physics. Consider a smart phone (Figure 1.3). Physics describes how electricity interacts with the various circuits inside the device. This knowledge helps engineers select the appropriate materials and circuit layout when building the smart phone. Next, consider a GPS system. Physics describes the relationship between the speed of an object, the distance over which it travels, and the time it takes to travel that distance. GPS relies on precise calculations that account for variations in the Earth's landscapes, the exact distance between orbiting satellites, and even the effect of a complex occurrence of time dilation. Most of these calculations are founded on algorithms developed by Gladys West, a mathematician and computer scientist who programmed the first computers capable of highly accurate remote sensing and positioning. When you use a GPS device, it utilizes these algorithms to
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8
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1 • Introduction: The Nature of Science and Physics
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recognize where you are and how your position relates to other objects on Earth.
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FIGURE 1.3 The Apple iPhone is a common smart phone with a GPS function, sophisticated camera, haptic (vibration) capabilities, and many other functions. Physics describes the way that electricity flows through the circuits of this device. Engineers use their knowledge of physics to construct an iPhone with features that consumers will enjoy. One specific feature of an iPhone is the GPS function. GPS uses physics equations to determine the driving time between two locations on a map. (credit: Tinh tế Photo/Flickr)
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Applications of Physics
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You need not be a scientist to use physics. On the contrary, knowledge of physics is useful in everyday situations as well as in nonscientific professions. It can help you understand how microwave ovens work, why metals should not be put into them, and why they might affect pacemakers. (See Figure 1.4 and Figure 1.5.) Physics allows you to understand the hazards of radiation and rationally evaluate these hazards more easily. Physics also explains the reason why a black car radiator helps remove heat in a car engine, and it explains why a white roof helps keep the inside of a house cool. Similarly, the operation of a car’s ignition system as well as the transmission of electrical signals through our body’s nervous system are much easier to understand when you think about them in terms of basic physics. Physics is the foundation of many important disciplines and contributes directly to others. Chemistry, for example—since it deals with the interactions of atoms and molecules—is rooted in atomic and molecular physics. Most branches of engineering are applied physics. In architecture, physics is at the heart of structural stability, and is involved in the acoustics, heating, lighting, and cooling of buildings. Parts of geology rely heavily on physics, such as radioactive dating of rocks, earthquake analysis, and heat transfer in the Earth. Some disciplines, such as biophysics and geophysics, are hybrids of physics and other disciplines. Physics has many applications in the biological sciences. On the microscopic level, it helps describe the properties of cell walls and cell membranes (Figure 1.6 and Figure 1.7). On the macroscopic level, it can explain the heat, work, and power associated with the human body. Physics is involved in medical diagnostics, such as x-rays, magnetic resonance imaging (MRI), and ultrasonic blood flow measurements. Medical therapy sometimes directly involves physics; for example, cancer radiotherapy uses ionizing radiation. Physics can also explain sensory phenomena, such as how musical instruments make sound, how the eye detects color, and how lasers can transmit information. It is not necessary to formally study all applications of physics. What is most useful is knowledge of the basic laws of physics and a skill in the analytical methods for applying them. The study of physics also can improve your problemsolving skills. Furthermore, physics has retained the most basic aspects of science, so it is used by all of the sciences, and the study of physics makes other sciences easier to understand.
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Access for free at openstax.org
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1.1 • Physics: An Introduction
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FIGURE 1.4 The laws of physics help us understand how common appliances work. For example, the laws of physics can help explain how microwave ovens heat up food, and they also help us understand why it is dangerous to place metal objects in a microwave oven. (credit: MoneyBlogNewz)
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FIGURE 1.5 These two applications of physics have more in common than meets the eye. Microwave ovens use electromagnetic waves to heat food. Magnetic resonance imaging (MRI) also uses electromagnetic waves to yield an image of the brain, from which the exact location of tumors can be determined. (credit: Rashmi Chawla, Daniel Smith, and Paul E. Marik)
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FIGURE 1.6 Physics, chemistry, and biology help describe the properties of cell walls in plant cells, such as the onion cells seen here. (credit: Umberto Salvagnin)
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1 • Introduction: The Nature of Science and Physics
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FIGURE 1.7 An artist’s rendition of the structure of a cell membrane. Membranes form the boundaries of animal cells and are complex in structure and function. Many of the most fundamental properties of life, such as the firing of nerve cells, are related to membranes. The disciplines of biology, chemistry, and physics all help us understand the membranes of animal cells. (credit: Mariana Ruiz)
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Models, Theories, and Laws; The Role of Experimentation
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The laws of nature are concise descriptions of the universe around us; they are human statements of the underlying laws or rules that all natural processes follow. Such laws are intrinsic to the universe; humans did not create them and so cannot change them. We can only discover and understand them. Their discovery is a very human endeavor, with all the elements of mystery, imagination, struggle, triumph, and disappointment inherent in any creative effort. (See Figure 1.8 and Figure 1.9.) The cornerstone of discovering natural laws is observation; science must describe the universe as it is, not as we may imagine it to be.
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FIGURE 1.8 Isaac Newton (1642–1727) was very reluctant to publish his revolutionary work and had to be convinced to do so. In his later years, he stepped down from his academic post and became exchequer of the Royal Mint. He took this post seriously, inventing reeding (or creating ridges) on the edge of coins to prevent unscrupulous people from trimming the silver off of them before using them as currency. (credit: Arthur Shuster and Arthur E. Shipley: Britain’s Heritage of Science. London, 1917.)
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1.1 • Physics: An Introduction
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11
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FIGURE 1.9 Marie Curie (1867–1934) sacrificed monetary assets to help finance her early research and damaged her physical well-being with radiation exposure. She is the only person to win Nobel prizes in both physics and chemistry. One of her daughters also won a Nobel Prize. (credit: Wikimedia Commons)
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We all are curious to some extent. We look around, make generalizations, and try to understand what we see—for example, we look up and wonder whether one type of cloud signals an oncoming storm. As we become serious about exploring nature, we become more organized and formal in collecting and analyzing data. We attempt greater precision, perform controlled experiments (if we can), and write down ideas about how the data may be organized and unified. We then formulate models, theories, and laws based on the data we have collected and analyzed to generalize and communicate the results of these experiments.
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A model is a representation of something that is often too difficult (or impossible) to display directly. While a model is justified with experimental proof, it is only accurate under limited situations. An example is the planetary model of the atom in which electrons are pictured as orbiting the nucleus, analogous to the way planets orbit the Sun. (See Figure 1.10.) We cannot observe electron orbits directly, but the mental image helps explain the observations we can make, such as the emission of light from hot gases (atomic spectra). Physicists use models for a variety of purposes. For example, models can help physicists analyze a scenario and perform a calculation, or they can be used to represent a situation in the form of a computer simulation. A theory is an explanation for patterns in nature that is supported by scientific evidence and verified multiple times by various groups of researchers. Some theories include models to help visualize phenomena, whereas others do not. Newton’s theory of gravity, for example, does not require a model or mental image, because we can observe the objects directly with our own senses. The kinetic theory of gases, on the other hand, is a model in which a gas is viewed as being composed of atoms and molecules. Atoms and molecules are too small to be observed directly with our senses—thus, we picture them mentally to understand what our instruments tell us about the behavior of gases.
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A law uses concise language to describe a generalized pattern in nature that is supported by scientific evidence and
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repeated experiments. Often, a law can be expressed in the form of a single mathematical equation. Laws and
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theories are similar in that they are both scientific statements that result from a tested hypothesis and are
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supported by scientific evidence. However, the designation law is reserved for a concise and very general statement
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that describes phenomena in nature, such as the law that energy is conserved during any process, or Newton’s
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second law of motion, which relates force, mass, and acceleration by the simple equation
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. A theory, in
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contrast, is a less concise statement of observed phenomena. For example, the Theory of Evolution and the Theory
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of Relativity cannot be expressed concisely enough to be considered a law. The biggest difference between a law
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and a theory is that a theory is much more complex and dynamic. A law describes a single action, whereas a theory
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explains an entire group of related phenomena. And, whereas a law is a postulate that forms the foundation of the
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scientific method, a theory is the end result of that process.
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Less broadly applicable statements are usually called principles (such as Pascal’s principle, which is applicable only in fluids), but the distinction between laws and principles often is not carefully made.
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12
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1 • Introduction: The Nature of Science and Physics
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FIGURE 1.10 What is a model? This planetary model of the atom shows electrons orbiting the nucleus. It is a drawing that we use to form a mental image of the atom that we cannot see directly with our eyes because it is too small.
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Models, Theories, and Laws
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Models, theories, and laws are used to help scientists analyze the data they have already collected. However, often after a model, theory, or law has been developed, it points scientists toward new discoveries they would not otherwise have made.
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The models, theories, and laws we devise sometimes imply the existence of objects or phenomena as yet unobserved. These predictions are remarkable triumphs and tributes to the power of science. It is the underlying order in the universe that enables scientists to make such spectacular predictions. However, if experiment does not verify our predictions, then the theory or law is wrong, no matter how elegant or convenient it is. Laws can never be known with absolute certainty because it is impossible to perform every imaginable experiment in order to confirm a law in every possible scenario. Physicists operate under the assumption that all scientific laws and theories are valid until a counterexample is observed. If a good-quality, verifiable experiment contradicts a well-established law, then the law must be modified or overthrown completely.
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The study of science in general and physics in particular is an adventure much like the exploration of uncharted ocean. Discoveries are made; models, theories, and laws are formulated; and the beauty of the physical universe is made more sublime for the insights gained.
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The Scientific Method
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Ibn al-Haytham (sometimes referred to as Alhazen), a 10th-11th century scientist working in Cairo, significantly advanced the understanding of optics and vision. But his contributions go much further. In demonstrating that previous approaches were incorrect, he emphasized that scientists must be ready to reject existing knowledge and become "the enemy" of everything they read; he expressed that scientists must trust only objective evidence. Al-Haytham emphasized repeated experimentation and validation, and acknowledged that senses and predisposition could lead to poor conclusions. His work was a precursor to the scientific method that we use today.
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As scientists inquire and gather information about the world, they follow a process called the scientific method. This process typically begins with an observation and question that the scientist will research. Next, the scientist typically performs some research about the topic and then devises a hypothesis. Then, the scientist will test the hypothesis by performing an experiment. Finally, the scientist analyzes the results of the experiment and draws a conclusion. Note that the scientific method can be applied to many situations that are not limited to science, and this method can be modified to suit the situation.
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Consider an example. Let us say that you try to turn on your car, but it will not start. You undoubtedly wonder: Why will the car not start? You can follow a scientific method to answer this question. First off, you may perform some research to determine a variety of reasons why the car will not start. Next, you will state a hypothesis. For example, you may believe that the car is not starting because it has no engine oil. To test this, you open the hood of the car and examine the oil level. You observe that the oil is at an acceptable level, and you thus conclude that the oil level is not contributing to your car issue. To troubleshoot the issue further, you may devise a new
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Access for free at openstax.org
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1.1 • Physics: An Introduction
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13
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hypothesis to test and then repeat the process again.
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The Evolution of Natural Philosophy into Modern Physics
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Physics was not always a separate and distinct discipline. It remains connected to other sciences to this day. The word physics comes from Greek, meaning nature. The study of nature came to be called “natural philosophy.” From ancient times through the Renaissance, natural philosophy encompassed many fields, including astronomy, biology, chemistry, physics, mathematics, and medicine. Over the last few centuries, the growth of knowledge has resulted in ever-increasing specialization and branching of natural philosophy into separate fields, with physics retaining the most basic facets. (See Figure 1.11, Figure 1.12, and Figure 1.13.) Physics as it developed from the Renaissance to the end of the 19th century is called classical physics. It was transformed into modern physics by revolutionary discoveries made starting at the beginning of the 20th century.
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FIGURE 1.11 Over the centuries, natural philosophy has evolved into more specialized disciplines, as illustrated by the contributions of some of the greatest minds in history. The Greek philosopher Aristotle (384–322 B.C.) wrote on a broad range of topics including physics, animals, the soul, politics, and poetry. (credit: Jastrow (2006)/Ludovisi Collection)
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FIGURE 1.12 Ibn al-Haytham (965–1040) and Galileo Galilei (1564–1642) were so critical to the advancement of scientific practice and to the fields of mathematics, optics, physics, and astronomy that a later prominent scientist featured them on the cover of his major work. (credit: Johannes Hevelius (author), Adolph Boÿ (artist), J. Falck (engraver); Houghton Library, Harvard University/Wikimedia Commons)
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14
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1 • Introduction: The Nature of Science and Physics
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FIGURE 1.13 Niels Bohr (1885–1962) made fundamental contributions to the development of quantum mechanics, one part of modern physics. (credit: United States Library of Congress Prints and Photographs Division)
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Classical physics is not an exact description of the universe, but it is an excellent approximation under the following conditions: Matter must be moving at speeds less than about 1% of the speed of light, the objects dealt with must be large enough to be seen with a microscope, and only weak gravitational fields, such as the field generated by the Earth, can be involved. Because humans live under such circumstances, classical physics seems intuitively reasonable, while many aspects of modern physics seem bizarre. This is why models are so useful in modern physics—they let us conceptualize phenomena we do not ordinarily experience. We can relate to models in human terms and visualize what happens when objects move at high speeds or imagine what objects too small to observe with our senses might be like. For example, we can understand an atom’s properties because we can picture it in our minds, although we have never seen an atom with our eyes. New tools, of course, allow us to better picture phenomena we cannot see. In fact, new instrumentation has allowed us in recent years to actually “picture” the atom.
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Limits on the Laws of Classical Physics
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For the laws of classical physics to apply, the following criteria must be met: Matter must be moving at speeds less than about 1% of the speed of light, the objects dealt with must be large enough to be seen with a microscope, and only weak gravitational fields (such as the field generated by the Earth) can be involved.
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FIGURE 1.14 Using a scanning tunneling microscope (STM), scientists can see the individual atoms that compose this sheet of gold. (credit: Erwinrossen)
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Some of the most spectacular advances in science have been made in modern physics. Many of the laws of classical physics have been modified or rejected, and revolutionary changes in technology, society, and our view of the universe have resulted. Like science fiction, modern physics is filled with fascinating objects beyond our normal experiences, but it has the advantage over science fiction of being very real. Why, then, is the majority of this text devoted to topics of classical physics? There are two main reasons: Classical physics gives an extremely accurate description of the universe under a wide range of everyday circumstances, and knowledge of classical physics is necessary to understand modern physics. Modern physics itself consists of the two revolutionary theories, relativity and quantum mechanics. These theories
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1.2 • Physical Quantities and Units
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15
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deal with the very fast and the very small, respectively. Relativity must be used whenever an object is traveling at greater than about 1% of the speed of light or experiences a strong gravitational field such as that near the Sun. Quantum mechanics must be used for objects smaller than can be seen with a microscope. The combination of these two theories is relativistic quantum mechanics, and it describes the behavior of small objects traveling at high speeds or experiencing a strong gravitational field. Relativistic quantum mechanics is the best universally applicable theory we have. Because of its mathematical complexity, it is used only when necessary, and the other theories are used whenever they will produce sufficiently accurate results. We will find, however, that we can do a great deal of modern physics with the algebra and trigonometry used in this text.
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CHECK YOUR UNDERSTANDING
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A friend tells you they have learned about a new law of nature. What can you know about the information even before your friend describes the law? How would the information be different if your friend told you they had learned about a scientific theory rather than a law?
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Solution
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Without knowing the details of the law, you can still infer that the information your friend has learned conforms to the requirements of all laws of nature: it will be a concise description of the universe around us; a statement of the underlying rules that all natural processes follow. If the information had been a theory, you would be able to infer that the information will be a large-scale, broadly applicable generalization.
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PHET EXPLORATIONS
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Equation Grapher
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Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves
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for the individual terms (e.g.
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) to see how they add to generate the polynomial curve.
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Click to view content (https://openstax.org/books/college-physics-2e/pages/1-1-physics-an-introduction)
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1.2 Physical Quantities and Units
|
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LEARNING OBJECTIVES By the end of this section, you will be able to:
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• Perform unit conversions both in the SI and English units. • Explain the most common prefixes in the SI units and be able to write them in scientific notation.
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FIGURE 1.15 The distance from Earth to the Moon may seem immense, but it is just a tiny fraction of the distances from Earth to other celestial bodies. (credit: NASA)
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The range of objects and phenomena studied in physics is immense. From the incredibly short lifetime of a nucleus to the age of the Earth, from the tiny sizes of sub-nuclear particles to the vast distance to the edges of the known universe, from the force exerted by a jumping flea to the force between Earth and the Sun, there are enough factors of 10 to challenge the imagination of even the most experienced scientist. Giving numerical values for physical quantities and equations for physical principles allows us to understand nature much more deeply than does qualitative description alone. To comprehend these vast ranges, we must also have accepted units in which to
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16
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1 • Introduction: The Nature of Science and Physics
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express them. And we shall find that (even in the potentially mundane discussion of meters, kilograms, and seconds) a profound simplicity of nature appears—most physical quantities can be expressed as combinations of only four fundamental physical quantities: length, mass, time, and electric current.
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We define a physical quantity either by specifying how it is measured or by stating how it is calculated from other measurements. For example, we define distance and time by specifying methods for measuring them, whereas we define average speed by stating that it is calculated as distance traveled divided by time of travel.
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Measurements of physical quantities are expressed in terms of units, which are standardized values. For example, the length of a race, which is a physical quantity, can be expressed in units of meters (for sprinters) or kilometers (for distance runners). Without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way. (See Figure 1.16.)
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FIGURE 1.16 Distances given in unknown units are maddeningly useless.
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There are two major systems of units used in the world: SI units (also known as the metric system) and English units (also known as the customary or imperial system). English units were historically used in nations once ruled by the British Empire and are still widely used in the United States. Virtually every other country in the world now uses SI units as the standard; the metric system is also the standard system agreed upon by scientists and mathematicians. The acronym “SI” is derived from the French Système International.
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SI Units: Fundamental and Derived Units
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Table 1.1 gives the fundamental SI units that are used throughout this textbook. This text uses non-SI units in a few applications where they are in very common use, such as the measurement of blood pressure in millimeters of mercury (mm Hg). Whenever non-SI units are discussed, they will be tied to SI units through conversions.
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Length
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Mass
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Time Electric Current
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meter (m) kilogram (kg) second (s) ampere (A)
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TABLE 1.1 Fundamental SI Units
|
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It is an intriguing fact that some physical quantities are more fundamental than others and that the most fundamental physical quantities can be defined only in terms of the procedure used to measure them. The units in which they are measured are thus called fundamental units. In this textbook, the fundamental physical quantities are taken to be length, mass, time, and electric current. (Note that electric current will not be introduced until much later in this text.) All other physical quantities, such as force and electric charge, can be expressed as algebraic combinations of length, mass, time, and current (for example, speed is length divided by time); these units are called derived units.
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Access for free at openstax.org
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1.2 • Physical Quantities and Units
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17
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Units of Time, Length, and Mass: The Second, Meter, and Kilogram
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The Second The SI unit for time, the second (abbreviated s), has a long history. For many years it was defined as 1/86,400 of a mean solar day. More recently, a new standard was adopted to gain greater accuracy and to define the second in terms of a non-varying, or constant, physical phenomenon (because the solar day is getting longer due to very gradual slowing of the Earth’s rotation). Cesium atoms can be made to vibrate in a very steady way, and these vibrations can be readily observed and counted. In 1967 the second was redefined as the time required for 9,192,631,770 of these vibrations. (See Figure 1.17.) Accuracy in the fundamental units is essential, because all measurements are ultimately expressed in terms of fundamental units and can be no more accurate than are the fundamental units themselves.
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FIGURE 1.17 An atomic clock such as this one uses the vibrations of cesium atoms to keep time to a precision of better than a microsecond per year. The fundamental unit of time, the second, is based on such clocks. This image is looking down from the top of an atomic fountain nearly 30 feet tall! (credit: Steve Jurvetson/Flickr)
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The Meter The SI unit for length is the meter (abbreviated m); its definition has also changed over time to become more accurate and precise. The meter was first defined in 1791 as 1/10,000,000 of the distance from the equator to the North Pole. This measurement was improved in 1889 by redefining the meter to be the distance between two engraved lines on a platinum-iridium bar now kept near Paris. By 1960, it had become possible to define the meter even more accurately in terms of the wavelength of light, so it was again redefined as 1,650,763.73 wavelengths of orange light emitted by krypton atoms. In 1983, the meter was given its present definition (partly for greater accuracy) as the distance light travels in a vacuum in 1/299,792,458 of a second. (See Figure 1.18.) This change defines the speed of light to be exactly 299,792,458 meters per second. The length of the meter will change if the speed of light is someday measured with greater accuracy.
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FIGURE 1.18 The meter is defined to be the distance light travels in 1/299,792,458 of a second in a vacuum. Distance traveled is speed multiplied by time.
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The Kilogram The SI unit for mass is the kilogram (abbreviated kg); it was previously defined to be the mass of a platinum-iridium cylinder kept with the old meter standard at the International Bureau of Weights and Measures near Paris. Exact replicas of the previously defined kilogram are also kept at the United States’ National Institute of Standards and Technology, or NIST, located in Gaithersburg, Maryland outside of Washington D.C., and at other locations around the world. The determination of all other masses could be ultimately traced to a comparison with the standard mass. Even though the platinum-iridium cylinder was resistant to corrosion, airborne contaminants were able to adhere to its surface, slightly changing its mass over time. In May 2019, the scientific community adopted a more
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18
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1 • Introduction: The Nature of Science and Physics
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stable definition of the kilogram. The kilogram is now defined in terms of the second, the meter, and Planck's constant, h (a quantum mechanical value that relates a photon's energy to its frequency).
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Electric current and its accompanying unit, the ampere, will be introduced in Introduction to Electric Current, Resistance, and Ohm's Law when electricity and magnetism are covered. The initial modules in this textbook are concerned with mechanics, fluids, heat, and waves. In these subjects all pertinent physical quantities can be expressed in terms of the fundamental units of length, mass, and time.
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Metric Prefixes
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SI units are part of the metric system. The metric system is convenient for scientific and engineering calculations because the units are categorized by factors of 10. Table 1.2 gives metric prefixes and symbols used to denote various factors of 10.
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Metric systems have the advantage that conversions of units involve only powers of 10. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. In nonmetric systems, such as the system of U.S. customary units, the relationships are not as simple—there are 12 inches in a foot, 5280 feet in a mile, and so on. Another advantage of the metric system is that the same unit can be used over extremely large ranges of values simply by using an appropriate metric prefix. For example, distances in meters are suitable in construction, while distances in kilometers are appropriate for air travel, and the tiny measure of nanometers are convenient in optical design. With the metric system there is no need to invent new units for particular applications.
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The term order of magnitude refers to the scale of a value expressed in the metric system. Each power of , and
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so forth are all different orders of magnitude. All quantities that can be expressed as a product of a specific power of
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are said to be of the same order of magnitude. For example, the number can be written as
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, and the
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number can be written as
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Thus, the numbers and are of the same order of magnitude:
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Order of magnitude can be thought of as a ballpark estimate for the scale of a value. The diameter of an atom is on
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the order of
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while the diameter of the Sun is on the order of
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The Quest for Microscopic Standards for Basic Units
|
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The fundamental units described in this chapter are those that produce the greatest accuracy and precision in measurement. There is a sense among physicists that, because there is an underlying microscopic substructure to matter, it would be most satisfying to base our standards of measurement on microscopic objects and fundamental physical phenomena such as the speed of light. A microscopic standard has been accomplished for the standard of time, which is based on the oscillations of the cesium atom.
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The standard for length was once based on the wavelength of light (a small-scale length) emitted by a certain type of atom, but it has been supplanted by the more precise measurement of the speed of light. If it becomes possible to measure the mass of atoms or a particular arrangement of atoms such as a silicon sphere to greater precision than the kilogram standard, it may become possible to base mass measurements on the small scale. There are also possibilities that electrical phenomena on the small scale may someday allow us to base a unit of charge on the charge of electrons and protons, but at present current and charge are related to large-scale currents and forces between wires.
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Prefix Symbol Value
|
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Example (some are approximate)
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exa
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E
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exameter Em
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distance light travels in a century
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peta P
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petasecond Ps
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30 million years
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tera T
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terawatt
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TW
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powerful laser output
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TABLE 1.2 Metric Prefixes for Powers of 10 and their Symbols
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Access for free at openstax.org
|
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1.2 • Physical Quantities and Units
|
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19
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Prefix Symbol Value
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Example (some are approximate)
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giga G
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gigahertz
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GHz
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a microwave frequency
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mega M
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megacurie MCi
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high radioactivity
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kilo k
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kilometer km
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about 6/10 mile
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hecto h
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hectoliter hL
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26 gallons
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deka da
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dekagram dag
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teaspoon of butter
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—
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—
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(=1)
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deci d
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deciliter
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dL
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less than half a soda
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centi c
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centimeter cm
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fingertip thickness
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milli m
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millimeter mm
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flea at its shoulders
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micro µ
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micrometer µm
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detail in microscope
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nano n
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nanogram ng
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small speck of dust
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pico p
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picofarad pF
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small capacitor in radio
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femto f
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femtometer fm
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size of a proton
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atto a
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attosecond as
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time light crosses an atom
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TABLE 1.2 Metric Prefixes for Powers of 10 and their Symbols
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|
Known Ranges of Length, Mass, and Time
|
|||
|
|
The vastness of the universe and the breadth over which physics applies are illustrated by the wide range of examples of known lengths, masses, and times in Table 1.3. Examination of this table will give you some feeling for the range of possible topics and numerical values. (See Figure 1.19 and Figure 1.20.)
|
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|
|||
|
|
FIGURE 1.19 Tiny phytoplankton swims among crystals of ice in the Antarctic Sea. They range from a few micrometers to as much as 2 millimeters in length. (credit: Prof. Gordon T. Taylor, Stony Brook University; NOAA Corps Collections)
|
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|
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|
|
20
|
|||
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|
|
|||
|
|
1 • Introduction: The Nature of Science and Physics
|
|||
|
|
|
|||
|
|
FIGURE 1.20 Galaxies collide 2.4 billion light years away from Earth. The tremendous range of observable phenomena in nature challenges the imagination. (credit: NASA/CXC/UVic./A. Mahdavi et al. Optical/lensing: CFHT/UVic./H. Hoekstra et al.)
|
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|
|
Unit Conversion and Dimensional Analysis
|
|||
|
|
It is often necessary to convert from one type of unit to another. For example, if you are reading a European cookbook, some quantities may be expressed in units of liters and you need to convert them to cups. Or, perhaps you are reading walking directions from one location to another and you are interested in how many miles you will be walking. In this case, you will need to convert units of feet to miles.
|
|||
|
|
Let us consider a simple example of how to convert units. Let us say that we want to convert 80 meters (m) to kilometers (km).
|
|||
|
|
The first thing to do is to list the units that you have and the units that you want to convert to. In this case, we have units in meters and we want to convert to kilometers.
|
|||
|
|
Next, we need to determine a conversion factor relating meters to kilometers. A conversion factor is a ratio expressing how many of one unit are equal to another unit. For example, there are 12 inches in 1 foot, 100 centimeters in 1 meter, 60 seconds in 1 minute, and so on. In this case, we know that there are 1,000 meters in 1 kilometer.
|
|||
|
|
Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor so that the units cancel out, as shown:
|
|||
|
|
1.1
|
|||
|
|
Note that the unwanted m unit cancels, leaving only the desired km unit. You can use this method to convert between any types of unit.
|
|||
|
|
Click Appendix C for a more complete list of conversion factors.
|
|||
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|
|||
|
|
Lengths in meters
|
|||
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|
|||
|
|
Masses in kilograms (more
|
|||
|
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|
|||
|
|
Times in seconds (more
|
|||
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|
|||
|
|
precise values in parentheses) precise values in parentheses)
|
|||
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|
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|
|
Present experimental limit to smallest observable detail
|
|||
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|
|||
|
|
Mass of an electron
|
|||
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|
|||
|
|
Time for light to cross a proton
|
|||
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|
|||
|
|
Diameter of a proton
|
|||
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|
|
|||
|
|
Mass of a hydrogen atom
|
|||
|
|
|
|||
|
|
Mean life of an extremely unstable nucleus
|
|||
|
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|
|||
|
|
TABLE 1.3 Approximate Values of Length, Mass, and Time
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
1.2 • Physical Quantities and Units
|
|||
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|
|||
|
|
21
|
|||
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|
|||
|
|
Lengths in meters
|
|||
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|
|||
|
|
Masses in kilograms (more
|
|||
|
|
|
|||
|
|
Times in seconds (more
|
|||
|
|
|
|||
|
|
precise values in parentheses) precise values in parentheses)
|
|||
|
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|
|||
|
|
Diameter of a uranium nucleus
|
|||
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|
|||
|
|
Mass of a bacterium
|
|||
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|
|||
|
|
Time for one oscillation of visible light
|
|||
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|
|||
|
|
Diameter of a hydrogen atom
|
|||
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|
|||
|
|
Mass of a mosquito
|
|||
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|
|||
|
|
Time for one vibration of an atom in a solid
|
|||
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|
|||
|
|
Thickness of membranes in cells of living organisms
|
|||
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|
|||
|
|
Mass of a hummingbird
|
|||
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|
|||
|
|
Time for one oscillation of an FM radio wave
|
|||
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|
|||
|
|
Wavelength of visible light
|
|||
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|
|||
|
|
Mass of a liter of water (about a quart)
|
|||
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|
|||
|
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Duration of a nerve impulse
|
|||
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|
|||
|
|
Size of a grain of sand
|
|||
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|
|||
|
|
Mass of a person
|
|||
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|
|||
|
|
Time for one heartbeat
|
|||
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|
|||
|
|
Height of a 4-year-old child
|
|||
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|
|||
|
|
Mass of a car
|
|||
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|
|||
|
|
One day
|
|||
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|
|||
|
|
Length of a football field
|
|||
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|
|||
|
|
Mass of a large ship
|
|||
|
|
|
|||
|
|
One year (y)
|
|||
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|
|
|||
|
|
Greatest ocean depth
|
|||
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|
|
|||
|
|
Mass of a large iceberg
|
|||
|
|
|
|||
|
|
Diameter of the Earth
|
|||
|
|
|
|||
|
|
Mass of the nucleus of a comet
|
|||
|
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|
|||
|
|
Distance from the Earth to the Sun
|
|||
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|
|||
|
|
Mass of the Moon
|
|||
|
|
|
|||
|
|
Distance traveled by light in 1 year (a light year)
|
|||
|
|
|
|||
|
|
Mass of the Earth
|
|||
|
|
|
|||
|
|
Diameter of the Milky Way galaxy
|
|||
|
|
|
|||
|
|
Mass of the Sun
|
|||
|
|
|
|||
|
|
Distance from the Earth to the nearest large galaxy (Andromeda)
|
|||
|
|
|
|||
|
|
Mass of the Milky Way galaxy (current upper limit)
|
|||
|
|
|
|||
|
|
Distance from the Earth to the edges of the known universe
|
|||
|
|
|
|||
|
|
Mass of the known universe (current upper limit)
|
|||
|
|
|
|||
|
|
TABLE 1.3 Approximate Values of Length, Mass, and Time
|
|||
|
|
|
|||
|
|
About half the life expectancy of a human Recorded history
|
|||
|
|
Age of the Earth
|
|||
|
|
Age of the universe
|
|||
|
|
|
|||
|
|
22
|
|||
|
|
|
|||
|
|
1 • Introduction: The Nature of Science and Physics
|
|||
|
|
|
|||
|
|
EXAMPLE 1.1
|
|||
|
|
Unit Conversions: A Short Drive Home
|
|||
|
|
Suppose that you drive the 10.0 km from your school to home in 20.0 min. Calculate your average speed (a) in kilometers per hour (km/h) and (b) in meters per second (m/s). (Note: Average speed is distance traveled divided by time of travel.) Strategy First we calculate the average speed using the given units. Then we can get the average speed into the desired units by picking the correct conversion factor and multiplying by it. The correct conversion factor is the one that cancels the unwanted unit and leaves the desired unit in its place. Solution for (a) (1) Calculate average speed. Average speed is distance traveled divided by time of travel. (Take this definition as a given for now—average speed and other motion concepts will be covered in a later module.) In equation form,
|
|||
|
|
1.2
|
|||
|
|
(2) Substitute the given values for distance and time.
|
|||
|
|
1.3
|
|||
|
|
(3) Convert km/min to km/h: multiply by the conversion factor that will cancel minutes and leave hours. That conversion factor is Discussion for (a) To check your answer, consider the following: (1) Be sure that you have properly cancelled the units in the unit conversion. If you have written the unit conversion factor upside down, the units will not cancel properly in the equation. If you accidentally get the ratio upside down, then the units will not cancel; rather, they will give you the wrong units as follows:
|
|||
|
|
1.4
|
|||
|
|
which are obviously not the desired units of km/h. (2) Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of km/h and we have indeed obtained these units. (3) Check the significant figures. Because each of the values given in the problem has three significant figures, the answer should also have three significant figures. The answer 30.0 km/hr does indeed have three significant figures, so this is appropriate. Note that the significant figures in the conversion factor are not relevant because an hour is defined to be 60 minutes, so the precision of the conversion factor is perfect. (4) Next, check whether the answer is reasonable. Let us consider some information from the problem—if you travel 10 km in a third of an hour (20 min), you would travel three times that far in an hour. The answer does seem reasonable. Solution for (b) There are several ways to convert the average speed into meters per second. (1) Start with the answer to (a) and convert km/h to m/s. Two conversion factors are needed—one to convert hours to seconds, and another to convert kilometers to meters. (2) Multiplying by these yields
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
1.3 • Accuracy, Precision, and Significant Figures
|
|||
|
|
|
|||
|
|
23
|
|||
|
|
|
|||
|
|
1.5
|
|||
|
|
1.6
|
|||
|
|
Discussion for (b) If we had started with 0.500 km/min, we would have needed different conversion factors, but the answer would have been the same: 8.33 m/s. You may have noted that the answers in the worked example just covered were given to three digits. Why? When do you need to be concerned about the number of digits in something you calculate? Why not write down all the digits your calculator produces? The module Accuracy, Precision, and Significant Figures will help you answer these questions.
|
|||
|
|
|
|||
|
|
Nonstandard Units
|
|||
|
|
While there are numerous types of units that we are all familiar with, there are others that are much more obscure. For example, a firkin is a unit of volume that was once used to measure beer. One firkin equals about 34 liters. To learn more about nonstandard units, use a dictionary or encyclopedia to research different “weights and measures.” Take note of any unusual units, such as a barleycorn, that are not listed in the text. Think about how the unit is defined and state its relationship to SI units.
|
|||
|
|
|
|||
|
|
CHECK YOUR UNDERSTANDING
|
|||
|
|
|
|||
|
|
Some hummingbirds beat their wings more than 50 times per second. A scientist is measuring the time it takes for a
|
|||
|
|
|
|||
|
|
hummingbird to beat its wings once. Which fundamental unit should the scientist use to describe the measurement?
|
|||
|
|
|
|||
|
|
Which factor of 10 is the scientist likely to use to describe the motion precisely? Identify the metric prefix that
|
|||
|
|
|
|||
|
|
corresponds to this factor of 10.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
The scientist will measure the time between each movement using the fundamental unit of seconds. Because the
|
|||
|
|
|
|||
|
|
wings beat so fast, the scientist will probably need to measure in milliseconds, or
|
|||
|
|
|
|||
|
|
seconds. (50 beats per
|
|||
|
|
|
|||
|
|
second corresponds to 20 milliseconds per beat.)
|
|||
|
|
|
|||
|
|
CHECK YOUR UNDERSTANDING
|
|||
|
|
One cubic centimeter is equal to one milliliter. What does this tell you about the different units in the SI metric system?
|
|||
|
|
Solution
|
|||
|
|
The fundamental unit of length (meter) is probably used to create the derived unit of volume (liter). The measure of a milliliter is dependent on the measure of a centimeter.
|
|||
|
|
|
|||
|
|
1.3 Accuracy, Precision, and Significant Figures
|
|||
|
|
LEARNING OBJECTIVES By the end of this section, you will be able to:
|
|||
|
|
• Determine the appropriate number of significant figures in both addition and subtraction, as well as multiplication and division calculations.
|
|||
|
|
• Calculate the percent uncertainty of a measurement.
|
|||
|
|
|
|||
|
|
24
|
|||
|
|
|
|||
|
|
1 • Introduction: The Nature of Science and Physics
|
|||
|
|
|
|||
|
|
FIGURE 1.21 A double-pan mechanical balance is used to compare different masses. Usually an object with unknown mass is placed in one pan and objects of known mass are placed in the other pan. When the bar that connects the two pans is horizontal, then the masses in both pans are equal. The “known masses” are typically metal cylinders of standard mass such as 1 gram, 10 grams, and 100 grams. (credit: Serge Melki)
|
|||
|
|
|
|||
|
|
FIGURE 1.22 Many mechanical balances, such as double-pan balances, have been replaced by digital scales, which can typically measure the mass of an object more precisely. Whereas a mechanical balance may only read the mass of an object to the nearest tenth of a gram, many digital scales can measure the mass of an object up to the nearest thousandth of a gram. (credit: Karel Jakubec)
|
|||
|
|
Accuracy and Precision of a Measurement
|
|||
|
|
Science is based on observation and experiment—that is, on measurements. Accuracy is how close a measurement is to the correct value for that measurement. For example, let us say that you are measuring the length of standard computer paper. The packaging in which you purchased the paper states that it is 11.0 inches long. You measure the length of the paper three times and obtain the following measurements: 11.1 in., 11.2 in., and 10.9 in. These measurements are quite accurate because they are very close to the correct value of 11.0 inches. In contrast, if you had obtained a measurement of 12 inches, your measurement would not be very accurate.
|
|||
|
|
The precision of a measurement system refers to how close the agreement is between repeated measurements (which are repeated under the same conditions). Consider the example of the paper measurements. The precision of the measurements refers to the spread of the measured values. One way to analyze the precision of the measurements would be to determine the range, or difference, between the lowest and the highest measured values. In that case, the lowest value was 10.9 in. and the highest value was 11.2 in. Thus, the measured values deviated from each other by at most 0.3 in. These measurements were relatively precise because they did not vary too much in value. However, if the measured values had been 10.9, 11.1, and 11.9, then the measurements would not be very precise because there would be significant variation from one measurement to another.
|
|||
|
|
The measurements in the paper example are both accurate and precise, but in some cases, measurements are accurate but not precise, or they are precise but not accurate. Let us consider an example of a GPS system that is attempting to locate the position of a restaurant in a city. Think of the restaurant location as existing at the center of a bull’s-eye target, and think of each GPS attempt to locate the restaurant as a black dot. In Figure 1.23, you can see
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
1.3 • Accuracy, Precision, and Significant Figures
|
|||
|
|
|
|||
|
|
25
|
|||
|
|
|
|||
|
|
that the GPS measurements are spread out far apart from each other, but they are all relatively close to the actual location of the restaurant at the center of the target. This indicates a low precision, high accuracy measuring system. However, in Figure 1.24, the GPS measurements are concentrated quite closely to one another, but they are far away from the target location. This indicates a high precision, low accuracy measuring system.
|
|||
|
|
|
|||
|
|
FIGURE 1.23 A GPS system attempts to locate a restaurant at the center of the bull’s-eye. The black dots represent each attempt to pinpoint the location of the restaurant. The dots are spread out quite far apart from one another, indicating low precision, but they are each rather close to the actual location of the restaurant, indicating high accuracy. (credit: Dark Evil)
|
|||
|
|
|
|||
|
|
FIGURE 1.24 In this figure, the dots are concentrated rather closely to one another, indicating high precision, but they are rather far away from the actual location of the restaurant, indicating low accuracy. (credit: Dark Evil)
|
|||
|
|
|
|||
|
|
Accuracy, Precision, and Uncertainty
|
|||
|
|
|
|||
|
|
The degree of accuracy and precision of a measuring system are related to the uncertainty in the measurements.
|
|||
|
|
|
|||
|
|
Uncertainty is a quantitative measure of how much your measured values deviate from a standard or expected
|
|||
|
|
|
|||
|
|
value. If your measurements are not very accurate or precise, then the uncertainty of your values will be very high.
|
|||
|
|
|
|||
|
|
In more general terms, uncertainty can be thought of as a disclaimer for your measured values. For example, if
|
|||
|
|
|
|||
|
|
someone asked you to provide the mileage on your car, you might say that it is 45,000 miles, plus or minus 500
|
|||
|
|
|
|||
|
|
miles. The plus or minus amount is the uncertainty in your value. That is, you are indicating that the actual mileage
|
|||
|
|
|
|||
|
|
of your car might be as low as 44,500 miles or as high as 45,500 miles, or anywhere in between. All measurements
|
|||
|
|
|
|||
|
|
contain some amount of uncertainty. In our example of measuring the length of the paper, we might say that the
|
|||
|
|
|
|||
|
|
length of the paper is 11 in., plus or minus 0.2 in. The uncertainty in a measurement, , is often denoted as
|
|||
|
|
|
|||
|
|
(“delta ”), so the measurement result would be recorded as
|
|||
|
|
|
|||
|
|
. In our paper example, the length of the paper
|
|||
|
|
|
|||
|
|
could be expressed as
|
|||
|
|
|
|||
|
|
The factors contributing to uncertainty in a measurement include:
|
|||
|
|
|
|||
|
|
1. Limitations of the measuring device, 2. The skill of the person making the measurement, 3. Irregularities in the object being measured, 4. Any other factors that affect the outcome (highly dependent on the situation).
|
|||
|
|
|
|||
|
|
In our example, such factors contributing to the uncertainty could be the following: the smallest division on the ruler is 0.1 in., the person using the ruler has bad eyesight, or one side of the paper is slightly longer than the other. At any rate, the uncertainty in a measurement must be based on a careful consideration of all the factors that might contribute and their possible effects.
|
|||
|
|
|
|||
|
|
26
|
|||
|
|
|
|||
|
|
1 • Introduction: The Nature of Science and Physics
|
|||
|
|
|
|||
|
|
Making Connections: Real-World Connections – Fevers or Chills?
|
|||
|
|
|
|||
|
|
Uncertainty is a critical piece of information, both in physics and in many other real-world applications. Imagine
|
|||
|
|
|
|||
|
|
you are caring for a sick child. You suspect the child has a fever, so you check their temperature with a
|
|||
|
|
|
|||
|
|
thermometer. What if the uncertainty of the thermometer were
|
|||
|
|
|
|||
|
|
? If the child’s temperature reading was
|
|||
|
|
|
|||
|
|
(which is normal body temperature), the “true” temperature could be anywhere from a hypothermic
|
|||
|
|
|
|||
|
|
to a dangerously high
|
|||
|
|
|
|||
|
|
. A thermometer with an uncertainty of
|
|||
|
|
|
|||
|
|
would be useless.
|
|||
|
|
|
|||
|
|
Percent Uncertainty One method of expressing uncertainty is as a percent of the measured value. If a measurement uncertainty, , the percent uncertainty (%unc) is defined to be
|
|||
|
|
|
|||
|
|
is expressed with
|
|||
|
|
|
|||
|
|
1.7
|
|||
|
|
|
|||
|
|
EXAMPLE 1.2
|
|||
|
|
|
|||
|
|
Calculating Percent Uncertainty: A Bag of Apples
|
|||
|
|
A grocery store sells bags of apples. You purchase four bags over the course of a month and weigh the apples each time. You obtain the following measurements:
|
|||
|
|
|
|||
|
|
• Week 1 weight: • Week 2 weight: • Week 3 weight: • Week 4 weight:
|
|||
|
|
|
|||
|
|
You determine that the weight of the bag’s weight?
|
|||
|
|
|
|||
|
|
bag has an uncertainty of
|
|||
|
|
|
|||
|
|
. What is the percent uncertainty of the
|
|||
|
|
|
|||
|
|
Strategy
|
|||
|
|
|
|||
|
|
First, observe that the expected value of the bag’s weight, , is 5 lb. The uncertainty in this value, , is 0.4 lb. We can use the following equation to determine the percent uncertainty of the weight:
|
|||
|
|
|
|||
|
|
1.8
|
|||
|
|
|
|||
|
|
Solution Plug the known values into the equation:
|
|||
|
|
|
|||
|
|
1.9
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
We can conclude that the weight of the apple bag is
|
|||
|
|
|
|||
|
|
. Consider how this percent uncertainty would change
|
|||
|
|
|
|||
|
|
if the bag of apples were half as heavy, but the uncertainty in the weight remained the same. Hint for future
|
|||
|
|
|
|||
|
|
calculations: when calculating percent uncertainty, always remember that you must multiply the fraction by 100%.
|
|||
|
|
|
|||
|
|
If you do not do this, you will have a decimal quantity, not a percent value.
|
|||
|
|
|
|||
|
|
Uncertainties in Calculations There is an uncertainty in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements going into the calculation have small uncertainties (a few percent or less), then the method of adding percents can be used for multiplication or division. This method says that the percent uncertainty in a quantity calculated by multiplication
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
1.3 • Accuracy, Precision, and Significant Figures
|
|||
|
|
|
|||
|
|
27
|
|||
|
|
|
|||
|
|
or division is the sum of the percent uncertainties in the items used to make the calculation. For example, if a floor
|
|||
|
|
|
|||
|
|
has a length of
|
|||
|
|
|
|||
|
|
and a width of
|
|||
|
|
|
|||
|
|
, with uncertainties of and , respectively, then the area of the
|
|||
|
|
|
|||
|
|
floor is
|
|||
|
|
|
|||
|
|
and has an uncertainty of . (Expressed as an area this is
|
|||
|
|
|
|||
|
|
, which we round to
|
|||
|
|
|
|||
|
|
since the area of the floor is given to a tenth of a square meter.)
|
|||
|
|
|
|||
|
|
CHECK YOUR UNDERSTANDING
|
|||
|
|
|
|||
|
|
A high school track coach has just purchased a new stopwatch. The stopwatch manual states that the stopwatch has an uncertainty of ±0.05 s. The team's top sprinter clocked a 100 meter sprint at 12.04 seconds last week and at 11.96 seconds this week. Can we conclude that this week's time was faster?
|
|||
|
|
Solution
|
|||
|
|
No, the uncertainty in the stopwatch is too great to effectively differentiate between the sprint times.
|
|||
|
|
|
|||
|
|
Precision of Measuring Tools and Significant Figures
|
|||
|
|
|
|||
|
|
An important factor in the accuracy and precision of measurements involves the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, a standard ruler can measure length to the nearest millimeter, while a caliper can measure length to the nearest 0.01 millimeter. The caliper is a more precise measuring tool because it can measure extremely small differences in length. The more precise the measuring tool, the more precise and accurate the measurements can be.
|
|||
|
|
|
|||
|
|
When we express measured values, we can only list as many digits as we initially measured with our measuring tool.
|
|||
|
|
|
|||
|
|
For example, if you use a standard ruler to measure the length of a stick, you may measure it to be
|
|||
|
|
|
|||
|
|
. You
|
|||
|
|
|
|||
|
|
could not express this value as
|
|||
|
|
|
|||
|
|
because your measuring tool was not precise enough to measure a
|
|||
|
|
|
|||
|
|
hundredth of a centimeter. It should be noted that the last digit in a measured value has been estimated in some
|
|||
|
|
|
|||
|
|
way by the person performing the measurement. For example, the person measuring the length of a stick with a
|
|||
|
|
|
|||
|
|
ruler notices that the stick length seems to be somewhere in between
|
|||
|
|
|
|||
|
|
and
|
|||
|
|
|
|||
|
|
, and they must
|
|||
|
|
|
|||
|
|
estimate the value of the last digit. Using the method of significant figures, the rule is that the last digit written
|
|||
|
|
|
|||
|
|
down in a measurement is the first digit with some uncertainty. In order to determine the number of significant
|
|||
|
|
|
|||
|
|
digits in a value, start with the first measured value at the left and count the number of digits through the last digit
|
|||
|
|
|
|||
|
|
written on the right. For example, the measured value
|
|||
|
|
|
|||
|
|
has three digits, or significant figures. Significant
|
|||
|
|
|
|||
|
|
figures indicate the precision of a measuring tool that was used to measure a value.
|
|||
|
|
|
|||
|
|
Zeros Special consideration is given to zeros when counting significant figures. The zeros in 0.053 are not significant, because they are only placekeepers that locate the decimal point. There are two significant figures in 0.053. The zeros in 10.053 are not placekeepers but are significant—this number has five significant figures. The zeros in 1300 may or may not be significant depending on the style of writing numbers. They could mean the number is known to the last digit, or they could be placekeepers. So 1300 could have two, three, or four significant figures. (To avoid this ambiguity, write 1300 in scientific notation.) Zeros are significant except when they serve only as placekeepers.
|
|||
|
|
|
|||
|
|
CHECK YOUR UNDERSTANDING
|
|||
|
|
|
|||
|
|
Determine the number of significant figures in the following measurements: a. 0.0009 b. 15,450.0 c. d. 87.990 e. 30.42
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
(a) 1; the zeros in this number are placekeepers that indicate the decimal point
|
|||
|
|
|
|||
|
|
(b) 6; here, the zeros indicate that a measurement was made to the 0.1 decimal point, so the zeros are significant
|
|||
|
|
|
|||
|
|
(c) 1; the value signifies the decimal place, not the number of measured values
|
|||
|
|
|
|||
|
|
28
|
|||
|
|
|
|||
|
|
1 • Introduction: The Nature of Science and Physics
|
|||
|
|
|
|||
|
|
(d) 5; the final zero indicates that a measurement was made to the 0.001 decimal point, so it is significant (e) 4; any zeros located in between significant figures in a number are also significant
|
|||
|
|
|
|||
|
|
Significant Figures in Calculations When combining measurements with different degrees of accuracy and precision, the number of significant digits in the final answer can be no greater than the number of significant digits in the least precise measured value. There are two different rules, one for multiplication and division and the other for addition and subtraction, as discussed below.
|
|||
|
|
|
|||
|
|
1. For multiplication and division: The result should have the same number of significant figures as the quantity
|
|||
|
|
|
|||
|
|
having the least significant figures entering into the calculation. For example, the area of a circle can be calculated
|
|||
|
|
|
|||
|
|
from its radius using
|
|||
|
|
|
|||
|
|
. Let us see how many significant figures the area has if the radius has only two—say,
|
|||
|
|
|
|||
|
|
. Then,
|
|||
|
|
|
|||
|
|
1.10
|
|||
|
|
|
|||
|
|
is what you would get using a calculator that has an eight-digit output. But because the radius has only two significant figures, it limits the calculated quantity to two significant figures or
|
|||
|
|
|
|||
|
|
1.11
|
|||
|
|
|
|||
|
|
even though is good to at least eight digits.
|
|||
|
|
|
|||
|
|
2. For addition and subtraction: The answer can contain no more decimal places than the least precise measurement. Suppose that you buy 7.56-kg of potatoes in a grocery store as measured with a scale with precision 0.01 kg. Then you drop off 6.052-kg of potatoes at your laboratory as measured by a scale with precision 0.001 kg. Finally, you go home and add 13.7 kg of potatoes as measured by a bathroom scale with precision 0.1 kg. How many kilograms of potatoes do you now have, and how many significant figures are appropriate in the answer? The mass is found by simple addition and subtraction:
|
|||
|
|
|
|||
|
|
1.12
|
|||
|
|
|
|||
|
|
Next, we identify the least precise measurement: 13.7 kg. This measurement is expressed to the 0.1 decimal place, so our final answer must also be expressed to the 0.1 decimal place. Thus, the answer is rounded to the tenths place, giving us 15.2 kg.
|
|||
|
|
|
|||
|
|
Significant Figures in this Text In this text, most numbers are assumed to have three significant figures. Furthermore, consistent numbers of significant figures are used in all worked examples. You will note that an answer given to three digits is based on input good to at least three digits, for example. If the input has fewer significant figures, the answer will also have fewer significant figures. Care is also taken that the number of significant figures is reasonable for the situation posed. In some topics, particularly in optics, more accurate numbers are needed and more than three significant figures will be used. Finally, if a number is exact, such as the two in the formula for the circumference of a circle,
|
|||
|
|
, it does not affect the number of significant figures in a calculation.
|
|||
|
|
|
|||
|
|
CHECK YOUR UNDERSTANDING
|
|||
|
|
|
|||
|
|
Perform the following calculations and express your answer using the correct number of significant digits.
|
|||
|
|
|
|||
|
|
(a) A woman has two bags weighing 13.5 pounds and one bag with a weight of 10.2 pounds. What is the total weight
|
|||
|
|
|
|||
|
|
of the bags?
|
|||
|
|
|
|||
|
|
(b) The force on an object is equal to its mass multiplied by its acceleration . If a wagon with mass 55 kg
|
|||
|
|
|
|||
|
|
accelerates at a rate of
|
|||
|
|
|
|||
|
|
, what is the force on the wagon? (The unit of force is called the newton, and it is
|
|||
|
|
|
|||
|
|
expressed with the symbol N.)
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
1.4 • Approximation
|
|||
|
|
|
|||
|
|
29
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
(a) 37.2 pounds; Because the number of bags is an exact value, it is not considered in the significant figures.
|
|||
|
|
(b) 1.4 N; Because the value 55 kg has only two significant figures, the final value must also contain two significant figures.
|
|||
|
|
1.4 Approximation
|
|||
|
|
LEARNING OBJECTIVES By the end of this section, you will be able to:
|
|||
|
|
• Make reasonable approximations based on given data.
|
|||
|
|
On many occasions, physicists, other scientists, and engineers need to make approximations or “guesstimates” for a particular quantity. What is the distance to a certain destination? What is the approximate density of a given item? About how large a current will there be in a circuit? Many approximate numbers are based on formulae in which the input quantities are known only to a limited accuracy. As you develop problem-solving skills (that can be applied to a variety of fields through a study of physics), you will also develop skills at approximating. You will develop these skills through thinking more quantitatively, and by being willing to take risks. As with any endeavor, experience helps, as well as familiarity with units. These approximations allow us to rule out certain scenarios or unrealistic numbers. Approximations also allow us to challenge others and guide us in our approaches to our scientific world. Let us do two examples to illustrate this concept.
|
|||
|
|
EXAMPLE 1.3
|
|||
|
|
Approximate the Height of a Building
|
|||
|
|
Can you approximate the height of one of the buildings on your campus, or in your neighborhood? Let us make an approximation based upon the height of a person. In this example, we will calculate the height of a 39-story building.
|
|||
|
|
Strategy
|
|||
|
|
Think about the average height of an adult male. We can approximate the height of the building by scaling up from the height of a person.
|
|||
|
|
Solution
|
|||
|
|
Based on information in the example, we know there are 39 stories in the building. If we use the fact that the height of one story is approximately equal to about the length of two adult humans (each human is about 2-m tall), then we can estimate the total height of the building to be
|
|||
|
|
1.13
|
|||
|
|
Discussion
|
|||
|
|
You can use known quantities to determine an approximate measurement of unknown quantities. If your hand measures 10 cm across, how many hand lengths equal the width of your desk? What other measurements can you approximate besides length?
|
|||
|
|
|
|||
|
|
30
|
|||
|
|
|
|||
|
|
1 • Introduction: The Nature of Science and Physics
|
|||
|
|
|
|||
|
|
EXAMPLE 1.4 Approximating Vast Numbers: a Trillion Dollars
|
|||
|
|
|
|||
|
|
FIGURE 1.25 A bank stack contains one-hundred $100 bills, and is worth $10,000. How many bank stacks make up a trillion dollars? (credit: Andrew Magill)
|
|||
|
|
The U.S. federal debt in 2021 was a little more than $28 trillion. Most of us do not have any concept of how much even one trillion actually is. Suppose that you were given a trillion dollars in $100 bills. If you made 100-bill stacks and used them to evenly cover a football field (between the end zones), make an approximation of how high the money pile would become. (We will use feet/inches rather than meters here because football fields are measured in yards.) One of your friends says 3 in., while another says 10 ft. What do you think?
|
|||
|
|
Strategy
|
|||
|
|
When you imagine the situation, you probably envision thousands of small stacks of 100 wrapped $100 bills, such as you might see in movies or at a bank. Since this is an easy-to-approximate quantity, let us start there. We can find the volume of a stack of 100 bills, find out how many stacks make up one trillion dollars, and then set this volume equal to the area of the football field multiplied by the unknown height.
|
|||
|
|
Solution
|
|||
|
|
(1) Calculate the volume of a stack of 100 bills. The dimensions of a single bill are approximately 3 in. by 6 in. A stack of 100 of these is about 0.5 in. thick. So the total volume of a stack of 100 bills is:
|
|||
|
|
|
|||
|
|
1.14
|
|||
|
|
|
|||
|
|
(2) Calculate the number of stacks. Note that a trillion dollars is equal to
|
|||
|
|
|
|||
|
|
and a stack of one-hundred
|
|||
|
|
|
|||
|
|
bills is equal to
|
|||
|
|
|
|||
|
|
or
|
|||
|
|
|
|||
|
|
. The number of stacks you will have is:
|
|||
|
|
|
|||
|
|
1.15
|
|||
|
|
|
|||
|
|
(3) Calculate the area of a football field in square inches. The area of a football field is Because we are working in inches, we need to convert square yards to square inches:
|
|||
|
|
|
|||
|
|
which gives
|
|||
|
|
|
|||
|
|
1.16
|
|||
|
|
|
|||
|
|
This conversion gives us these calculations.)
|
|||
|
|
|
|||
|
|
for the area of the field. (Note that we are using only one significant figure in
|
|||
|
|
|
|||
|
|
(4) Calculate the total volume of the bills. The volume of all the -bill stacks is
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
1.4 • Approximation
|
|||
|
|
|
|||
|
|
31
|
|||
|
|
|
|||
|
|
(5) Calculate the height. To determine the height of the bills, use the equation:
|
|||
|
|
|
|||
|
|
1.17
|
|||
|
|
|
|||
|
|
The height of the money will be about 100 in. high. Converting this value to feet gives
|
|||
|
|
1.18
|
|||
|
|
Discussion The final approximate value is much higher than the early estimate of 3 in., but the other early estimate of 10 ft (120 in.) was roughly correct. How did the approximation measure up to your first guess? What can this exercise tell you in terms of rough “guesstimates” versus carefully calculated approximations?
|
|||
|
|
|
|||
|
|
CHECK YOUR UNDERSTANDING
|
|||
|
|
|
|||
|
|
Using mental math and your understanding of fundamental units, approximate the area of a regulation basketball
|
|||
|
|
|
|||
|
|
court. Describe the process you used to arrive at your final approximation.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
An average male is about two meters tall. It would take approximately 15 males laid out end to end to cover the
|
|||
|
|
|
|||
|
|
length, and about 7 to cover the width. That gives an approximate area of
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
32 1 • Glossary
|
|||
|
|
|
|||
|
|
Glossary
|
|||
|
|
accuracy the degree to which a measured value agrees with correct value for that measurement
|
|||
|
|
approximation an estimated value based on prior experience and reasoning
|
|||
|
|
classical physics physics that was developed from the Renaissance to the end of the 19th century
|
|||
|
|
conversion factor a ratio expressing how many of one unit are equal to another unit
|
|||
|
|
derived units units that can be calculated using algebraic combinations of the fundamental units
|
|||
|
|
English units system of measurement used in the United States; includes units of measurement such as feet, gallons, and pounds
|
|||
|
|
fundamental units units that can only be expressed relative to the procedure used to measure them
|
|||
|
|
kilogram the SI unit for mass, abbreviated (kg) law a description, using concise language or a
|
|||
|
|
mathematical formula, a generalized pattern in nature that is supported by scientific evidence and repeated experiments meter the SI unit for length, abbreviated (m) method of adding percents the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation metric system a system in which values can be calculated in factors of 10 model representation of something that is often too difficult (or impossible) to display directly modern physics the study of relativity, quantum mechanics, or both order of magnitude refers to the size of a quantity as it relates to a power of 10 percent uncertainty the ratio of the uncertainty of a measurement to the measured value, expressed as a percentage
|
|||
|
|
Section Summary
|
|||
|
|
1.1 Physics: An Introduction
|
|||
|
|
• Science seeks to discover and describe the underlying order and simplicity in nature.
|
|||
|
|
• Physics is the most basic of the sciences, concerning itself with energy, matter, space and time, and their interactions.
|
|||
|
|
• Scientific laws and theories express the general truths of nature and the body of knowledge they encompass. These laws of nature are rules that all natural processes appear to follow.
|
|||
|
|
1.2 Physical Quantities and Units
|
|||
|
|
• Physical quantities are a characteristic or property
|
|||
|
|
|
|||
|
|
physical quantity a characteristic or property of an object that can be measured or calculated from other measurements
|
|||
|
|
physics the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
|
|||
|
|
precision the degree to which repeated measurements agree with each other
|
|||
|
|
quantum mechanics the study of objects smaller than can be seen with a microscope
|
|||
|
|
relativity the study of objects moving at speeds greater than about 1% of the speed of light, or of objects being affected by a strong gravitational field
|
|||
|
|
scientific method a method that typically begins with an observation and question that the scientist will research; next, the scientist typically performs some research about the topic and then devises a hypothesis; then, the scientist will test the hypothesis by performing an experiment; finally, the scientist analyzes the results of the experiment and draws a conclusion
|
|||
|
|
second the SI unit for time, abbreviated (s) SI units the international system of units that
|
|||
|
|
scientists in most countries have agreed to use; includes units such as meters, liters, and grams significant figures express the precision of a measuring tool used to measure a value theory an explanation for patterns in nature that is supported by scientific evidence and verified multiple times by various groups of researchers uncertainty a quantitative measure of how much your measured values deviate from a standard or expected value units a standard used for expressing and comparing measurements
|
|||
|
|
of an object that can be measured or calculated from other measurements. • Units are standards for expressing and comparing the measurement of physical quantities. All units can be expressed as combinations of four fundamental units. • The four fundamental units we will use in this text are the meter (for length), the kilogram (for mass), the second (for time), and the ampere (for electric current). These units are part of the metric system, which uses powers of 10 to relate quantities over the vast ranges encountered in nature. • The four fundamental units are abbreviated as follows: meter, m; kilogram, kg; second, s; and
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
1 • Conceptual Questions 33
|
|||
|
|
|
|||
|
|
ampere, A. The metric system also uses a standard set of prefixes to denote each order of magnitude greater than or lesser than the fundamental unit itself. • Unit conversions involve changing a value expressed in one type of unit to another type of unit. This is done by using conversion factors, which are ratios relating equal quantities of different units.
|
|||
|
|
|
|||
|
|
1.3 Accuracy, Precision, and Significant Figures
|
|||
|
|
|
|||
|
|
• Accuracy of a measured value refers to how close a measurement is to the correct value. The uncertainty in a measurement is an estimate of the amount by which the measurement result may differ from this value.
|
|||
|
|
• Precision of measured values refers to how close
|
|||
|
|
|
|||
|
|
Conceptual Questions
|
|||
|
|
|
|||
|
|
1.1 Physics: An Introduction
|
|||
|
|
|
|||
|
|
1. Models are particularly useful in relativity and quantum mechanics, where conditions are outside those normally encountered by humans. What is a model?
|
|||
|
|
2. How does a model differ from a theory? 3. If two different theories describe experimental
|
|||
|
|
observations equally well, can one be said to be more valid than the other (assuming both use accepted rules of logic)? 4. What determines the validity of a theory? 5. Certain criteria must be satisfied if a measurement or observation is to be believed. Will the criteria necessarily be as strict for an expected result as for an unexpected result? 6. Can the validity of a model be limited, or must it be universally valid? How does this compare to the required validity of a theory or a law? 7. Classical physics is a good approximation to modern physics under certain circumstances. What are they? 8. When is it necessary to use relativistic quantum mechanics?
|
|||
|
|
|
|||
|
|
Problems & Exercises
|
|||
|
|
|
|||
|
|
1.2 Physical Quantities and Units
|
|||
|
|
|
|||
|
|
1. The speed limit on some interstate highways is
|
|||
|
|
|
|||
|
|
roughly 100 km/h. (a) What is this in meters per
|
|||
|
|
|
|||
|
|
second? (b) How many miles per hour is this?
|
|||
|
|
|
|||
|
|
2. A car is traveling at a speed of
|
|||
|
|
|
|||
|
|
. (a) What is
|
|||
|
|
|
|||
|
|
its speed in kilometers per hour? (b) Is it exceeding
|
|||
|
|
|
|||
|
|
the
|
|||
|
|
|
|||
|
|
speed limit?
|
|||
|
|
|
|||
|
|
the agreement is between repeated measurements. • The precision of a measuring tool is related to the size of its measurement increments. The smaller the measurement increment, the more precise the tool. • Significant figures express the precision of a measuring tool. • When multiplying or dividing measured values, the final answer can contain only as many significant figures as the least precise value. • When adding or subtracting measured values, the final answer cannot contain more decimal places than the least precise value.
|
|||
|
|
1.4 Approximation
|
|||
|
|
Scientists often approximate the values of quantities to perform calculations and analyze systems.
|
|||
|
|
9. Can classical physics be used to accurately describe a satellite moving at a speed of 7500 m/s? Explain why or why not.
|
|||
|
|
1.2 Physical Quantities and Units
|
|||
|
|
10. Identify some advantages of metric units.
|
|||
|
|
1.3 Accuracy, Precision, and Significant Figures
|
|||
|
|
11. What is the relationship between the accuracy and uncertainty of a measurement?
|
|||
|
|
12. Prescriptions for vision correction are given in units called diopters (D). Determine the meaning of that unit. Obtain information (perhaps by calling an optometrist or performing an internet search) on the minimum uncertainty with which corrections in diopters are determined and the accuracy with which corrective lenses can be produced. Discuss the sources of uncertainties in both the prescription and accuracy in the manufacture of lenses.
|
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|
|||
|
|
3. Show that
|
|||
|
|
|
|||
|
|
. Hint: Show the
|
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|
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|
|||
|
|
explicit steps involved in converting
|
|||
|
|
|
|||
|
|
4. American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.)
|
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|
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|
34 1 • Problems & Exercises
|
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|
|
5. Soccer fields vary in size. A large soccer field is 115
|
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|
|||
|
|
m long and 85 m wide. What are its dimensions in
|
|||
|
|
|
|||
|
|
feet and inches? (Assume that 1 meter equals
|
|||
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|
|||
|
|
3.281 feet.)
|
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6. What is the height in meters of a person who is 6 ft
|
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|
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|
|
1.0 in. tall? (Assume that 1 meter equals 39.37 in.)
|
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|
7. Mount Everest, at 29,028 feet, is the tallest
|
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|
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|
|
mountain on the Earth. What is its height in
|
|||
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|
|
|||
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|
kilometers? (Assume that 1 kilometer equals 3,281
|
|||
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|
|||
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|
feet.)
|
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8. The speed of sound is measured to be
|
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|
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|
on a
|
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|
|||
|
|
certain day. What is this in km/h?
|
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|
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|
|
9. Tectonic plates are large segments of the Earth’s
|
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|
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|
|
crust that move slowly. Suppose that one such
|
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|
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|
|
plate has an average speed of 4.0 cm/year. (a) What
|
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|
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|
|
distance does it move in 1 s at this speed? (b) What
|
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|
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|
|
is its speed in kilometers per million years?
|
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10. (a) Refer to Table 1.3 to determine the average
|
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|
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distance between the Earth and the Sun. Then
|
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|
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|
calculate the average speed of the Earth in its
|
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|
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|
|
orbit in kilometers per second. (b) What is this in
|
|||
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|
|
|||
|
|
meters per second?
|
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|
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|
|
1.3 Accuracy, Precision, and Significant Figures
|
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|
|||
|
|
Express your answers to problems in this section to the correct number of significant figures and proper units.
|
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|
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|
|
11. Suppose that your bathroom scale reads your
|
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|
|||
|
|
mass as 65 kg with a 3% uncertainty. What is the
|
|||
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|
|
|||
|
|
uncertainty in your mass (in kilograms)?
|
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|
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|
|
12. A good-quality measuring tape can be off by 0.50
|
|||
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|
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|
|
cm over a distance of 20 m. What is its percent
|
|||
|
|
|
|||
|
|
uncertainty?
|
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|
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|
|
13. (a) A car speedometer has a
|
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|
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|
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uncertainty.
|
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|
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|
|
What is the range of possible speeds when it reads
|
|||
|
|
|
|||
|
|
? (b) Convert this range to miles per hour.
|
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|
|||
|
|
14. An infant’s pulse rate is measured to be
|
|||
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|
|||
|
|
beats/min. What is the percent uncertainty in this
|
|||
|
|
|
|||
|
|
measurement?
|
|||
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|
|||
|
|
15. (a) Suppose that a person has an average heart
|
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|
|||
|
|
rate of 72.0 beats/min. How many beats do they
|
|||
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|
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|
|
have in 2.0 y? (b) In 2.00 y? (c) In 2.000 y?
|
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|
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|
|
16. A can contains 375 mL of soda. How much is left
|
|||
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|
|||
|
|
after 308 mL is removed?
|
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|
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|
|
17. State how many significant figures are proper in
|
|||
|
|
|
|||
|
|
the results of the following calculations: (a)
|
|||
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|
|||
|
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(b)
|
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|
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|
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(c)
|
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|
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|
|
.
|
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|
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|
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18. (a) How many significant figures are in the
|
|||
|
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|
|||
|
|
numbers 99 and 100? (b) If the uncertainty in
|
|||
|
|
|
|||
|
|
each number is 1, what is the percent uncertainty
|
|||
|
|
|
|||
|
|
in each? (c) Which is a more meaningful way to
|
|||
|
|
|
|||
|
|
express the accuracy of these two numbers,
|
|||
|
|
|
|||
|
|
significant figures or percent uncertainties?
|
|||
|
|
|
|||
|
|
19. (a) If your speedometer has an uncertainty of
|
|||
|
|
|
|||
|
|
at a speed of
|
|||
|
|
|
|||
|
|
, what is the
|
|||
|
|
|
|||
|
|
percent uncertainty? (b) If it has the same percent
|
|||
|
|
|
|||
|
|
uncertainty when it reads
|
|||
|
|
|
|||
|
|
, what is the
|
|||
|
|
|
|||
|
|
range of speeds you could be going?
|
|||
|
|
|
|||
|
|
20. (a) A person’s blood pressure is measured to be
|
|||
|
|
|
|||
|
|
. What is its percent uncertainty?
|
|||
|
|
|
|||
|
|
(b) Assuming the same percent uncertainty, what
|
|||
|
|
|
|||
|
|
is the uncertainty in a blood pressure
|
|||
|
|
|
|||
|
|
measurement of
|
|||
|
|
|
|||
|
|
?
|
|||
|
|
|
|||
|
|
21. A person measures their heart rate by counting
|
|||
|
|
|
|||
|
|
the number of beats in . If
|
|||
|
|
|
|||
|
|
beats are
|
|||
|
|
|
|||
|
|
counted in
|
|||
|
|
|
|||
|
|
, what is the heart rate and
|
|||
|
|
|
|||
|
|
its uncertainty in beats per minute?
|
|||
|
|
|
|||
|
|
22. What is the area of a circle
|
|||
|
|
|
|||
|
|
in diameter?
|
|||
|
|
|
|||
|
|
23. If a marathon runner averages 9.5 mi/h, how long
|
|||
|
|
|
|||
|
|
does it take him or her to run a 26.22-mi
|
|||
|
|
|
|||
|
|
marathon?
|
|||
|
|
|
|||
|
|
24. A marathon runner completes a
|
|||
|
|
|
|||
|
|
course in , 30 min, and . There is an
|
|||
|
|
|
|||
|
|
uncertainty of
|
|||
|
|
|
|||
|
|
in the distance traveled and
|
|||
|
|
|
|||
|
|
an uncertainty of 1 s in the elapsed time. (a)
|
|||
|
|
|
|||
|
|
Calculate the percent uncertainty in the distance.
|
|||
|
|
|
|||
|
|
(b) Calculate the uncertainty in the elapsed time.
|
|||
|
|
|
|||
|
|
(c) What is the average speed in meters per
|
|||
|
|
|
|||
|
|
second? (d) What is the uncertainty in the average
|
|||
|
|
|
|||
|
|
speed?
|
|||
|
|
|
|||
|
|
25. The sides of a small rectangular box are measured
|
|||
|
|
|
|||
|
|
to be
|
|||
|
|
|
|||
|
|
,
|
|||
|
|
|
|||
|
|
long. Calculate
|
|||
|
|
|
|||
|
|
its volume and uncertainty in cubic centimeters.
|
|||
|
|
|
|||
|
|
26. When non-metric units were used in the United
|
|||
|
|
|
|||
|
|
Kingdom, a unit of mass called the pound-mass
|
|||
|
|
|
|||
|
|
(lbm) was employed, where
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
(a) If there is an uncertainty of
|
|||
|
|
|
|||
|
|
in the
|
|||
|
|
|
|||
|
|
pound-mass unit, what is its percent uncertainty?
|
|||
|
|
|
|||
|
|
(b) Based on that percent uncertainty, what mass
|
|||
|
|
|
|||
|
|
in pound-mass has an uncertainty of 1 kg when
|
|||
|
|
|
|||
|
|
converted to kilograms?
|
|||
|
|
|
|||
|
|
27. The length and width of a rectangular room are
|
|||
|
|
|
|||
|
|
measured to be
|
|||
|
|
|
|||
|
|
and
|
|||
|
|
|
|||
|
|
. Calculate the area of the room
|
|||
|
|
|
|||
|
|
and its uncertainty in square meters.
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
1 • Problems & Exercises 35
|
|||
|
|
|
|||
|
|
28. A car engine moves a piston with a circular cross
|
|||
|
|
|
|||
|
|
section of
|
|||
|
|
|
|||
|
|
diameter a distance
|
|||
|
|
|
|||
|
|
of
|
|||
|
|
|
|||
|
|
to compress the gas in the
|
|||
|
|
|
|||
|
|
cylinder. (a) By what amount is the gas decreased
|
|||
|
|
|
|||
|
|
in volume in cubic centimeters? (b) Find the
|
|||
|
|
|
|||
|
|
uncertainty in this volume.
|
|||
|
|
|
|||
|
|
1.4 Approximation
|
|||
|
|
|
|||
|
|
29. How many heartbeats are there in a lifetime?
|
|||
|
|
|
|||
|
|
30. A generation is about one-third of a lifetime.
|
|||
|
|
|
|||
|
|
Approximately how many generations have
|
|||
|
|
|
|||
|
|
passed since the year 0 AD?
|
|||
|
|
|
|||
|
|
31. How many times longer than the mean life of an
|
|||
|
|
|
|||
|
|
extremely unstable atomic nucleus is the lifetime
|
|||
|
|
|
|||
|
|
of a human? (Hint: The lifetime of an unstable
|
|||
|
|
|
|||
|
|
atomic nucleus is on the order of
|
|||
|
|
|
|||
|
|
.)
|
|||
|
|
|
|||
|
|
32. Calculate the approximate number of atoms in a
|
|||
|
|
|
|||
|
|
bacterium. Assume that the average mass of an
|
|||
|
|
|
|||
|
|
atom in the bacterium is ten times the mass of a
|
|||
|
|
|
|||
|
|
hydrogen atom. (Hint: The mass of a hydrogen
|
|||
|
|
|
|||
|
|
atom is on the order of
|
|||
|
|
|
|||
|
|
and the mass of
|
|||
|
|
|
|||
|
|
a bacterium is on the order of
|
|||
|
|
|
|||
|
|
)
|
|||
|
|
|
|||
|
|
FIGURE 1.26 This color-enhanced photo shows Salmonella typhimurium (red) attacking human cells. These bacteria are commonly known for causing foodborne illness. Can you estimate the number of atoms in each bacterium? (credit: Rocky Mountain Laboratories, NIAID, NIH)
|
|||
|
|
33. Approximately how many atoms thick is a cell membrane, assuming all atoms there average about twice the size of a hydrogen atom?
|
|||
|
|
34. (a) What fraction of Earth’s diameter is the greatest ocean depth? (b) The greatest mountain height?
|
|||
|
|
35. (a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is ten times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a human?
|
|||
|
|
36. Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second?
|
|||
|
|
|
|||
|
|
36 1 • Problems & Exercises Access for free at openstax.org
|
|||
|
|
|
|||
|
|
CHAPTER 2
|
|||
|
|
Kinematics
|
|||
|
|
FIGURE 2.1 The motion of an American kestrel through the air can be described by the bird’s displacement, speed, velocity, and acceleration. When it flies in a straight line without any change in direction, its motion is said to be one dimensional. (credit: Vince Maidens, Wikimedia Commons)
|
|||
|
|
CHAPTER OUTLINE 2.1 Displacement 2.2 Vectors, Scalars, and Coordinate Systems 2.3 Time, Velocity, and Speed 2.4 Acceleration 2.5 Motion Equations for Constant Acceleration in One Dimension 2.6 Problem-Solving Basics for One-Dimensional Kinematics 2.7 Falling Objects 2.8 Graphical Analysis of One-Dimensional Motion
|
|||
|
|
INTRODUCTION TO ONE-DIMENSIONAL KINEMATICS Objects are in motion everywhere we look. Everything from a tennis game to a space-probe flyby of the planet Neptune involves motion. When you are resting, your heart moves blood through your veins. And even in inanimate objects, there is continuous motion in the vibrations of atoms and molecules. Questions about motion are interesting in and of themselves: How long will it take for a space probe to get to Mars? Where will a football land if it is thrown at a certain angle? But an understanding of motion is also key to understanding other concepts in physics. An understanding of acceleration, for example, is crucial to the study of force. Our formal study of physics begins with kinematics which is defined as the study of motion without considering its causes. The word “kinematics” comes from a Greek term meaning motion and is related to other English words such as “cinema” (movies) and “kinesiology” (the study of human motion). In one-dimensional kinematics and TwoDimensional Kinematics we will study only the motion of a football, for example, without worrying about what forces cause or change its motion. Such considerations come in other chapters. In this chapter, we examine the simplest
|
|||
|
|
|
|||
|
|
38
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
type of motion—namely, motion along a straight line, or one-dimensional motion. In Two-Dimensional Kinematics, we apply concepts developed here to study motion along curved paths (two- and three-dimensional motion); for example, that of a car rounding a curve.
|
|||
|
|
Click to view content (https://openstax.org/books/college-physics-2e/pages/2-introduction-to-onedimensionalkinematics)
|
|||
|
|
2.1 Displacement
|
|||
|
|
LEARNING OBJECTIVES By the end of this section, you will be able to:
|
|||
|
|
• Define position, displacement, distance, and distance traveled. • Explain the relationship between position and displacement. • Distinguish between displacement and distance traveled. • Calculate displacement and distance given initial position, final position, and the path between the two.
|
|||
|
|
|
|||
|
|
FIGURE 2.2 These cyclists in Vietnam can be described by their position relative to buildings and a canal. Their motion can be described by their change in position, or displacement, in the frame of reference. (credit: Suzan Black, Fotopedia)
|
|||
|
|
Position
|
|||
|
|
In order to describe the motion of an object, you must first be able to describe its position—where it is at any particular time. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a rocket launch would be described in terms of the position of the rocket with respect to the Earth as a whole, while a professor’s position could be described in terms of where she is in relation to the nearby white board. (See Figure 2.3.) In other cases, we use reference frames that are not stationary but are in motion relative to the Earth. To describe the position of a person in an airplane, for example, we use the airplane, not the Earth, as the reference frame. (See Figure 2.4.)
|
|||
|
|
Displacement
|
|||
|
|
If an object moves relative to a reference frame (for example, if a professor moves to the right relative to a white board or a passenger moves toward the rear of an airplane), then the object’s position changes. This change in position is known as displacement. The word “displacement” implies that an object has moved, or has been displaced.
|
|||
|
|
Displacement
|
|||
|
|
Displacement is the change in position of an object: 2.1
|
|||
|
|
where is displacement, is the final position, and is the initial position.
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.1 • Displacement
|
|||
|
|
|
|||
|
|
39
|
|||
|
|
|
|||
|
|
In this text the upper case Greek letter (delta) always means “change in” whatever quantity follows it; thus, means change in position. Always solve for displacement by subtracting initial position from final position .
|
|||
|
|
Note that the SI unit for displacement is the meter (m) (see Physical Quantities and Units), but sometimes kilometers, miles, feet, and other units of length are used. Keep in mind that when units other than the meter are used in a problem, you may need to convert them into meters to complete the calculation.
|
|||
|
|
|
|||
|
|
FIGURE 2.3 A professor paces left and right while lecturing. Her position relative to Earth is given by . The professor relative to Earth is represented by an arrow pointing to the right.
|
|||
|
|
|
|||
|
|
displacement of the
|
|||
|
|
|
|||
|
|
FIGURE 2.4 A passenger moves from his seat to the back of the plane. His location relative to the airplane is given by . The displacement of the passenger relative to the plane is represented by an arrow toward the rear of the plane. Notice that the arrow representing his displacement is twice as long as the arrow representing the displacement of the professor (he moves twice as far) in Figure 2.3.
|
|||
|
|
|
|||
|
|
Note that displacement has a direction as well as a magnitude. The professor’s displacement is 2.0 m to the right,
|
|||
|
|
|
|||
|
|
and the airline passenger’s displacement is 4.0 m toward the rear. In one-dimensional motion, direction can be
|
|||
|
|
|
|||
|
|
specified with a plus or minus sign. When you begin a problem, you should select which direction is positive (usually
|
|||
|
|
|
|||
|
|
that will be to the right or up, but you are free to select positive as being any direction). The professor’s initial
|
|||
|
|
|
|||
|
|
position is
|
|||
|
|
|
|||
|
|
and her final position is
|
|||
|
|
|
|||
|
|
. Thus her displacement is
|
|||
|
|
|
|||
|
|
2.2
|
|||
|
|
|
|||
|
|
40
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
In this coordinate system, motion to the right is positive, whereas motion to the left is negative. Similarly, the
|
|||
|
|
|
|||
|
|
airplane passenger’s initial position is
|
|||
|
|
|
|||
|
|
and his final position is
|
|||
|
|
|
|||
|
|
, so his displacement is
|
|||
|
|
|
|||
|
|
2.3
|
|||
|
|
|
|||
|
|
His displacement is negative because his motion is toward the rear of the plane, or in the negative direction in our coordinate system.
|
|||
|
|
|
|||
|
|
Distance
|
|||
|
|
|
|||
|
|
Although displacement is described in terms of direction, distance is not. Distance is defined to be the magnitude or size of displacement between two positions. Note that the distance between two positions is not the same as the distance traveled between them. Distance traveled is the total length of the path traveled between two positions. Distance has no direction and, thus, no sign. For example, the distance the professor walks is 2.0 m. The distance the airplane passenger walks is 4.0 m.
|
|||
|
|
|
|||
|
|
Misconception Alert: Distance Traveled vs. Magnitude of Displacement
|
|||
|
|
It is important to note that the distance traveled, however, can be greater than the magnitude of the displacement (by magnitude, we mean just the size of the displacement without regard to its direction; that is, just a number with a unit). For example, the professor could pace back and forth many times, perhaps walking a distance of 150 m during a lecture, yet still end up only 2.0 m to the right of her starting point. In this case her displacement would be +2.0 m, the magnitude of her displacement would be 2.0 m, but the distance she traveled would be 150 m. In kinematics we nearly always deal with displacement and magnitude of displacement, and almost never with distance traveled. One way to think about this is to assume you marked the start of the motion and the end of the motion. The displacement is simply the difference in the position of the two marks and is independent of the path taken in traveling between the two marks. The distance traveled, however, is the total length of the path taken between the two marks.
|
|||
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|
|||
|
|
CHECK YOUR UNDERSTANDING
|
|||
|
|
A cyclist rides 3 km west and then turns around and rides 2 km east. (a) What is their displacement? (b) What distance do they ride? (c) What is the magnitude of their displacement?
|
|||
|
|
Solution
|
|||
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|
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|
|
(a) The rider’s displacement is positive and west to be negative.)
|
|||
|
|
(b) The distance traveled is
|
|||
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|
(c) The magnitude of the displacement is
|
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|
|||
|
|
FIGURE 2.5
|
|||
|
|
. (The displacement is negative because we take east to be
|
|||
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|
. .
|
|||
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|
|||
|
|
2.2 Vectors, Scalars, and Coordinate Systems
|
|||
|
|
LEARNING OBJECTIVES By the end of this section, you will be able to:
|
|||
|
|
• Define and distinguish between scalar and vector quantities. • Assign a coordinate system for a scenario involving one-dimensional motion.
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
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|
|||
|
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2.2 • Vectors, Scalars, and Coordinate Systems
|
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41
|
|||
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FIGURE 2.6 The motion of this Eclipse Concept jet can be described in terms of the distance it has traveled (a scalar quantity) or its displacement in a specific direction (a vector quantity). In order to specify the direction of motion, its displacement must be described based on a coordinate system. In this case, it may be convenient to choose motion toward the left as positive motion (it is the forward direction for the plane), although in many cases, the -coordinate runs from left to right, with motion to the right as positive and motion to the left as negative. (credit: Armchair Aviator, Flickr)
|
|||
|
|
What is the difference between distance and displacement? Whereas displacement is defined by both direction and magnitude, distance is defined only by magnitude. Displacement is an example of a vector quantity. Distance is an example of a scalar quantity. A vector is any quantity with both magnitude and direction. Other examples of vectors include a velocity of 90 km/h east and a force of 500 newtons straight down.
|
|||
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|
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|
|
The direction of a vector in one-dimensional motion is given simply by a plus or minus sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector’s magnitude (e.g., the larger the magnitude, the longer the length of the vector) and points in the same direction as the vector.
|
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|
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|
|
Some physical quantities, like distance, either have no direction or none is specified. A scalar is any quantity that
|
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|
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has a magnitude, but no direction. For example, a
|
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|
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|
|
temperature, the 250 kilocalories (250 Calories) of energy in
|
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|
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|
|
a candy bar, a 90 km/h speed limit, a person’s 1.8 m height, and a distance of 2.0 m are all scalars—quantities with
|
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|
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|
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no specified direction. Note, however, that a scalar can be negative, such as a
|
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|
|||
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|
temperature. In this case, the
|
|||
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|
|||
|
|
minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.
|
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|
|
Coordinate Systems for One-Dimensional Motion
|
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|
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In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one-dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however, as with the jet in Figure 2.6, it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it.
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42
|
|||
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2 • Kinematics
|
|||
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|
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|
FIGURE 2.7 It is usually convenient to consider motion upward or to the right as positive negative .
|
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|
|||
|
|
and motion downward or to the left as
|
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|
|
CHECK YOUR UNDERSTANDING
|
|||
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|
|
|||
|
|
A person’s speed can stay the same as they round a corner and changes direction. Given this information, is speed a scalar or a vector quantity? Explain.
|
|||
|
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Solution
|
|||
|
|
Speed is a scalar quantity. It does not change at all with direction changes; therefore, it has magnitude only. If it were a vector quantity, it would change as direction changes (even if its magnitude remained constant).
|
|||
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|
|||
|
|
2.3 Time, Velocity, and Speed
|
|||
|
|
LEARNING OBJECTIVES By the end of this section, you will be able to:
|
|||
|
|
• Explain the relationships between instantaneous velocity, average velocity, instantaneous speed, average speed, displacement, and time.
|
|||
|
|
• Calculate velocity and speed given initial position, initial time, final position, and final time. • Derive a graph of velocity vs. time given a graph of position vs. time. • Interpret a graph of velocity vs. time.
|
|||
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|
|||
|
|
FIGURE 2.8 The motion of these racing snails can be described by their speeds and their velocities. (credit: tobitasflickr, Flickr)
|
|||
|
|
There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner’s speed?” cannot be answered without an understanding of other concepts. In this section we add definitions of time, velocity, and speed to expand our description of motion.
|
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Time
|
|||
|
|
As discussed in Physical Quantities and Units, the most fundamental physical quantities are defined by how they are measured. This is the case with time. Every measurement of time involves measuring a change in some physical quantity. It may be a number on a digital clock, a heartbeat, or the position of the Sun in the sky. In physics, the definition of time is simple—time is change, or the interval over which change occurs. It is impossible to know that
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.3 • Time, Velocity, and Speed
|
|||
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|
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43
|
|||
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|
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|
time has passed unless something changes.
|
|||
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|
|||
|
|
The amount of time or change is calibrated by comparison with a standard. The SI unit for time is the second, abbreviated s. We might, for example, observe that a certain pendulum makes one full swing every 0.75 s. We could then use the pendulum to measure time by counting its swings or, of course, by connecting the pendulum to a clock mechanism that registers time on a dial. This allows us to not only measure the amount of time, but also to determine a sequence of events.
|
|||
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|
|||
|
|
How does time relate to motion? We are usually interested in elapsed time for a particular motion, such as how long it takes an airplane passenger to get from his seat to the back of the plane. To find elapsed time, we note the time at the beginning and end of the motion and subtract the two. For example, a lecture may start at 11:00 A.M. and end at 11:50 A.M., so that the elapsed time would be 50 min. Elapsed time is the difference between the ending time and beginning time,
|
|||
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|
|||
|
|
2.4
|
|||
|
|
|
|||
|
|
where is the change in time or elapsed time, is the time at the end of the motion, and is the time at the beginning of the motion. (As usual, the delta symbol, , means the change in the quantity that follows it.)
|
|||
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|
|||
|
|
Life is simpler if the beginning time is taken to be zero, as when we use a stopwatch. If we were using a
|
|||
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|
|||
|
|
stopwatch, it would simply read zero at the start of the lecture and 50 min at the end. If
|
|||
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|
|||
|
|
, then
|
|||
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|
|||
|
|
.
|
|||
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|
|||
|
|
In this text, for simplicity’s sake,
|
|||
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|
|||
|
|
• motion starts at time equal to zero • the symbol is used for elapsed time unless otherwise specified
|
|||
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|
|||
|
|
Velocity
|
|||
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|
|||
|
|
Your notion of velocity is probably the same as its scientific definition. You know that if you have a large displacement in a small amount of time you have a large velocity, and that velocity has units of distance divided by time, such as miles per hour or kilometers per hour.
|
|||
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|
|||
|
|
Average Velocity
|
|||
|
|
Average velocity is displacement (change in position) divided by the time of travel,
|
|||
|
|
2.5
|
|||
|
|
where is the average (indicated by the bar over the ) velocity, is the change in position (or displacement), and and are the final and beginning positions at times and , respectively. If the starting time is taken to be zero, then the average velocity is simply
|
|||
|
|
2.6
|
|||
|
|
|
|||
|
|
Notice that this definition indicates that velocity is a vector because displacement is a vector. It has both magnitude and direction. The SI unit for velocity is meters per second or m/s, but many other units, such as km/h, mi/h (also written as mph), and cm/s, are in common use. Suppose, for example, an airplane passenger took 5 seconds to move −4 m (the negative sign indicates that displacement is toward the back of the plane). His average velocity would be
|
|||
|
|
2.7
|
|||
|
|
The minus sign indicates the average velocity is also toward the rear of the plane.
|
|||
|
|
The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point, however. For example, we cannot tell from average velocity whether the airplane passenger stops momentarily or backs up before he goes to the back of the plane. To get more details, we must consider smaller
|
|||
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|
|||
|
|
44
|
|||
|
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|
|||
|
|
2 • Kinematics
|
|||
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|
|||
|
|
segments of the trip over smaller time intervals.
|
|||
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|
|||
|
|
FIGURE 2.9 A more detailed record of an airplane passenger heading toward the back of the plane, showing smaller segments of his trip.
|
|||
|
|
The smaller the time intervals considered in a motion, the more detailed the information. When we carry this process to its logical conclusion, we are left with an infinitesimally small interval. Over such an interval, the average velocity becomes the instantaneous velocity or the velocity at a specific instant. A car’s speedometer, for example, shows the magnitude (but not the direction) of the instantaneous velocity of the car. (Police give tickets based on instantaneous velocity, but when calculating how long it will take to get from one place to another on a road trip, you need to use average velocity.) Instantaneous velocity is the average velocity at a specific instant in time (or over an infinitesimally small time interval). Mathematically, finding instantaneous velocity, , at a precise instant can involve taking a limit, a calculus operation beyond the scope of this text. However, under many circumstances, we can find precise values for instantaneous velocity without calculus.
|
|||
|
|
Speed
|
|||
|
|
In everyday language, most people use the terms “speed” and “velocity” interchangeably. In physics, however, they do not have the same meaning and they are distinct concepts. One major difference is that speed has no direction. Thus speed is a scalar. Just as we need to distinguish between instantaneous velocity and average velocity, we also need to distinguish between instantaneous speed and average speed. Instantaneous speed is the magnitude of instantaneous velocity. For example, suppose the airplane passenger at one instant had an instantaneous velocity of −3.0 m/s (the minus meaning toward the rear of the plane). At that same time his instantaneous speed was 3.0 m/s. Or suppose that at one time during a shopping trip your instantaneous velocity is 40 km/h due north. Your instantaneous speed at that instant would be 40 km/h—the same magnitude but without a direction. Average speed, however, is very different from average velocity. Average speed is the distance traveled divided by elapsed time. We have noted that distance traveled can be greater than the magnitude of displacement. So average speed can be greater than average velocity, which is displacement divided by time. For example, if you drive to a store and return home in half an hour, and your car’s odometer shows the total distance traveled was 6 km, then your average speed was 12 km/h. Your average velocity, however, was zero, because your displacement for the round trip is zero. (Displacement is change in position and, thus, is zero for a round trip.) Thus average speed is not simply the magnitude of average velocity.
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.3 • Time, Velocity, and Speed
|
|||
|
|
|
|||
|
|
45
|
|||
|
|
|
|||
|
|
FIGURE 2.10 During a 30-minute round trip to the store, the total distance traveled is 6 km. The average speed is 12 km/h. The displacement for the round trip is zero, since there was no net change in position. Thus the average velocity is zero.
|
|||
|
|
Another way of visualizing the motion of an object is to use a graph. A plot of position or of velocity as a function of time can be very useful. For example, for this trip to the store, the position, velocity, and speed-vs.-time graphs are displayed in Figure 2.11. (Note that these graphs depict a very simplified model of the trip. We are assuming that speed is constant during the trip, which is unrealistic given that we’ll probably stop at the store. But for simplicity’s sake, we will model it with no stops or changes in speed. We are also assuming that the route between the store and the house is a perfectly straight line.)
|
|||
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|
|||
|
|
46
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
FIGURE 2.11 Position vs. time, velocity vs. time, and speed vs. time on a trip. Note that the velocity for the return trip is negative.
|
|||
|
|
Making Connections: Take-Home Investigation—Getting a Sense of Speed
|
|||
|
|
If you have spent much time driving, you probably have a good sense of speeds between about 10 and 70 miles per hour. But what are these in meters per second? What do we mean when we say that something is moving at 10 m/s? To get a better sense of what these values really mean, do some observations and calculations on your own:
|
|||
|
|
• calculate typical car speeds in meters per second • estimate jogging and walking speed by timing yourself; convert the measurements into both m/s and mi/h
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.4 • Acceleration
|
|||
|
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|
|||
|
|
47
|
|||
|
|
|
|||
|
|
• determine the speed of an ant, snail, or falling leaf
|
|||
|
|
|
|||
|
|
CHECK YOUR UNDERSTANDING
|
|||
|
|
|
|||
|
|
A commuter train travels from Baltimore to Washington, DC, and back in 1 hour and 45 minutes. The distance
|
|||
|
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|
|||
|
|
between the two stations is approximately 40 miles. What is (a) the average velocity of the train, and (b) the average
|
|||
|
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|
|||
|
|
speed of the train in m/s?
|
|||
|
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|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
(a) The average velocity of the train is zero because
|
|||
|
|
|
|||
|
|
; the train ends up at the same place it starts.
|
|||
|
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|
|||
|
|
(b) The average speed of the train is calculated below. Note that the train travels 40 miles one way and 40 miles back, for a total distance of 80 miles.
|
|||
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|
|||
|
|
2.8
|
|||
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|
|
|||
|
|
2.9
|
|||
|
|
|
|||
|
|
2.4 Acceleration
|
|||
|
|
LEARNING OBJECTIVES By the end of this section, you will be able to:
|
|||
|
|
• Define and distinguish between instantaneous acceleration, average acceleration, and deceleration. • Calculate acceleration given initial time, initial velocity, final time, and final velocity.
|
|||
|
|
|
|||
|
|
FIGURE 2.12 A plane decelerates, or slows down, as it comes in for landing in St. Maarten. Its acceleration is opposite in direction to its velocity. (credit: Steve Conry, Flickr)
|
|||
|
|
In everyday conversation, to accelerate means to speed up. The accelerator in a car can in fact cause it to speed up. The greater the acceleration, the greater the change in velocity over a given time. The formal definition of acceleration is consistent with these notions, but more inclusive.
|
|||
|
|
Average Acceleration
|
|||
|
|
Average Acceleration is the rate at which velocity changes,
|
|||
|
|
2.10
|
|||
|
|
where is average acceleration, is velocity, and is time. (The bar over the means average acceleration.)
|
|||
|
|
Because acceleration is velocity in m/s divided by time in s, the SI units for acceleration are , meters per second squared or meters per second per second, which literally means by how many meters per second the velocity changes every second.
|
|||
|
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|
|||
|
|
48
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
Recall that velocity is a vector—it has both magnitude and direction. This means that a change in velocity can be a change in magnitude (or speed), but it can also be a change in direction. For example, if a car turns a corner at constant speed, it is accelerating because its direction is changing. The quicker you turn, the greater the acceleration. So there is an acceleration when velocity changes either in magnitude (an increase or decrease in speed) or in direction, or both.
|
|||
|
|
Acceleration as a Vector
|
|||
|
|
Acceleration is a vector in the same direction as the change in velocity, . Since velocity is a vector, it can change either in magnitude or in direction. Acceleration is therefore a change in either speed or direction, or both.
|
|||
|
|
Keep in mind that although acceleration is in the direction of the change in velocity, it is not always in the direction of motion. When an object slows down, its acceleration is opposite to the direction of its motion. This is known as deceleration.
|
|||
|
|
|
|||
|
|
FIGURE 2.13 A subway train in Sao Paulo, Brazil, decelerates as it comes into a station. It is accelerating in a direction opposite to its direction of motion. (credit: Yusuke Kawasaki, Flickr)
|
|||
|
|
Misconception Alert: Deceleration vs. Negative Acceleration
|
|||
|
|
Deceleration always refers to acceleration in the direction opposite to the direction of the velocity. Deceleration always reduces speed. Negative acceleration, however, is acceleration in the negative direction in the chosen coordinate system. Negative acceleration may or may not be deceleration, and deceleration may or may not be considered negative acceleration. For example, consider Figure 2.14.
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.4 • Acceleration
|
|||
|
|
|
|||
|
|
49
|
|||
|
|
|
|||
|
|
FIGURE 2.14 (a) This car is speeding up as it moves toward the right. It therefore has positive acceleration in our coordinate system. (b) This car is slowing down as it moves toward the right. Therefore, it has negative acceleration in our coordinate system, because its acceleration is toward the left. The car is also decelerating: the direction of its acceleration is opposite to its direction of motion. (c) This car is moving toward the left, but slowing down over time. Therefore, its acceleration is positive in our coordinate system because it is toward the right. However, the car is decelerating because its acceleration is opposite to its motion. (d) This car is speeding up as it moves toward the left. It has negative acceleration because it is accelerating toward the left. However, because its acceleration is in the same direction as its motion, it is speeding up (not decelerating).
|
|||
|
|
EXAMPLE 2.1
|
|||
|
|
Calculating Acceleration: A Racehorse Leaves the Gate
|
|||
|
|
A racehorse coming out of the gate accelerates from rest to a velocity of 15.0 m/s due west in 1.80 s. What is its average acceleration?
|
|||
|
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|
|||
|
|
50
|
|||
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|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
FIGURE 2.15 (credit: Jon Sullivan, PD Photo.org)
|
|||
|
|
Strategy
|
|||
|
|
First we draw a sketch and assign a coordinate system to the problem. This is a simple problem, but it always helps to visualize it. Notice that we assign east as positive and west as negative. Thus, in this case, we have negative velocity.
|
|||
|
|
|
|||
|
|
We can solve this problem by identifying acceleration directly from the equation
|
|||
|
|
|
|||
|
|
FIGURE 2.16
|
|||
|
|
and from the given information and then calculating the average .
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the knowns.
|
|||
|
|
|
|||
|
|
,
|
|||
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|
|||
|
|
.
|
|||
|
|
|
|||
|
|
(the negative sign indicates direction toward the west),
|
|||
|
|
|
|||
|
|
2. Find the change in velocity. Since the horse is going from zero to
|
|||
|
|
|
|||
|
|
velocity:
|
|||
|
|
|
|||
|
|
.
|
|||
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|
|||
|
|
, its change in velocity equals its final
|
|||
|
|
|
|||
|
|
3. Plug in the known values ( and ) and solve for the unknown .
|
|||
|
|
|
|||
|
|
2.11
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
The negative sign for acceleration indicates that acceleration is toward the west. An acceleration of
|
|||
|
|
|
|||
|
|
due
|
|||
|
|
|
|||
|
|
west means that the horse increases its velocity by 8.33 m/s due west each second, that is, 8.33 meters per second
|
|||
|
|
|
|||
|
|
per second, which we write as
|
|||
|
|
|
|||
|
|
. This is truly an average acceleration, because the ride is not smooth. We
|
|||
|
|
|
|||
|
|
shall see later that an acceleration of this magnitude would require the rider to hang on with a force nearly equal to
|
|||
|
|
|
|||
|
|
his weight.
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.4 • Acceleration
|
|||
|
|
|
|||
|
|
51
|
|||
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|
|||
|
|
Instantaneous Acceleration
|
|||
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|
|
|||
|
|
Instantaneous acceleration , or the acceleration at a specific instant in time, is obtained by the same process as
|
|||
|
|
|
|||
|
|
discussed for instantaneous velocity in Time, Velocity, and Speed—that is, by considering an infinitesimally small
|
|||
|
|
|
|||
|
|
interval of time. How do we find instantaneous acceleration using only algebra? The answer is that we choose an
|
|||
|
|
|
|||
|
|
average acceleration that is representative of the motion. Figure 2.17 shows graphs of instantaneous acceleration
|
|||
|
|
|
|||
|
|
versus time for two very different motions. In Figure 2.17(a), the acceleration varies slightly and the average over
|
|||
|
|
|
|||
|
|
the entire interval is nearly the same as the instantaneous acceleration at any time. In this case, we should treat this
|
|||
|
|
|
|||
|
|
motion as if it had a constant acceleration equal to the average (in this case about
|
|||
|
|
|
|||
|
|
). In Figure 2.17(b), the
|
|||
|
|
|
|||
|
|
acceleration varies drastically over time. In such situations it is best to consider smaller time intervals and choose
|
|||
|
|
|
|||
|
|
an average acceleration for each. For example, we could consider motion over the time intervals from 0 to 1.0 s and
|
|||
|
|
|
|||
|
|
from 1.0 to 3.0 s as separate motions with accelerations of
|
|||
|
|
|
|||
|
|
and
|
|||
|
|
|
|||
|
|
, respectively.
|
|||
|
|
|
|||
|
|
FIGURE 2.17 Graphs of instantaneous acceleration versus time for two different one-dimensional motions. (a) Here acceleration varies only slightly and is always in the same direction, since it is positive. The average over the interval is nearly the same as the acceleration at any given time. (b) Here the acceleration varies greatly, perhaps representing a package on a post office conveyor belt that is accelerated forward and backward as it bumps along. It is necessary to consider small time intervals (such as from 0 to 1.0 s) with constant or nearly constant acceleration in such a situation.
|
|||
|
|
The next several examples consider the motion of the subway train shown in Figure 2.18. In (a) the shuttle moves to the right, and in (b) it moves to the left. The examples are designed to further illustrate aspects of motion and to illustrate some of the reasoning that goes into solving problems.
|
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|
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|
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52
|
|||
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|
|||
|
|
2 • Kinematics
|
|||
|
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|
|||
|
|
FIGURE 2.18 One-dimensional motion of a subway train considered in Example 2.2, Example 2.3, Example 2.4, Example 2.5, Example 2.6,
|
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|
|||
|
|
and Example 2.7. Here we have chosen the -axis so that + means to the right and means to the left for displacements, velocities, and
|
|||
|
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|
|||
|
|
accelerations. (a) The subway train moves to the right from to . Its displacement is +2.0 km. (b) The train moves to the left from
|
|||
|
|
|
|||
|
|
to . Its displacement is
|
|||
|
|
|
|||
|
|
. (Note that the prime symbol (′) is used simply to distinguish between displacement in the two
|
|||
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|
|||
|
|
different situations. The distances of travel and the size of the cars are on different scales to fit everything into the diagram.)
|
|||
|
|
|
|||
|
|
EXAMPLE 2.2
|
|||
|
|
|
|||
|
|
Calculating Displacement: A Subway Train
|
|||
|
|
What are the magnitude and sign of displacements for the motions of the subway train shown in parts (a) and (b) of Figure 2.18?
|
|||
|
|
|
|||
|
|
Strategy
|
|||
|
|
|
|||
|
|
A drawing with a coordinate system is already provided, so we don’t need to make a sketch, but we should analyze it
|
|||
|
|
|
|||
|
|
to make sure we understand what it is showing. Pay particular attention to the coordinate system. To find
|
|||
|
|
|
|||
|
|
displacement, we use the equation
|
|||
|
|
|
|||
|
|
. This is straightforward since the initial and final positions are
|
|||
|
|
|
|||
|
|
given.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the knowns. In the figure we see that
|
|||
|
|
|
|||
|
|
and
|
|||
|
|
|
|||
|
|
for part (a), and
|
|||
|
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|
|||
|
|
and
|
|||
|
|
|
|||
|
|
for part (b).
|
|||
|
|
|
|||
|
|
2. Solve for displacement in part (a).
|
|||
|
|
|
|||
|
|
2.12
|
|||
|
|
|
|||
|
|
3. Solve for displacement in part (b).
|
|||
|
|
|
|||
|
|
2.13
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
The direction of the motion in (a) is to the right and therefore its displacement has a positive sign, whereas motion in (b) is to the left and thus has a negative sign.
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.4 • Acceleration
|
|||
|
|
|
|||
|
|
53
|
|||
|
|
|
|||
|
|
EXAMPLE 2.3
|
|||
|
|
|
|||
|
|
Comparing Distance Traveled with Displacement: A Subway Train
|
|||
|
|
What are the distances traveled for the motions shown in parts (a) and (b) of the subway train in Figure 2.18?
|
|||
|
|
|
|||
|
|
Strategy
|
|||
|
|
|
|||
|
|
To answer this question, think about the definitions of distance and distance traveled, and how they are related to displacement. Distance between two positions is defined to be the magnitude of displacement, which was found in Example 2.2. Distance traveled is the total length of the path traveled between the two positions. (See Displacement.) In the case of the subway train shown in Figure 2.18, the distance traveled is the same as the distance between the initial and final positions of the train.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. The displacement for part (a) was +2.00 km. Therefore, the distance between the initial and final positions was 2.00 km, and the distance traveled was 2.00 km.
|
|||
|
|
|
|||
|
|
2. The displacement for part (b) was
|
|||
|
|
|
|||
|
|
Therefore, the distance between the initial and final positions was
|
|||
|
|
|
|||
|
|
1.50 km, and the distance traveled was 1.50 km.
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
Distance is a scalar. It has magnitude but no sign to indicate direction.
|
|||
|
|
|
|||
|
|
EXAMPLE 2.4
|
|||
|
|
Calculating Acceleration: A Subway Train Speeding Up
|
|||
|
|
Suppose the train in Figure 2.18(a) accelerates from rest to 30.0 km/h in the first 20.0 s of its motion. What is its average acceleration during that time interval? Strategy It is worth it at this point to make a simple sketch:
|
|||
|
|
|
|||
|
|
FIGURE 2.19
|
|||
|
|
|
|||
|
|
This problem involves three steps. First we must determine the change in velocity, then we must determine the change in time, and finally we use these values to calculate the acceleration.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the knowns.
|
|||
|
|
|
|||
|
|
(the trains starts at rest),
|
|||
|
|
|
|||
|
|
, and
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Calculate . Since the train starts from rest, its change in velocity is means velocity to the right.
|
|||
|
|
|
|||
|
|
, where the plus sign
|
|||
|
|
|
|||
|
|
3. Plug in known values and solve for the unknown, .
|
|||
|
|
|
|||
|
|
54
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
2.14
|
|||
|
|
4. Since the units are mixed (we have both hours and seconds for time), we need to convert everything into SI units of meters and seconds. (See Physical Quantities and Units for more guidance.)
|
|||
|
|
2.15
|
|||
|
|
Discussion The plus sign means that acceleration is to the right. This is reasonable because the train starts from rest and ends up with a velocity to the right (also positive). So acceleration is in the same direction as the change in velocity, as is always the case.
|
|||
|
|
|
|||
|
|
EXAMPLE 2.5
|
|||
|
|
Calculate Acceleration: A Subway Train Slowing Down
|
|||
|
|
Now suppose that at the end of its trip, the train in Figure 2.18(a) slows to a stop from a speed of 30.0 km/h in 8.00 s. What is its average acceleration while stopping?
|
|||
|
|
Strategy
|
|||
|
|
|
|||
|
|
FIGURE 2.20
|
|||
|
|
|
|||
|
|
In this case, the train is decelerating and its acceleration is negative because it is toward the left. As in the previous example, we must find the change in velocity and the change in time and then solve for acceleration.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the knowns.
|
|||
|
|
|
|||
|
|
,
|
|||
|
|
|
|||
|
|
(the train is stopped, so its velocity is 0), and
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Solve for the change in velocity, .
|
|||
|
|
|
|||
|
|
2.16
|
|||
|
|
|
|||
|
|
3. Plug in the knowns, and , and solve for .
|
|||
|
|
|
|||
|
|
2.17
|
|||
|
|
|
|||
|
|
4. Convert the units to meters and seconds.
|
|||
|
|
|
|||
|
|
2.18
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
The minus sign indicates that acceleration is to the left. This sign is reasonable because the train initially has a positive velocity in this problem, and a negative acceleration would oppose the motion. Again, acceleration is in the same direction as the change in velocity, which is negative here. This acceleration can be called a deceleration because it has a direction opposite to the velocity.
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.4 • Acceleration
|
|||
|
|
|
|||
|
|
55
|
|||
|
|
|
|||
|
|
The graphs of position, velocity, and acceleration vs. time for the trains in Example 2.4 and Example 2.5 are displayed in Figure 2.21. (We have taken the velocity to remain constant from 20 to 40 s, after which the train decelerates.)
|
|||
|
|
|
|||
|
|
FIGURE 2.21 (a) Position of the train over time. Notice that the train’s position changes slowly at the beginning of the journey, then more and more quickly as it picks up speed. Its position then changes more slowly as it slows down at the end of the journey. In the middle of the journey, while the velocity remains constant, the position changes at a constant rate. (b) Velocity of the train over time. The train’s velocity increases as it accelerates at the beginning of the journey. It remains the same in the middle of the journey (where there is no acceleration). It decreases as the train decelerates at the end of the journey. (c) The acceleration of the train over time. The train has positive acceleration as it speeds up at the beginning of the journey. It has no acceleration as it travels at constant velocity in the middle of the journey. Its acceleration is negative as it slows down at the end of the journey.
|
|||
|
|
|
|||
|
|
56
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
EXAMPLE 2.6
|
|||
|
|
Calculating Average Velocity: The Subway Train
|
|||
|
|
What is the average velocity of the train in part b of Example 2.2, and shown again below, if it takes 5.00 min to make its trip?
|
|||
|
|
|
|||
|
|
FIGURE 2.22
|
|||
|
|
|
|||
|
|
Strategy
|
|||
|
|
|
|||
|
|
Average velocity is displacement divided by time. It will be negative here, since the train moves to the left and has a negative displacement.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the knowns.
|
|||
|
|
|
|||
|
|
,
|
|||
|
|
|
|||
|
|
,
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Determine displacement, . We found to be
|
|||
|
|
|
|||
|
|
in Example 2.2.
|
|||
|
|
|
|||
|
|
3. Solve for average velocity.
|
|||
|
|
|
|||
|
|
2.19
|
|||
|
|
|
|||
|
|
4. Convert units.
|
|||
|
|
|
|||
|
|
2.20
|
|||
|
|
|
|||
|
|
Discussion The negative velocity indicates motion to the left.
|
|||
|
|
|
|||
|
|
EXAMPLE 2.7
|
|||
|
|
Calculating Deceleration: The Subway Train
|
|||
|
|
Finally, suppose the train in Figure 2.22 slows to a stop from a velocity of 20.0 km/h in 10.0 s. What is its average acceleration? Strategy Once again, let’s draw a sketch:
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.4 • Acceleration
|
|||
|
|
|
|||
|
|
57
|
|||
|
|
|
|||
|
|
FIGURE 2.23
|
|||
|
|
|
|||
|
|
As before, we must find the change in velocity and the change in time to calculate average acceleration.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the knowns.
|
|||
|
|
|
|||
|
|
,
|
|||
|
|
|
|||
|
|
,
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Calculate . The change in velocity here is actually positive, since
|
|||
|
|
|
|||
|
|
3. Solve for .
|
|||
|
|
|
|||
|
|
4. Convert units.
|
|||
|
|
|
|||
|
|
2.21 2.22
|
|||
|
|
|
|||
|
|
2.23
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
The plus sign means that acceleration is to the right. This is reasonable because the train initially has a negative velocity (to the left) in this problem and a positive acceleration opposes the motion (and so it is to the right). Again, acceleration is in the same direction as the change in velocity, which is positive here. As in Example 2.5, this acceleration can be called a deceleration since it is in the direction opposite to the velocity.
|
|||
|
|
|
|||
|
|
Sign and Direction
|
|||
|
|
Perhaps the most important thing to note about these examples is the signs of the answers. In our chosen coordinate system, plus means the quantity is to the right and minus means it is to the left. This is easy to imagine for displacement and velocity. But it is a little less obvious for acceleration. Most people interpret negative acceleration as the slowing of an object. This was not the case in Example 2.7, where a positive acceleration slowed a negative velocity. The crucial distinction was that the acceleration was in the opposite direction from the velocity. In fact, a negative acceleration will increase a negative velocity. For example, the train moving to the left in Figure 2.22 is sped up by an acceleration to the left. In that case, both and are negative. The plus and minus signs give the directions of the accelerations. If acceleration has the same sign as the velocity, the object is speeding up. If acceleration has the opposite sign as the velocity, the object is slowing down.
|
|||
|
|
CHECK YOUR UNDERSTANDING
|
|||
|
|
An airplane lands on a runway traveling east. Describe its acceleration.
|
|||
|
|
Solution
|
|||
|
|
If we take east to be positive, then the airplane has negative acceleration, as it is accelerating toward the west. It is also decelerating: its acceleration is opposite in direction to its velocity.
|
|||
|
|
|
|||
|
|
58
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
PHET EXPLORATIONS
|
|||
|
|
Moving Man Simulation
|
|||
|
|
Learn about position, velocity, and acceleration graphs. Move the little man back and forth with the mouse and plot his motion. Set the position, velocity, or acceleration and let the simulation move the man for you.
|
|||
|
|
Click to view content (https://openstax.org/l/02moving_man).
|
|||
|
|
|
|||
|
|
2.5 Motion Equations for Constant Acceleration in One Dimension
|
|||
|
|
LEARNING OBJECTIVES By the end of this section, you will be able to:
|
|||
|
|
• Calculate displacement of an object that is not accelerating, given initial position and velocity. • Calculate final velocity of an accelerating object, given initial velocity, acceleration, and time. • Calculate displacement and final position of an accelerating object, given initial position, initial velocity,
|
|||
|
|
time, and acceleration.
|
|||
|
|
|
|||
|
|
FIGURE 2.24 Kinematic equations can help us describe and predict the motion of moving objects such as these kayaks racing in Newbury, England. (credit: Barry Skeates, Flickr)
|
|||
|
|
|
|||
|
|
We might know that the greater the acceleration of, say, a car moving away from a stop sign, the greater the displacement in a given time. But we have not developed a specific equation that relates acceleration and displacement. In this section, we develop some convenient equations for kinematic relationships, starting from the definitions of displacement, velocity, and acceleration already covered.
|
|||
|
|
|
|||
|
|
Notation: t, x, v, a
|
|||
|
|
|
|||
|
|
First, let us make some simplifications in notation. Taking the initial time to be zero, as if time is measured with a
|
|||
|
|
|
|||
|
|
stopwatch, is a great simplification. Since elapsed time is
|
|||
|
|
|
|||
|
|
, taking
|
|||
|
|
|
|||
|
|
means that
|
|||
|
|
|
|||
|
|
, the final
|
|||
|
|
|
|||
|
|
time on the stopwatch. When initial time is taken to be zero, we use the subscript 0 to denote initial values of
|
|||
|
|
|
|||
|
|
position and velocity. That is, is the initial position and is the initial velocity. We put no subscripts on the final
|
|||
|
|
|
|||
|
|
values. That is, is the final time, is the final position, and is the final velocity. This gives a simpler expression for
|
|||
|
|
|
|||
|
|
elapsed time—now,
|
|||
|
|
|
|||
|
|
. It also simplifies the expression for displacement, which is now
|
|||
|
|
|
|||
|
|
. Also, it
|
|||
|
|
|
|||
|
|
simplifies the expression for change in velocity, which is now
|
|||
|
|
|
|||
|
|
. To summarize, using the simplified
|
|||
|
|
|
|||
|
|
notation, with the initial time taken to be zero,
|
|||
|
|
|
|||
|
|
2.24
|
|||
|
|
|
|||
|
|
where the subscript 0 denotes an initial value and the absence of a subscript denotes a final value in whatever motion is under consideration.
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.5 • Motion Equations for Constant Acceleration in One Dimension
|
|||
|
|
|
|||
|
|
59
|
|||
|
|
|
|||
|
|
We now make the important assumption that acceleration is constant. This assumption allows us to avoid using calculus to find instantaneous acceleration. Since acceleration is constant, the average and instantaneous accelerations are equal. That is,
|
|||
|
|
2.25
|
|||
|
|
so we use the symbol for acceleration at all times. Assuming acceleration to be constant does not seriously limit the situations we can study nor degrade the accuracy of our treatment. For one thing, acceleration is constant in a great number of situations. Furthermore, in many other situations we can accurately describe motion by assuming a constant acceleration equal to the average acceleration for that motion. Finally, in motions where acceleration changes drastically, such as a car accelerating to top speed and then braking to a stop, the motion can be considered in separate parts, each of which has its own constant acceleration.
|
|||
|
|
|
|||
|
|
Solving for Displacement ( ) and Final Position ( ) from Average Velocity when Acceleration ( ) is Constant
|
|||
|
|
To get our first two new equations, we start with the definition of average velocity:
|
|||
|
|
|
|||
|
|
2.26
|
|||
|
|
|
|||
|
|
Substituting the simplified notation for and yields
|
|||
|
|
|
|||
|
|
2.27
|
|||
|
|
|
|||
|
|
Solving for yields where the average velocity is
|
|||
|
|
|
|||
|
|
2.28
|
|||
|
|
|
|||
|
|
2.29
|
|||
|
|
|
|||
|
|
The equation
|
|||
|
|
|
|||
|
|
reflects the fact that, when acceleration is constant, is just the simple average of the
|
|||
|
|
|
|||
|
|
initial and final velocities. For example, if you steadily increase your velocity (that is, with constant acceleration) from 30 to 60 km/h, then your average velocity during this steady increase is 45 km/h. Using the equation
|
|||
|
|
|
|||
|
|
to check this, we see that
|
|||
|
|
|
|||
|
|
which seems logical.
|
|||
|
|
|
|||
|
|
2.30
|
|||
|
|
|
|||
|
|
EXAMPLE 2.8
|
|||
|
|
Calculating Displacement: How Far does the Jogger Run?
|
|||
|
|
A jogger runs down a straight stretch of road with an average velocity of 4.00 m/s for 2.00 min. What is his final position, taking his initial position to be zero? Strategy Draw a sketch.
|
|||
|
|
|
|||
|
|
60
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
FIGURE 2.25
|
|||
|
|
The final position is given by the equation 2.31
|
|||
|
|
To find , we identify the values of , , and from the statement of the problem and substitute them into the equation.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the knowns.
|
|||
|
|
|
|||
|
|
,
|
|||
|
|
|
|||
|
|
, and
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Enter the known values into the equation.
|
|||
|
|
|
|||
|
|
2.32
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
Velocity and final displacement are both positive, which means they are in the same direction.
|
|||
|
|
|
|||
|
|
The equation
|
|||
|
|
|
|||
|
|
gives insight into the relationship between displacement, average velocity, and time. It
|
|||
|
|
|
|||
|
|
shows, for example, that displacement is a linear function of average velocity. (By linear function, we mean that
|
|||
|
|
|
|||
|
|
displacement depends on rather than on raised to some other power, such as . When graphed, linear
|
|||
|
|
|
|||
|
|
functions look like straight lines with a constant slope.) On a car trip, for example, we will get twice as far in a given
|
|||
|
|
|
|||
|
|
time if we average 90 km/h than if we average 45 km/h.
|
|||
|
|
|
|||
|
|
FIGURE 2.26 There is a linear relationship between displacement and average velocity. For a given time , an object moving twice as fast as another object will move twice as far as the other object.
|
|||
|
|
Solving for Final Velocity
|
|||
|
|
We can derive another useful equation by manipulating the definition of acceleration.
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.5 • Motion Equations for Constant Acceleration in One Dimension
|
|||
|
|
|
|||
|
|
61
|
|||
|
|
|
|||
|
|
Substituting the simplified notation for and gives us Solving for yields
|
|||
|
|
|
|||
|
|
2.33 2.34 2.35
|
|||
|
|
|
|||
|
|
EXAMPLE 2.9
|
|||
|
|
|
|||
|
|
Calculating Final Velocity: An Airplane Slowing Down after Landing
|
|||
|
|
An airplane lands with an initial velocity of 70.0 m/s and then decelerates at velocity?
|
|||
|
|
|
|||
|
|
for 40.0 s. What is its final
|
|||
|
|
|
|||
|
|
Strategy
|
|||
|
|
|
|||
|
|
Draw a sketch. We draw the acceleration vector in the direction opposite the velocity vector because the plane is decelerating.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
FIGURE 2.27
|
|||
|
|
|
|||
|
|
1. Identify the knowns.
|
|||
|
|
|
|||
|
|
,
|
|||
|
|
|
|||
|
|
,
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Identify the unknown. In this case, it is final velocity, .
|
|||
|
|
|
|||
|
|
3. Determine which equation to use. We can calculate the final velocity using the equation
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
4. Plug in the known values and solve.
|
|||
|
|
|
|||
|
|
2.36
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
The final velocity is much less than the initial velocity, as desired when slowing down, but still positive. With jet engines, reverse thrust could be maintained long enough to stop the plane and start moving it backward. That would be indicated by a negative final velocity, which is not the case here.
|
|||
|
|
|
|||
|
|
FIGURE 2.28 The airplane lands with an initial velocity of 70.0 m/s and slows to a final velocity of 10.0 m/s before heading for the terminal.
|
|||
|
|
|
|||
|
|
62
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
Note that the acceleration is negative because its direction is opposite to its velocity, which is positive.
|
|||
|
|
|
|||
|
|
In addition to being useful in problem solving, the equation
|
|||
|
|
|
|||
|
|
gives us insight into the relationships
|
|||
|
|
|
|||
|
|
among velocity, acceleration, and time. From it we can see, for example, that
|
|||
|
|
|
|||
|
|
• final velocity depends on how large the acceleration is and how long it lasts • if the acceleration is zero, then the final velocity equals the initial velocity
|
|||
|
|
constant) • if is negative, then the final velocity is less than the initial velocity
|
|||
|
|
|
|||
|
|
, as expected (i.e., velocity is
|
|||
|
|
|
|||
|
|
(All of these observations fit our intuition, and it is always useful to examine basic equations in light of our intuition and experiences to check that they do indeed describe nature accurately.)
|
|||
|
|
|
|||
|
|
Making Connections: Real-World Connection
|
|||
|
|
|
|||
|
|
FIGURE 2.29 The Space Shuttle Endeavor blasts off from the Kennedy Space Center in February 2010. (credit: Matthew Simantov, Flickr)
|
|||
|
|
An intercontinental ballistic missile (ICBM) has a larger average acceleration than the Space Shuttle and achieves a greater velocity in the first minute or two of flight (actual ICBM burn times are classified—short-burntime missiles are more difficult for an enemy to destroy). But the Space Shuttle obtains a greater final velocity, so that it can orbit the earth rather than come directly back down as an ICBM does. The Space Shuttle does this by accelerating for a longer time.
|
|||
|
|
|
|||
|
|
Solving for Final Position When Velocity is Not Constant (
|
|||
|
|
|
|||
|
|
)
|
|||
|
|
|
|||
|
|
We can combine the equations above to find a third equation that allows us to calculate the final position of an object experiencing constant acceleration. We start with
|
|||
|
|
|
|||
|
|
2.37
|
|||
|
|
|
|||
|
|
Adding to each side of this equation and dividing by 2 gives
|
|||
|
|
|
|||
|
|
2.38
|
|||
|
|
|
|||
|
|
Since
|
|||
|
|
|
|||
|
|
for constant acceleration, then
|
|||
|
|
|
|||
|
|
Now we substitute this expression for into the equation for displacement,
|
|||
|
|
|
|||
|
|
, yielding
|
|||
|
|
|
|||
|
|
2.39 2.40
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.5 • Motion Equations for Constant Acceleration in One Dimension
|
|||
|
|
|
|||
|
|
63
|
|||
|
|
|
|||
|
|
EXAMPLE 2.10
|
|||
|
|
|
|||
|
|
Calculating Displacement of an Accelerating Object: Dragsters
|
|||
|
|
|
|||
|
|
Dragsters can achieve average accelerations of rate for 5.56 s. How far does it travel in this time?
|
|||
|
|
|
|||
|
|
. Suppose such a dragster accelerates from rest at this
|
|||
|
|
|
|||
|
|
FIGURE 2.30 U.S. Army Top Fuel pilot Tony “The Sarge” Schumacher begins a race with a controlled burnout. (credit: Lt. Col. William Thurmond. Photo Courtesy of U.S. Army.)
|
|||
|
|
Strategy
|
|||
|
|
Draw a sketch.
|
|||
|
|
|
|||
|
|
FIGURE 2.31
|
|||
|
|
We are asked to find displacement, which is if we take to be zero. (Think about it like the starting line of a race. It can be anywhere, but we call it 0 and measure all other positions relative to it.) We can use the equation
|
|||
|
|
once we identify , , and from the statement of the problem.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the knowns. Starting from rest means that
|
|||
|
|
|
|||
|
|
, is given as
|
|||
|
|
|
|||
|
|
2. Plug the known values into the equation to solve for the unknown :
|
|||
|
|
|
|||
|
|
and is given as 5.56 s.
|
|||
|
|
|
|||
|
|
2.41
|
|||
|
|
|
|||
|
|
Since the initial position and velocity are both zero, this simplifies to
|
|||
|
|
|
|||
|
|
2.42
|
|||
|
|
|
|||
|
|
Substituting the identified values of and gives
|
|||
|
|
|
|||
|
|
2.43
|
|||
|
|
|
|||
|
|
yielding
|
|||
|
|
|
|||
|
|
2.44
|
|||
|
|
|
|||
|
|
64
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
If we convert 402 m to miles, we find that the distance covered is very close to one quarter of a mile, the standard distance for drag racing. So the answer is reasonable. This is an impressive displacement in only 5.56 s, but topnotch dragsters can do a quarter mile in even less time than this.
|
|||
|
|
|
|||
|
|
What else can we learn by examining the equation
|
|||
|
|
|
|||
|
|
We see that:
|
|||
|
|
|
|||
|
|
• displacement depends on the square of the elapsed time when acceleration is not zero. In Example 2.10, the dragster covers only one fourth of the total distance in the first half of the elapsed time
|
|||
|
|
|
|||
|
|
• if acceleration is zero, then the initial velocity equals average velocity (
|
|||
|
|
|
|||
|
|
) and
|
|||
|
|
|
|||
|
|
becomes
|
|||
|
|
|
|||
|
|
Solving for Final Velocity when Velocity Is Not Constant (
|
|||
|
|
|
|||
|
|
)
|
|||
|
|
|
|||
|
|
A fourth useful equation can be obtained from another algebraic manipulation of previous equations.
|
|||
|
|
|
|||
|
|
If we solve
|
|||
|
|
|
|||
|
|
for , we get
|
|||
|
|
|
|||
|
|
2.45
|
|||
|
|
|
|||
|
|
Substituting this and
|
|||
|
|
|
|||
|
|
into
|
|||
|
|
|
|||
|
|
, we get
|
|||
|
|
|
|||
|
|
2.46
|
|||
|
|
|
|||
|
|
EXAMPLE 2.11
|
|||
|
|
Calculating Final Velocity: Dragsters
|
|||
|
|
Calculate the final velocity of the dragster in Example 2.10 without using information about time. Strategy Draw a sketch.
|
|||
|
|
|
|||
|
|
FIGURE 2.32
|
|||
|
|
|
|||
|
|
The equation
|
|||
|
|
|
|||
|
|
is ideally suited to this task because it relates velocities, acceleration, and
|
|||
|
|
|
|||
|
|
displacement, and no time information is required.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the known values. We know that
|
|||
|
|
|
|||
|
|
, since the dragster starts from rest. Then we note that
|
|||
|
|
|
|||
|
|
(this was the answer in Example 2.10). Finally, the average acceleration was given to be
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Plug the knowns into the equation
|
|||
|
|
|
|||
|
|
and solve for
|
|||
|
|
|
|||
|
|
2.47
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.5 • Motion Equations for Constant Acceleration in One Dimension
|
|||
|
|
|
|||
|
|
65
|
|||
|
|
|
|||
|
|
Thus To get , we take the square root:
|
|||
|
|
|
|||
|
|
2.48
|
|||
|
|
|
|||
|
|
2.49
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
145 m/s is about 522 km/h or about 324 mi/h, but even this breakneck speed is short of the record for the quarter mile. Also, note that a square root has two values; we took the positive value to indicate a velocity in the same direction as the acceleration.
|
|||
|
|
|
|||
|
|
An examination of the equation among physical quantities:
|
|||
|
|
|
|||
|
|
can produce further insights into the general relationships
|
|||
|
|
|
|||
|
|
• The final velocity depends on how large the acceleration is and the distance over which it acts • For a fixed deceleration, a car that is going twice as fast doesn’t simply stop in twice the distance—it takes much
|
|||
|
|
further to stop. (This is why we have reduced speed zones near schools.)
|
|||
|
|
|
|||
|
|
Putting Equations Together
|
|||
|
|
|
|||
|
|
In the following examples, we further explore one-dimensional motion, but in situations requiring slightly more algebraic manipulation. The examples also give insight into problem-solving techniques. The box below provides easy reference to the equations needed.
|
|||
|
|
|
|||
|
|
Summary of Kinematic Equations (constant )
|
|||
|
|
2.50 2.51 2.52 2.53
|
|||
|
|
|
|||
|
|
2.54
|
|||
|
|
|
|||
|
|
EXAMPLE 2.12
|
|||
|
|
|
|||
|
|
Calculating Displacement: How Far Does a Car Go When Coming to a Halt?
|
|||
|
|
|
|||
|
|
On dry concrete, a car can decelerate at a rate of
|
|||
|
|
|
|||
|
|
, whereas on wet concrete it can decelerate at only
|
|||
|
|
|
|||
|
|
. Find the distances necessary to stop a car moving at 30.0 m/s (about 110 km/h) (a) on dry concrete and (b) on wet concrete. (c) Repeat both calculations, finding the displacement from the point where the driver sees a traffic light turn red, taking into account his reaction time of 0.500 s to get his foot on the brake.
|
|||
|
|
|
|||
|
|
Strategy
|
|||
|
|
|
|||
|
|
Draw a sketch.
|
|||
|
|
|
|||
|
|
66
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
FIGURE 2.33
|
|||
|
|
In order to determine which equations are best to use, we need to list all of the known values and identify exactly what we need to solve for. We shall do this explicitly in the next several examples, using tables to set them off.
|
|||
|
|
|
|||
|
|
Solution for (a)
|
|||
|
|
|
|||
|
|
1. Identify the knowns and what we want to solve for. We know that
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
( is
|
|||
|
|
|
|||
|
|
negative because it is in a direction opposite to velocity). We take to be 0. We are looking for displacement , or
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Identify the equation that will help up solve the problem. The best equation to use is
|
|||
|
|
|
|||
|
|
2.55
|
|||
|
|
|
|||
|
|
This equation is best because it includes only one unknown, . We know the values of all the other variables in this equation. (There are other equations that would allow us to solve for , but they require us to know the stopping time, , which we do not know. We could use them but it would entail additional calculations.)
|
|||
|
|
|
|||
|
|
3. Rearrange the equation to solve for .
|
|||
|
|
|
|||
|
|
2.56
|
|||
|
|
|
|||
|
|
4. Enter known values.
|
|||
|
|
|
|||
|
|
2.57
|
|||
|
|
|
|||
|
|
Thus,
|
|||
|
|
|
|||
|
|
2.58
|
|||
|
|
|
|||
|
|
Solution for (b)
|
|||
|
|
|
|||
|
|
This part can be solved in exactly the same manner as Part A. The only difference is that the deceleration is . The result is
|
|||
|
|
|
|||
|
|
2.59
|
|||
|
|
|
|||
|
|
Solution for (c)
|
|||
|
|
|
|||
|
|
Once the driver reacts, the stopping distance is the same as it is in Parts A and B for dry and wet concrete. So to answer this question, we need to calculate how far the car travels during the reaction time, and then add that to the stopping time. It is reasonable to assume that the velocity remains constant during the driver’s reaction time.
|
|||
|
|
|
|||
|
|
1. Identify the knowns and what we want to solve for. We know that
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
We take
|
|||
|
|
|
|||
|
|
to be 0. We are looking for
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Identify the best equation to use.
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.5 • Motion Equations for Constant Acceleration in One Dimension
|
|||
|
|
|
|||
|
|
67
|
|||
|
|
|
|||
|
|
works well because the only unknown value is , which is what we want to solve for.
|
|||
|
|
|
|||
|
|
3. Plug in the knowns to solve the equation.
|
|||
|
|
|
|||
|
|
2.60
|
|||
|
|
|
|||
|
|
This means the car travels 15.0 m while the driver reacts, making the total displacements in the two cases of dry and wet concrete 15.0 m greater than if he reacted instantly.
|
|||
|
|
|
|||
|
|
4. Add the displacement during the reaction time to the displacement when braking.
|
|||
|
|
|
|||
|
|
2.61
|
|||
|
|
|
|||
|
|
a. 64.3 m + 15.0 m = 79.3 m when dry b. 90.0 m + 15.0 m = 105 m when wet
|
|||
|
|
|
|||
|
|
FIGURE 2.34 The distance necessary to stop a car varies greatly, depending on road conditions and driver reaction time. Shown here are the braking distances for dry and wet pavement, as calculated in this example, for a car initially traveling at 30.0 m/s. Also shown are the total distances traveled from the point where the driver first sees a light turn red, assuming a 0.500 s reaction time.
|
|||
|
|
Discussion
|
|||
|
|
The displacements found in this example seem reasonable for stopping a fast-moving car. It should take longer to stop a car on wet rather than dry pavement. It is interesting that reaction time adds significantly to the displacements. But more important is the general approach to solving problems. We identify the knowns and the quantities to be determined and then find an appropriate equation. There is often more than one way to solve a problem. The various parts of this example can in fact be solved by other methods, but the solutions presented above are the shortest.
|
|||
|
|
|
|||
|
|
EXAMPLE 2.13
|
|||
|
|
|
|||
|
|
Calculating Time: A Car Merges into Traffic
|
|||
|
|
|
|||
|
|
Suppose a car merges into freeway traffic on a 200-m-long ramp. If its initial velocity is 10.0 m/s and it accelerates
|
|||
|
|
|
|||
|
|
at
|
|||
|
|
|
|||
|
|
, how long does it take to travel the 200 m up the ramp? (Such information might be useful to a traffic
|
|||
|
|
|
|||
|
|
engineer.)
|
|||
|
|
|
|||
|
|
Strategy
|
|||
|
|
|
|||
|
|
Draw a sketch.
|
|||
|
|
|
|||
|
|
68
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
FIGURE 2.35
|
|||
|
|
We are asked to solve for the time . As before, we identify the known quantities in order to choose a convenient physical relationship (that is, an equation with one unknown, ).
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the knowns and what we want to solve for. We know that
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
; and
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. We need to solve for . Choose the best equation. the equation is the variable for which we need to solve.
|
|||
|
|
|
|||
|
|
works best because the only unknown in
|
|||
|
|
|
|||
|
|
3. We will need to rearrange the equation to solve for . In this case, it will be easier to plug in the knowns first.
|
|||
|
|
|
|||
|
|
2.62
|
|||
|
|
|
|||
|
|
4. Simplify the equation. The units of meters (m) cancel because they are in each term. We can get the units of
|
|||
|
|
|
|||
|
|
seconds (s) to cancel by taking
|
|||
|
|
|
|||
|
|
, where is the magnitude of time and s is the unit. Doing so leaves
|
|||
|
|
|
|||
|
|
2.63
|
|||
|
|
|
|||
|
|
5. Use the quadratic formula to solve for .
|
|||
|
|
|
|||
|
|
(a) Rearrange the equation to get 0 on one side of the equation.
|
|||
|
|
|
|||
|
|
2.64
|
|||
|
|
|
|||
|
|
This is a quadratic equation of the form
|
|||
|
|
|
|||
|
|
2.65
|
|||
|
|
|
|||
|
|
where the constants are
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
(b) Its solutions are given by the quadratic formula:
|
|||
|
|
|
|||
|
|
2.66
|
|||
|
|
|
|||
|
|
This yields two solutions for , which are
|
|||
|
|
|
|||
|
|
2.67
|
|||
|
|
|
|||
|
|
In this case, then, the time is
|
|||
|
|
|
|||
|
|
in seconds, or
|
|||
|
|
|
|||
|
|
2.68
|
|||
|
|
|
|||
|
|
A negative value for time is unreasonable, since it would mean that the event happened 20 s before the motion began. We can discard that solution. Thus,
|
|||
|
|
|
|||
|
|
2.69
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
Whenever an equation contains an unknown squared, there will be two solutions. In some problems both solutions
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.6 • Problem-Solving Basics for One-Dimensional Kinematics
|
|||
|
|
|
|||
|
|
69
|
|||
|
|
|
|||
|
|
are meaningful, but in others, such as the above, only one solution is reasonable. The 10.0 s answer seems reasonable for a typical freeway on-ramp.
|
|||
|
|
|
|||
|
|
With the basics of kinematics established, we can go on to many other interesting examples and applications. In the process of developing kinematics, we have also glimpsed a general approach to problem solving that produces both correct answers and insights into physical relationships. Problem-Solving Basics discusses problem-solving basics and outlines an approach that will help you succeed in this invaluable task.
|
|||
|
|
|
|||
|
|
Making Connections: Take-Home Experiment—Breaking News
|
|||
|
|
|
|||
|
|
We have been using SI units of meters per second squared to describe some examples of acceleration or
|
|||
|
|
|
|||
|
|
deceleration of cars, runners, and trains. To achieve a better feel for these numbers, one can measure the
|
|||
|
|
|
|||
|
|
braking deceleration of a car doing a slow (and safe) stop. Recall that, for average acceleration,
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
While traveling in a car, slowly apply the brakes as you come up to a stop sign. Have a passenger note the initial
|
|||
|
|
|
|||
|
|
speed in miles per hour and the time taken (in seconds) to stop. From this, calculate the deceleration in miles
|
|||
|
|
|
|||
|
|
per hour per second. Convert this to meters per second squared and compare with other decelerations
|
|||
|
|
|
|||
|
|
mentioned in this chapter. Calculate the distance traveled in braking.
|
|||
|
|
|
|||
|
|
CHECK YOUR UNDERSTANDING
|
|||
|
|
|
|||
|
|
A rocket accelerates at a rate of
|
|||
|
|
|
|||
|
|
during launch. How long does it take the rocket to reach a velocity of 400
|
|||
|
|
|
|||
|
|
m/s?
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
To answer this, choose an equation that allows you to solve for time , given only , , and .
|
|||
|
|
|
|||
|
|
2.70
|
|||
|
|
|
|||
|
|
Rearrange to solve for .
|
|||
|
|
|
|||
|
|
2.71
|
|||
|
|
|
|||
|
|
2.6 Problem-Solving Basics for One-Dimensional Kinematics
|
|||
|
|
LEARNING OBJECTIVES By the end of this section, you will be able to:
|
|||
|
|
• Apply problem-solving steps and strategies to solve problems of one-dimensional kinematics. • Apply strategies to determine whether or not the result of a problem is reasonable, and if not, determine
|
|||
|
|
the cause.
|
|||
|
|
|
|||
|
|
FIGURE 2.36 Problem-solving skills are essential to your success in Physics. (credit: scui3asteveo, Flickr)
|
|||
|
|
Problem-solving skills are obviously essential to success in a quantitative course in physics. More importantly, the ability to apply broad physical principles, usually represented by equations, to specific situations is a very powerful
|
|||
|
|
|
|||
|
|
70
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations, whereas a list of facts cannot be made long enough to contain every possible circumstance. Such analytical skills are useful both for solving problems in this text and for applying physics in everyday and professional life.
|
|||
|
|
Problem-Solving Steps
|
|||
|
|
While there is no simple step-by-step method that works for every problem, the following general procedures facilitate problem solving and make it more meaningful. A certain amount of creativity and insight is required as well.
|
|||
|
|
Step 1 Examine the situation to determine which physical principles are involved. It often helps to draw a simple sketch at the outset. You will also need to decide which direction is positive and note that on your sketch. Once you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless.
|
|||
|
|
Step 2 Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Many problems are stated very succinctly and require some inspection to determine what is known. A sketch can also be very useful at this point. Formally identifying the knowns is of particular importance in applying physics to real-world situations. Remember, “stopped” means velocity is zero, and we often can take initial time and position as zero.
|
|||
|
|
Step 3 Identify exactly what needs to be determined in the problem (identify the unknowns). In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help.
|
|||
|
|
Step 4 Find an equation or set of equations that can help you solve the problem. Your list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all of the other variables are known, so you can easily solve for the unknown. If the equation contains more than one unknown, then an additional equation is needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations. You may have to use two (or more) different equations to get the final answer.
|
|||
|
|
Step 5 Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units. This step produces the numerical answer; it also provides a check on units that can help you find errors. If the units of the answer are incorrect, then an error has been made. However, be warned that correct units do not guarantee that the numerical part of the answer is also correct.
|
|||
|
|
Step 6 Check the answer to see if it is reasonable: Does it make sense? This final step is extremely important—the goal of physics is to accurately describe nature. To see if the answer is reasonable, check both its magnitude and its sign, in addition to its units. Your judgment will improve as you solve more and more physics problems, and it will become possible for you to make finer and finer judgments regarding whether nature is adequately described by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to mechanically solve a problem.
|
|||
|
|
When solving problems, we often perform these steps in different order, and we also tend to do several steps simultaneously. There is no rigid procedure that will work every time. Creativity and insight grow with experience, and the basics of problem solving become almost automatic. One way to get practice is to work out the text’s examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and progressing to the more difficult. Once you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.7 • Falling Objects
|
|||
|
|
|
|||
|
|
71
|
|||
|
|
|
|||
|
|
in many of the applications in this text.
|
|||
|
|
Unreasonable Results
|
|||
|
|
Physics must describe nature accurately. Some problems have results that are unreasonable because one premise is unreasonable or because certain premises are inconsistent with one another. The physical principle applied correctly then produces an unreasonable result. For example, if a person starting a foot race accelerates at
|
|||
|
|
for 100 s, his final speed will be 40 m/s (about 150 km/h)—clearly unreasonable because the time of 100 s is an unreasonable premise. The physics is correct in a sense, but there is more to describing nature than just manipulating equations correctly. Checking the result of a problem to see if it is reasonable does more than help uncover errors in problem solving—it also builds intuition in judging whether nature is being accurately described.
|
|||
|
|
Use the following strategies to determine whether an answer is reasonable and, if it is not, to determine what is the cause.
|
|||
|
|
Step 1 Solve the problem using strategies as outlined and in the format followed in the worked examples in the text. In the example given in the preceding paragraph, you would identify the givens as the acceleration and time and use the equation below to find the unknown final velocity. That is,
|
|||
|
|
2.72
|
|||
|
|
Step 2 Check to see if the answer is reasonable. Is it too large or too small, or does it have the wrong sign, improper units, …? In this case, you may need to convert meters per second into a more familiar unit, such as miles per hour.
|
|||
|
|
|
|||
|
|
2.73
|
|||
|
|
|
|||
|
|
This velocity is about four times greater than a person can run—so it is too large.
|
|||
|
|
|
|||
|
|
Step 3
|
|||
|
|
|
|||
|
|
If the answer is unreasonable, look for what specifically could cause the identified difficulty. In the example of the
|
|||
|
|
|
|||
|
|
runner, there are only two assumptions that are suspect. The acceleration could be too great or the time too long.
|
|||
|
|
|
|||
|
|
First look at the acceleration and think about what the number means. If someone accelerates at
|
|||
|
|
|
|||
|
|
, their
|
|||
|
|
|
|||
|
|
velocity is increasing by 0.4 m/s each second. Does this seem reasonable? If so, the time must be too long. It is not
|
|||
|
|
|
|||
|
|
possible for someone to accelerate at a constant rate of
|
|||
|
|
|
|||
|
|
for 100 s (almost two minutes).
|
|||
|
|
|
|||
|
|
2.7 Falling Objects
|
|||
|
|
|
|||
|
|
LEARNING OBJECTIVES By the end of this section, you will be able to:
|
|||
|
|
• Describe the effects of gravity on objects in motion. • Describe the motion of objects that are in free fall. • Calculate the position and velocity of objects in free fall.
|
|||
|
|
|
|||
|
|
Falling objects form an interesting class of motion problems. For example, we can estimate the depth of a vertical mine shaft by dropping a rock into it and listening for the rock to hit the bottom. By applying the kinematics developed so far to falling objects, we can examine some interesting situations and learn much about gravity in the process.
|
|||
|
|
|
|||
|
|
Gravity
|
|||
|
|
|
|||
|
|
The most remarkable and unexpected fact about falling objects is that, if air resistance and friction are negligible, then in a given location all objects fall toward the center of Earth with the same constant acceleration, independent of their mass. This experimentally determined fact is unexpected, because we are so accustomed to the effects of air resistance and friction that we expect light objects to fall slower than heavy ones.
|
|||
|
|
|
|||
|
|
72
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
FIGURE 2.37 A hammer and a feather will fall with the same constant acceleration if air resistance is considered negligible. This is a
|
|||
|
|
|
|||
|
|
general characteristic of gravity not unique to Earth, as astronaut David R. Scott demonstrated on the Moon in 1971, where the acceleration
|
|||
|
|
|
|||
|
|
due to gravity is only
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
In the real world, air resistance can cause a lighter object to fall slower than a heavier object of the same size. A tennis ball will reach the ground after a hard baseball dropped at the same time. (It might be difficult to observe the difference if the height is not large.) Air resistance opposes the motion of an object through the air, while friction between objects—such as between clothes and a laundry chute or between a stone and a pool into which it is dropped—also opposes motion between them. For the ideal situations of these first few chapters, an object falling without air resistance or friction is defined to be in free-fall.
|
|||
|
|
|
|||
|
|
The force of gravity causes objects to fall toward the center of Earth. The acceleration of free-falling objects is therefore called the acceleration due to gravity. The acceleration due to gravity is constant, which means we can apply the kinematics equations to any falling object where air resistance and friction are negligible. This opens a broad class of interesting situations to us. The acceleration due to gravity is so important that its magnitude is given its own symbol, . It is constant at any given location on Earth and has the average value
|
|||
|
|
|
|||
|
|
2.74
|
|||
|
|
|
|||
|
|
Although varies from
|
|||
|
|
|
|||
|
|
to
|
|||
|
|
|
|||
|
|
, depending on latitude, altitude, underlying geological formations,
|
|||
|
|
|
|||
|
|
and local topography, the average value of
|
|||
|
|
|
|||
|
|
will be used in this text unless otherwise specified. The
|
|||
|
|
|
|||
|
|
direction of the acceleration due to gravity is downward (towards the center of Earth). In fact, its direction defines
|
|||
|
|
|
|||
|
|
what we call vertical. Note that whether the acceleration in the kinematic equations has the value or
|
|||
|
|
|
|||
|
|
depends on how we define our coordinate system. If we define the upward direction as positive, then
|
|||
|
|
|
|||
|
|
, and if we define the downward direction as positive, then
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
One-Dimensional Motion Involving Gravity
|
|||
|
|
|
|||
|
|
The best way to see the basic features of motion involving gravity is to start with the simplest situations and then progress toward more complex ones. So we start by considering straight up and down motion with no air resistance or friction. These assumptions mean that the velocity (if there is any) is vertical. If the object is dropped, we know the initial velocity is zero. Once the object has left contact with whatever held or threw it, the object is in free-fall. Under these circumstances, the motion is one-dimensional and has constant acceleration of magnitude . We will also represent vertical displacement with the symbol and use for horizontal displacement.
|
|||
|
|
|
|||
|
|
Kinematic Equations for Objects in Free-Fall where Acceleration = -g
|
|||
|
|
2.75
|
|||
|
|
2.76
|
|||
|
|
|
|||
|
|
2.77
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.7 • Falling Objects
|
|||
|
|
|
|||
|
|
73
|
|||
|
|
|
|||
|
|
EXAMPLE 2.14
|
|||
|
|
Calculating Position and Velocity of a Falling Object: A Rock Thrown Upward
|
|||
|
|
A person standing on the edge of a high cliff throws a rock straight up with an initial velocity of 13.0 m/s. The rock misses the edge of the cliff as it falls back to earth. Calculate the position and velocity of the rock 1.00 s, 2.00 s, and 3.00 s after it is thrown, neglecting the effects of air resistance.
|
|||
|
|
Strategy
|
|||
|
|
Draw a sketch.
|
|||
|
|
|
|||
|
|
FIGURE 2.38
|
|||
|
|
We are asked to determine the position at various times. It is reasonable to take the initial position to be zero. This problem involves one-dimensional motion in the vertical direction. We use plus and minus signs to indicate direction, with up being positive and down negative. Since up is positive, and the rock is thrown upward, the initial velocity must be positive too. The acceleration due to gravity is downward, so is negative. It is crucial that the initial velocity and the acceleration due to gravity have opposite signs. Opposite signs indicate that the acceleration due to gravity opposes the initial motion and will slow and eventually reverse it.
|
|||
|
|
Since we are asked for values of position and velocity at three times, we will refer to these as and ; and ; and and .
|
|||
|
|
Solution for Position
|
|||
|
|
|
|||
|
|
1. Identify the knowns. We know that
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
2. Identify the best equation to use. We will use , here), which is the value we want to find.
|
|||
|
|
|
|||
|
|
3. Plug in the known values and solve for .
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
The rock is 8.10 m above its starting point at
|
|||
|
|
|
|||
|
|
s, since
|
|||
|
|
|
|||
|
|
to tell is to calculate and find out if it is positive or negative.
|
|||
|
|
|
|||
|
|
Solution for Velocity
|
|||
|
|
|
|||
|
|
; and
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
because it includes only one unknown, (or
|
|||
|
|
|
|||
|
|
2.78 . It could be moving up or down; the only way
|
|||
|
|
|
|||
|
|
1. Identify the knowns. We know that
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
from the solution above that
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Identify the best equation to use. The most straightforward is ).
|
|||
|
|
|
|||
|
|
3. Plug in the knowns and solve.
|
|||
|
|
|
|||
|
|
; and
|
|||
|
|
|
|||
|
|
. We also know
|
|||
|
|
|
|||
|
|
(from
|
|||
|
|
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|
, where
|
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|
2.79
|
|||
|
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|
|||
|
|
74
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
The positive value for means that the rock is still heading upward at original 13.0 m/s, as expected.
|
|||
|
|
|
|||
|
|
Solution for Remaining Times
|
|||
|
|
|
|||
|
|
The procedures for calculating the position and velocity at
|
|||
|
|
|
|||
|
|
and
|
|||
|
|
|
|||
|
|
results are summarized in Table 2.1 and illustrated in Figure 2.39.
|
|||
|
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|
|||
|
|
. However, it has slowed from its are the same as those above. The
|
|||
|
|
|
|||
|
|
Time, t Position, y Velocity, v Acceleration, a
|
|||
|
|
|
|||
|
|
TABLE 2.1 Results Graphing the data helps us understand it more clearly.
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.7 • Falling Objects
|
|||
|
|
|
|||
|
|
75
|
|||
|
|
|
|||
|
|
FIGURE 2.39 Vertical position, vertical velocity, and vertical acceleration vs. time for a rock thrown vertically up at the edge of a cliff. Notice that velocity changes linearly with time and that acceleration is constant. Misconception Alert! Notice that the position vs. time graph shows vertical position only. It is easy to get the impression that the graph shows some horizontal motion—the shape of the graph looks like the path of a projectile. But this is not the case; the horizontal axis is time, not space. The actual path of the rock in space is straight up, and straight down.
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
The interpretation of these results is important. At 1.00 s the rock is above its starting point and heading upward,
|
|||
|
|
|
|||
|
|
since and are both positive. At 2.00 s, the rock is still above its starting point, but the negative velocity means
|
|||
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|
|||
|
|
it is moving downward. At 3.00 s, both and are negative, meaning the rock is below its starting point and
|
|||
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|
|||
|
|
continuing to move downward. Notice that when the rock is at its highest point (at 1.5 s), its velocity is zero, but its
|
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|
|
|||
|
|
acceleration is still
|
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|
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|
|
. Its acceleration is
|
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|
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|
|
for the whole trip—while it is moving up and while it
|
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|
|||
|
|
is moving down. Note that the values for are the positions (or displacements) of the rock, not the total distances
|
|||
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|
|||
|
|
traveled. Finally, note that free-fall applies to upward motion as well as downward. Both have the same
|
|||
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|
|||
|
|
acceleration—the acceleration due to gravity, which remains constant the entire time. Astronauts training in the
|
|||
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|
|||
|
|
famous Vomit Comet, for example, experience free-fall while arcing up as well as down, as we will discuss in more
|
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|
|||
|
|
detail later.
|
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|
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|
76
|
|||
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|
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|
2 • Kinematics
|
|||
|
|
|
|||
|
|
Making Connections: Take-Home Experiment—Reaction Time
|
|||
|
|
A simple experiment can be done to determine your reaction time. Have a friend hold a ruler between your thumb and index finger, separated by about 1 cm. Note the mark on the ruler that is right between your fingers. Have your friend drop the ruler unexpectedly, and try to catch it between your two fingers. Note the new reading on the ruler. Assuming acceleration is that due to gravity, calculate your reaction time. How far would you travel in a car (moving at 30 m/s) if the time it took your foot to go from the gas pedal to the brake was twice this reaction time?
|
|||
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|
|
|||
|
|
EXAMPLE 2.15
|
|||
|
|
Calculating Velocity of a Falling Object: A Rock Thrown Down
|
|||
|
|
What happens if the person on the cliff throws the rock straight down, instead of straight up? To explore this question, calculate the velocity of the rock when it is 5.10 m below the starting point, and has been thrown downward with an initial speed of 13.0 m/s.
|
|||
|
|
Strategy
|
|||
|
|
Draw a sketch.
|
|||
|
|
|
|||
|
|
FIGURE 2.40
|
|||
|
|
Since up is positive, the final position of the rock will be negative because it finishes below the starting point at . Similarly, the initial velocity is downward and therefore negative, as is the acceleration due to gravity. We
|
|||
|
|
expect the final velocity to be negative since the rock will continue to move downward.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the knowns.
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Choose the kinematic equation that makes it easiest to solve the problem. The equation works well because the only unknown in it is . (We will plug in for .)
|
|||
|
|
|
|||
|
|
3. Enter the known values
|
|||
|
|
|
|||
|
|
2.80
|
|||
|
|
|
|||
|
|
where we have retained extra significant figures because this is an intermediate result.
|
|||
|
|
|
|||
|
|
Taking the square root, and noting that a square root can be positive or negative, gives The negative root is chosen to indicate that the rock is still heading down. Thus, Discussion
|
|||
|
|
|
|||
|
|
2.81 2.82
|
|||
|
|
|
|||
|
|
Note that this is exactly the same velocity the rock had at this position when it was thrown straight upward with the same initial speed. (See Example 2.14 and Figure 2.41(a).) This is not a coincidental result. Because we only consider the acceleration due to gravity in this problem, the speed of a falling object depends only on its initial
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.7 • Falling Objects
|
|||
|
|
|
|||
|
|
77
|
|||
|
|
|
|||
|
|
speed and its vertical position relative to the starting point. For example, if the velocity of the rock is calculated at a
|
|||
|
|
|
|||
|
|
height of 8.10 m above the starting point (using the method from Example 2.14) when the initial velocity is 13.0 m/s
|
|||
|
|
|
|||
|
|
straight up, a result of
|
|||
|
|
|
|||
|
|
is obtained. Here both signs are meaningful; the positive value occurs when the
|
|||
|
|
|
|||
|
|
rock is at 8.10 m and heading up, and the negative value occurs when the rock is at 8.10 m and heading back down.
|
|||
|
|
|
|||
|
|
It has the same speed but the opposite direction.
|
|||
|
|
|
|||
|
|
FIGURE 2.41 (a) A person throws a rock straight up, as explored in Example 2.14. The arrows are velocity vectors at 0, 1.00, 2.00, and 3.00 s. (b) A person throws a rock straight down from a cliff with the same initial speed as before, as in Example 2.15. Note that at the same distance below the point of release, the rock has the same velocity in both cases.
|
|||
|
|
|
|||
|
|
Another way to look at it is this: In Example 2.14, the rock is thrown up with an initial velocity of
|
|||
|
|
|
|||
|
|
. It rises
|
|||
|
|
|
|||
|
|
and then falls back down. When its position is
|
|||
|
|
|
|||
|
|
on its way back down, its velocity is
|
|||
|
|
|
|||
|
|
. That is, it has
|
|||
|
|
|
|||
|
|
the same speed on its way down as on its way up. We would then expect its velocity at a position of
|
|||
|
|
|
|||
|
|
to
|
|||
|
|
|
|||
|
|
be the same whether we have thrown it upwards at
|
|||
|
|
|
|||
|
|
or thrown it downwards at
|
|||
|
|
|
|||
|
|
. The velocity of
|
|||
|
|
|
|||
|
|
the rock on its way down from
|
|||
|
|
|
|||
|
|
is the same whether we have thrown it up or down to start with, as long as the
|
|||
|
|
|
|||
|
|
speed with which it was initially thrown is the same.
|
|||
|
|
|
|||
|
|
78
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
EXAMPLE 2.16
|
|||
|
|
Find g from Data on a Falling Object
|
|||
|
|
The acceleration due to gravity on Earth differs slightly from place to place, depending on topography (e.g., whether you are on a hill or in a valley) and subsurface geology (whether there is dense rock like iron ore as opposed to light rock like salt beneath you.) The precise acceleration due to gravity can be calculated from data taken in an introductory physics laboratory course. An object, usually a metal ball for which air resistance is negligible, is dropped and the time it takes to fall a known distance is measured. See, for example, Figure 2.42. Very precise results can be produced with this method if sufficient care is taken in measuring the distance fallen and the elapsed time.
|
|||
|
|
|
|||
|
|
FIGURE 2.42 Positions and velocities of a metal ball released from rest when air resistance is negligible. Velocity is seen to increase linearly with time while displacement increases with time squared. Acceleration is a constant and is equal to gravitational acceleration.
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.7 • Falling Objects
|
|||
|
|
|
|||
|
|
79
|
|||
|
|
|
|||
|
|
Suppose the ball falls 1.0000 m in 0.45173 s. Assuming the ball is not affected by air resistance, what is the precise acceleration due to gravity at this location?
|
|||
|
|
Strategy
|
|||
|
|
Draw a sketch.
|
|||
|
|
|
|||
|
|
FIGURE 2.43
|
|||
|
|
We need to solve for acceleration . Note that in this case, displacement is downward and therefore negative, as is acceleration.
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
1. Identify the knowns.
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
;
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
2. Choose the equation that allows you to solve for using the known values.
|
|||
|
|
|
|||
|
|
2.83
|
|||
|
|
|
|||
|
|
3. Substitute 0 for and rearrange the equation to solve for . Substituting 0 for yields
|
|||
|
|
|
|||
|
|
2.84
|
|||
|
|
|
|||
|
|
Solving for gives
|
|||
|
|
|
|||
|
|
2.85
|
|||
|
|
|
|||
|
|
4. Substitute known values yields
|
|||
|
|
|
|||
|
|
2.86
|
|||
|
|
|
|||
|
|
so, because
|
|||
|
|
|
|||
|
|
with the directions we have chosen,
|
|||
|
|
|
|||
|
|
2.87
|
|||
|
|
|
|||
|
|
Discussion
|
|||
|
|
|
|||
|
|
The negative value for indicates that the gravitational acceleration is downward, as expected. We expect the value
|
|||
|
|
|
|||
|
|
to be somewhere around the average value of
|
|||
|
|
|
|||
|
|
, so
|
|||
|
|
|
|||
|
|
makes sense. Since the data going into
|
|||
|
|
|
|||
|
|
the calculation are relatively precise, this value for is more precise than the average value of
|
|||
|
|
|
|||
|
|
; it
|
|||
|
|
|
|||
|
|
represents the local value for the acceleration due to gravity.
|
|||
|
|
|
|||
|
|
CHECK YOUR UNDERSTANDING
|
|||
|
|
|
|||
|
|
A chunk of ice breaks off a glacier and falls 30.0 meters before it hits the water. Assuming it falls freely (there is no
|
|||
|
|
|
|||
|
|
air resistance), how long does it take to hit the water?
|
|||
|
|
|
|||
|
|
Solution
|
|||
|
|
|
|||
|
|
We know that initial position
|
|||
|
|
|
|||
|
|
, final position
|
|||
|
|
|
|||
|
|
, and
|
|||
|
|
|
|||
|
|
. We can then use the
|
|||
|
|
|
|||
|
|
equation
|
|||
|
|
|
|||
|
|
to solve for . Inserting
|
|||
|
|
|
|||
|
|
, we obtain
|
|||
|
|
|
|||
|
|
80
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
2.88
|
|||
|
|
|
|||
|
|
where we take the positive value as the physically relevant answer. Thus, it takes about 2.5 seconds for the piece of ice to hit the water.
|
|||
|
|
|
|||
|
|
PHET EXPLORATIONS
|
|||
|
|
|
|||
|
|
Equation Grapher
|
|||
|
|
|
|||
|
|
Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves
|
|||
|
|
|
|||
|
|
for the individual terms (e.g.
|
|||
|
|
|
|||
|
|
) to see how they add to generate the polynomial curve.
|
|||
|
|
|
|||
|
|
Click to view content (https://openstax.org/books/college-physics-2e/pages/2-7-falling-objects)
|
|||
|
|
|
|||
|
|
2.8 Graphical Analysis of One-Dimensional Motion
|
|||
|
|
LEARNING OBJECTIVES By the end of this section, you will be able to:
|
|||
|
|
• Describe a straight-line graph in terms of its slope and y-intercept. • Determine average velocity or instantaneous velocity from a graph of position vs. time. • Determine average or instantaneous acceleration from a graph of velocity vs. time. • Derive a graph of velocity vs. time from a graph of position vs. time. • Derive a graph of acceleration vs. time from a graph of velocity vs. time.
|
|||
|
|
A graph, like a picture, is worth a thousand words. Graphs not only contain numerical information; they also reveal relationships between physical quantities. This section uses graphs of position, velocity, and acceleration versus time to illustrate one-dimensional kinematics.
|
|||
|
|
Slopes and General Relationships
|
|||
|
|
First note that graphs in this text have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against one another in such a graph, the horizontal axis is usually considered to be an independent variable and the vertical axis a dependent variable. If we call the horizontal axis the -axis and the vertical axis the -axis, as in Figure 2.44, a straight-line graph has the general form
|
|||
|
|
2.89
|
|||
|
|
Here is the slope, defined to be the rise divided by the run (as seen in the figure) of the straight line. The letter is used for the y-intercept, which is the point at which the line crosses the vertical axis.
|
|||
|
|
|
|||
|
|
FIGURE 2.44 A straight-line graph. The equation for a straight line is
|
|||
|
|
|
|||
|
|
.
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|
|||
|
|
2.8 • Graphical Analysis of One-Dimensional Motion
|
|||
|
|
|
|||
|
|
81
|
|||
|
|
|
|||
|
|
Graph of Position vs. Time (a = 0, so v is constant)
|
|||
|
|
Time is usually an independent variable that other quantities, such as position, depend upon. A graph of position versus time would, thus, have on the vertical axis and on the horizontal axis. Figure 2.45 is just such a straightline graph. It shows a graph of position versus time for a jet-powered car on a very flat dry lake bed in Nevada.
|
|||
|
|
|
|||
|
|
FIGURE 2.45 Graph of position versus time for a jet-powered car on the Bonneville Salt Flats.
|
|||
|
|
Using the relationship between dependent and independent variables, we see that the slope in the graph above is average velocity and the intercept is position at time zero—that is, . Substituting these symbols into gives
|
|||
|
|
2.90 or
|
|||
|
|
2.91 Thus a graph of position versus time gives a general relationship among displacement(change in position), velocity, and time, as well as giving detailed numerical information about a specific situation.
|
|||
|
|
The Slope of x vs. t
|
|||
|
|
The slope of the graph of position vs. time is velocity .
|
|||
|
|
2.92
|
|||
|
|
Notice that this equation is the same as that derived algebraically from other motion equations in Motion Equations for Constant Acceleration in One Dimension.
|
|||
|
|
From the figure we can see that the car has a position of 525 m at 0.50 s and 2000 m at 6.40 s. Its position at other times can be read from the graph; furthermore, information about its velocity and acceleration can also be obtained from the graph.
|
|||
|
|
EXAMPLE 2.17
|
|||
|
|
Determining Average Velocity from a Graph of Position versus Time: Jet Car
|
|||
|
|
Find the average velocity of the car whose position is graphed in Figure 2.45. Strategy The slope of a graph of vs. is average velocity, since slope equals rise over run. In this case, rise = change in position and run = change in time, so that
|
|||
|
|
|
|||
|
|
82
|
|||
|
|
|
|||
|
|
2 • Kinematics
|
|||
|
|
|
|||
|
|
2.93
|
|||
|
|
|
|||
|
|
Since the slope is constant here, any two points on the graph can be used to find the slope. (Generally speaking, it is most accurate to use two widely separated points on the straight line. This is because any error in reading data from the graph is proportionally smaller if the interval is larger.)
|
|||
|
|
Solution
|
|||
|
|
1. Choose two points on the line. In this case, we choose the points labeled on the graph: (6.4 s, 2000 m) and (0.50 s, 525 m). (Note, however, that you could choose any two points.)
|
|||
|
|
2. Substitute the and values of the chosen points into the equation. Remember in calculating change we always use final value minus initial value.
|
|||
|
|
|
|||
|
|
2.94
|
|||
|
|
|
|||
|
|
yielding Discussion
|
|||
|
|
|
|||
|
|
2.95
|
|||
|
|
|
|||
|
|
This is an impressively large land speed (900 km/h, or about 560 mi/h): much greater than the typical highway speed limit of 60 mi/h (27 m/s or 96 km/h), but considerably shy of the record of 343 m/s (1234 km/h or 766 mi/h) set in 1997.
|
|||
|
|
|
|||
|
|
Graphs of Motion when is constant but
|
|||
|
|
The graphs in Figure 2.46 below represent the motion of the jet-powered car as it accelerates toward its top speed, but only during the time when its acceleration is constant. Time starts at zero for this motion (as if measured with a stopwatch), and the position and velocity are initially 200 m and 15 m/s, respectively.
|
|||
|
|
|
|||
|
|
Access for free at openstax.org
|
|||
|
|
|