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Some Considerations on the Ballistics of a Gun of Seventy-Five-Mile Range Author(s): Arthur Gordon Webster Source: Proceedings of the American Philosophical Society , 1919, Vol. 58, No. 6 (1919), pp. 373-381 Published by: American Philosophical Society Stable URL: https://www.jstor.org/stable/984261
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SOME CONSIDERATIONS ON THE BALLISTICS OF A GUN OF SEVENTY-FIVE-MILE RANGE.1
By ARTHUR GORDON WEBSTER, Ph.D., Sc.D., LL.D.
(Read April 20, IQ18. Received May 2, 1919.)
On the afternoon of March 23, 1918, the civilized world was astounded by the news that the Germans were bombarding Paris. Inasmuch as it was known that the nearest point at which the German lines approached Paris was over seventy miles distant, curiosity was universal as to how this result was accomplished, and the most unlikely and absurd hypotheses were suggested.
The writer, among others, was asked whether or not the result was likely to be due to an aerial torpedo. The next day revealed the truth, which was, simply, that the Germans had really built a gun which carried a projectile this hitherto unheard-of distance.
It has since been determined that the gun is situated in the Forest of St. Gobain in the neighborhood of Laon, at a distance of about 120 kilometers or seventy miles.
At this distance the curvature of the earth causes one end of such a line to be about a half mile below the horizon at the other end, so that it is impossible to see the target from the gun or vice versa; there being no mountains of any such height in the whole region, visual aiming would be quite out of the question.
I wish, first, to call the attention not only to the remarkable ballistic achievement of the Germans in so far surpassing previous ranges but also to the unique opportunity possessed. It is obvious that precision of aim at such a distance is well-nigh impossible and that the only hope of effecting any damage lies in the possession of a very large and valuable target. Little has been allowed to come through the cable as to the damage done by these long-range shots, but enough has been learned in order to see their terrible potential-
1 Contribution from the Ballistic Institute, Clark University, No. 2. 373
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374 WEBSTER?BALLISTICS OF A GUN
ity. I shall remind you by a few lantern slid of monuments of civilization to such a degr
nowhere else in the world.
The gun was obviously aimed to strike the Cathedral of Notre Dame, cathedrals being the German specialty in this war. The church in which eighty people were killed on Good Friday is easily identified as the church of St. Gervais, which is across the street from the Hotel de Ville, the slides of which shown are taken from the top of that church. Other objects nearly in the line of the "axe" are the Louvre, the Sorbonne, the Pantheon, the Bon March? (a department store) ; I will not fatigue you with others.
Even if the so-called ellipse of dispersion should be very large it is easily seen that, if the shot should fall anywhere within a circle of perhaps twenty-five miles in diameter, the moral effect would be very great.
The longest previous range used during the war was about
twenty-two miles with the gun with which the Germans bombarded Dunkirk. In an article in Nature, March 28, 1918, which I have just seen this morning, my friend Sir George Greenhill, the author of the article on Ballistics in the Encyclopedia Brittanica, says :
"In the language of sport the German gunner has 'wiped the eye' of our artillery experts and defied all the timid preconceived notions of our oldfashioned traditions. The Jubilee long-range artillery experiments of 30 years ago were considered the ne plus ultra of our authorities, and we were stopped at that as they were declared of no military value. Today we have the arrears to make up of those years of delay, but the Germans watched our experiments with great interest, resumed them where we had left off and carried the idea forward until it has culminated today in his latest achievement of artillery of a gun to fire 75 miles and bombard Paris from the
frontier."
The Jubilee gun, referred to, fired a shell weighing 380 pounds at an elevation of 400 with a muzzle velocity of 2,400 feet per second, giving a range of 22,000 yards or ?2t/2 miles.2
2 It is a singular coincidence that as I left my laboratory to attend the meeting of the American Philosophical Society I remarked to my assistant, " Nothing is likely to ' queer ' this paper except the fact that Sir George Greenhill may have calculated the trajectory and published it in Nature." As I entered the room, the Secretary, Dr. Hays, called my attention to Sir George's paper and I was greatly relieved to find that it contained no figures of the trajectory.
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OF SEVENTY-FIVE-MILE RANGE. 375
Inasmuch as the only thing that has mad ballistic achievement is the taking advanta resistance due to the decreasing density of th it is first necessary to make some assumpt decrease with the height. I have here assumed is constant, giving the isothermal law of Lapl
(i) 8 = 80r^,
where 80= i.2932kg./mete being expressed in kilom about forty kilometers, or t tion of density of about si upper part of the traject parabola as investigated means true, as certain discussions that I have seen would seem to indicate, that there is a region about two miles high at which the effect of the atmosphere suddenly stops.
The second thing that must be known is the so-called ballistic coefficient of the projectile, involving its mass and diameter, the resistance of the air being supposed to be proportional to the square of the diameter while the acceleration is inversely proportional to
the mass.
For lack of sufficient information at the time that this paper was prepared, it was assumed that the mass of the projectile was 300 kilograms and its diameter twenty-two cenitmeters, or eight and one half inches. It is also necessary to know the form factor, which
depends upon the sharpness of the projectile. In the calculations made here the number .9, which is that of old-fashioned, rather blunt projectiles, is used. As, however, reports on the shell have
shown that it is furnished with a long pointed cap of sheet metal, the form factor should be considerably reduced. If, however, we take a mass of 180 kilograms, or 396.8 pounds, the results given here will be exact if we assume a form factor of .54, which is undoubtedly
much nearer the correct value. Finally, if we assume the mass to be 120 kilograms, this will give the same trajectory with a form factor of .36, which is smaller than that of any shot with which I am acquainted.
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376

WEBSTER?BALLISTICS OF A GUN

The method of calculation is as follows : The first a by successive arcs indicated by Gen. Siacci is used of I20 kilometers is divided into twelve equal parts, is drawn, making use of the velocity obtained, at th
preceding chord. We make use of the familiar eq
ballistics,
(2) ds = pde,

ds

(3)

dt1

(4) on = &lp = ? g COS 0,
, ? dVr
(5) -? = -F),

in the last of which, which is the only dynamical equation involved, we put

(6)

dt =

??? d(v cos T) c r/ N

g cos ? ' de

Instead of differentials we use finite differences. The table shows
the computation required for Curve 2, Fig. 4.

COMPUTATION FOR CURyE 2

Vr%

At \ Dx \??-*??\

PEG. | RAD.

KG./METl

SEC.

661.X

-51.8 315 1550 6.8

8.0 105.0 108.5

17 7 952 453.0

MASS ? 300 KG. INJTIAL VELOCfT?Y-013O0=05?2{?? 2At=?7i SEC. APPROX.

The meaning of the symbols is as follows :

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OF SEVENTY-FIVE-MILE RANGE. 377
The number of the chord is denoted by r; the projectile is denoted by v; and its jr-componen clination of the tangent at the beginning of the a away from the tangent, due to gravity, which is o of smallness with respect to the length of the arc, The mean ordinate of a given element is indica sistance R is found from the graphical table (F in the horizontal component of the velocity is Av for the projectile to traverse one arc is At. Dx rep back" computed from the resistance, representin which the projectile falls short of the assumed h owing to the resistance of the air. The height re
of the arc is
y=(x ? Dx) tan0 ? Ay.
The most important thing in a ballistic calculation is the knowledge of the ballistic, or resistance, function f(v). We have here made use of the famous result of the greatest of recent ballisticians,
1 Air Resistance Curve of Slacci 1 /o'k?v?- to'-M 3 Air Resistance Curve of F. Krupp
General Siacci, in his papers in the Rivista di Artiglieria e Genio (1896), from which it appears from the results of thousands of shots made by Bashforth in England, Mayevski in Russia (shots
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378

WEBSTER?BALLISTICS OF A GUN

made at Krupp's), Hojel in Holland, and Krupp in for velocities of over 300 meters per second the law with great exactness by a straight line. Although e sults do not go above velocities of 1,200 meters/sec hesitation in extrapolating for such values as ar Fig. 1 are shown, in Curve 1, the values given by S 2, the values of the function K(v)=f(v)/v2, and values of K(v) as given by Krupp. It is only fair results of the French Commission de G?vre more n Curve 3 than Curve 1.
The use of the linear law (originally suggested by been recommended by the Comte de Sparre in a pa
1901.

Velocity
Fig. 2.
In order to expedite matters, a graphical chart was prepare (Fig. 2) with abscissas denoting the velocity in meters per second containing straight lines and inclinations proportional to the densi ties of the air at different heights. On the right is a scale giving the height above the earth and by following the line correspondin
the resistance R is read off on the scale at the left.
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OF SEVENTY-FIVE-MILE RANGE.

379

The equations used are as follows :

(7)

(8) Avx=^vf(y)Ad,

Ax

gAx

(9)

(io)

?T = - -

Vxr

Avx

(ii)

= Vr+U

COS ?t+?

AvAt
(12) D* =

The calculation is made as follows: An arbitrary value of ? is selected, the drop Ay is calculated from (7), the change in angle from (10), the "set-back" from (12), and, finally, the corrected value of y corresponding to the given x-Dx is obtained. One row of values is obtained for each element of the trajectory.

Fig. 3 shows the details of the graphical method. After the chords are drawn by means of a flexible ruler, the trajectory is drawn through the vertices of the polygon constructed.
In Fig. 4 are shown some of the most striking results. The heavy black curve shows the surface of the earth and the change of
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380

WEBSTER?BALLISTICS OF A GUN

rarige, amounting to about a mile, that it would cause the conversions of the verticals).
The method of choosing was frankly one of guessin an angle of departure of 45o was assumed with an init of 1,300 meters/sec. This gave Curve 1. The angle o was then increased to 52o, which gave Curve 2, with a
kilometers. In order to show the enormous effect of the resistance

i &=4Se ff-C<f(yl-fW

3 T?*45* <ty)*c * e.?45* flrO

^Vrj3oo usi see

3 Sur/ace of Ear i h

?

100 Kilometers

Fig. 4.

of the air, No. 4 is drawn showing the parabolic or vacuum trajectory, with a range of 1744, and, finally, No. 3, on the assumption that the air has the constant density found at the surface of the earth. Further, the trajectory that we should have in case the density had the constant value, taken at a height two thirds of the maximum, as suggested by Colonel Ingalls for shots nearer the surface of the earth, was calculated. The range obtained was approximately 56 kilometers, quite different from the correct value of
Curve 1.
Note, May 2, 1919.?The foregoing paper, which was read over a year ago, has, of course, lost the timeliness that it had at that moment. In fact, I have been advised by a high ballistic authority not to publish it, as such calculations are now "a matter of routine."

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OF SEVENTY-FIVE-MILE RANGE. 381
To this I have replied, that, although they may not when I read the paper, and even now are n In fact, when I inquired of a high naval authorit take to calculate such a trajectory, he replied,
I said that we did it with two men in an hour.
Since then far better methods have been developed, but as I have seen only one publication of a trajectory, viz., by Major J. MaitlandAddison, R. A. (Journal of the Royal Artillery, Vol. XLV., No. 4), which confirms my results, I publish the paper as read, in the interest of the history of the matter, regretting that more pressing matters have so delayed the publication.
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