# Page:EB1911 - Volume 03.djvu/288

271
BALLISTICS

of the gun, and the investigation is carried out of the requisite charge of powder to secure the initial velocity of the projectile, without straining the gun unduly. The calculation of the stress in the various parts of the gun due to the powder pressure is dealt with in the article Ordnance.

I. Exterior Ballistics.

In the ancient theory due to Galileo, the resistance of the air is ignored, and, as shown in the article on Mechanics (§ 13), the trajectory is now a parabola. But this theory is very far from being of practical value for most purposes of gunnery; so that a first requirement is an accurate experimental knowledge of the resistance of the air to the projectiles employed, at all velocities useful in artillery. The theoretical assumptions of Newton and Euler (hypotheses magis mathematicae quam naturales) of a resistance varying as some simple power of the velocity, for instance, as the square or cube of the velocity (the quadratic or cubic law), lead to results of great analytical complexity, and are useful only for provisional extrapolation at high or low velocity, pending further experiment.

The foundation of our knowledge of the resistance of the air, as employed in the construction of ballistic tables, is the series of experiments carried out between 1864 and 1880 by the Rev. F. Bashforth, B.D. (Report on the Experiments made with the Bashforth Chronograph, &c., 1865-1870; Final Report, &c., 1878-1880; The Bashforth Chronograph, Cambridge, 1890). According to these experiments, the resistance of the air can be represented by no simple algebraical law over a large range of velocity. Abandoning therefore all a priori theoretical assumption, Bashforth set to work to measure experimentally the velocity of shot and the resistance of the air by means of equidistant electric screens furnished with vertical threads or wire, and by a chronograph which measured the instants of time at which the screens were cut by a shot flying nearly horizontally. Formulae of the calculus of finite differences enable us from the chronograph records to infer the velocity and retardation of the shot, and thence the resistance of the air.

As a first result of experiment it was found that the resistance of similar shot was proportional, at the same velocity, to the surface or cross section, or square of the diameter. The resistance R can thus be divided into two factors, one of which is d², where d denotes the diameter of the shot in inches, and the other factor is denoted by p, where p is the resistance in pounds at the same velocity to a similar 1-in, projectile; thus R =d²p, and the value of p, for velocity ranging from 1600 to 2150 ft. per second (f/s) is given in the second column of the extract from the abridged ballistic table below.

These values of p refer to a standard density of the air, of 534·22 grains per cubic foot, which is the density of dry air at sea-level in the latitude of Greenwich, at a temperature of 62° F. and a barometric height of 30 in.

But in consequence of the humidity of the climate of England it is better to suppose the air to be (on the average) two-thirds saturated with aqueous vapour, and then the standard temperature will be reduced to 60°F., so as to secure the same standard density; the density of the air being reduced perceptibly by the presence of the aqueous vapour.

It is further assumed, as the result of experiment, that the resistance is proportional to the density of the air; so that if the standard density changes from unity to any other relative density denoted by τ, then R = τd²p, and τ is called the coefficient of tenuity.

The factor τ becomes of importance in long range high angle fire, where the shot reaches the higher attenuated strata of the atmosphere; on the other hand, we must take τ about 800 in a calculation of shooting under water.

The resistance of the air is reduced considerably in modern projectiles by giving them a greater length and a sharper point, and by the omission of projecting studs, a factor κ, called the coefficient of shape, being introduced to allow for this change.

For a projectile in which the ogival head is struck with a radius of 2 diameters, Bashforth puts κ= 0·975; on the other hand, for a flat-headed projectile, as required at proof-butts, κ= 1·8, say 2 on the average.

For spherical shot κ is not constant, and a separate ballistic table must be constructed; but κ may be taken as 1·7 on the average.

Lastly, to allow for the superior centering of the shot obtainable with the breech-loading system, Bashforth introduces a factor σ, called the coefficient of steadiness.

This steadiness may vary during the flight of the projectile, as the shot may be unsteady for some distance after leaving the muzzle, afterwards steadying down, like a spinning-top. Again, σ may increase as the gun wears out, after firing a number of rounds.

Collecting all the coefficients, τ, κ, σ, into one, we put

(1) ${\displaystyle R=nd^{2}p=nd^{2}f(v),\,}$ where
(2) ${\displaystyle n=\kappa \sigma \tau ,\,}$

and n is called the coefficient of reduction.

By means of a well-chosen value of n, determined by a few experiments, it is possible, pending further experiment, with the most recent design, to Utilize Bashforth's experimental results carried out with old-fashioned projectiles fired from muzzle-loading guns. For instance, n= 0·8 or even less is considered a good average for the modern rifle bullet.

Starting with the experimental values of p, for a standard projectile, fired under standard conditions in air of standard density, we proceed to the construction of the ballistic table. We first determine the time t in seconds required for the velocity of a shot, d inches in diameter and weighing w lb to fall from any initial velocity V(f/s) to any final velocity v(f/s). The shot is supposed to move horizontally, and the curving effect of gravity is ignored.

If Δt seconds is the time during which the resistance of the air, R lb, causes the velocity of the shot to fall Δv(f/s), so that the velocity drops from v+½Δv to v-½Δv in passing through the mean velocity v, then

 (3) ${\displaystyle R\Delta t\,}$ ${\displaystyle =\,}$ ${\displaystyle {\mbox{loss of momentum in second-pounds,}}\,}$ ${\displaystyle =\,}$ ${\displaystyle w(v+{\frac {1}{2}}\Delta v)/g-w(v-{\frac {1}{2}}\Delta v)/g=w\Delta v/gp}$

so that with the value of R in (1),

(4) ${\displaystyle \Delta t=w\Delta v/nd^{2}pg.\,}$

We put

(5) ${\displaystyle w/nd^{2}=C,\,}$

and call C the ballistic coefficient (driving power) of the shot, so that

(6) ${\displaystyle \Delta =C\Delta T,{\mbox{ where}}\,}$
(7) ${\displaystyle \Delta T=\Delta v/gp,\,}$

and ΔT is the time in seconds for the velocity to drop Δv of the standard shot for which C=1, and for which the ballistic table is calculated.

Since p is determined experimentally and tabulated as a function of v, the velocity is taken as the argument of the ballistic table; and taking δv=10, the average value of p in the interval is used to determine ΔT.

Denoting the value of T at any velocity v by T (v), then

(8) ${\displaystyle T(v)\,}$=sum of all the preceding values of ΔT plus an arbitrary constant, expressed by the notation
(9) ${\displaystyle T(v)=\Sigma (\Delta v)/gp+{\mbox{ a constant, or}}\int dv/gp+{\mbox{ a constant}},}$ in which p is supposed known as a function of v.

The constant may be any arbitrary number, as in using the table the difference only is required of two tabular values for an initial velocity V and final velocity v; and thus

(10) ${\displaystyle T(V)-T(v)=\Sigma _{v}^{V}\Delta v/gp{\mbox{ or }}\int dv/gp;}$

and for a shot whose ballistic coefficient is C

(11) ${\displaystyle t=C\left\lbrack T(V)T(v)\right\rbrack .}$

To save the trouble of proportional parts the value of T(v) for unit increment of v is interpolated in a full-length extended ballistic table for T.

Next, if the shot advances a distance Δs ft. in the time Δt, during which the velocity falls from vv to vv we have

 (12) ${\displaystyle R\Delta s\,}$ ${\displaystyle =\,}$ ${\displaystyle {\mbox{loss of kinetic energy in foot-pounds,}}\,}$ ${\displaystyle =\,}$ ${\displaystyle w(v+{\frac {1}{2}}\Delta v)^{2}/g-w(v-{\frac {1}{2}}\Delta v)^{2}/g=wv\Delta v/g,{\mbox{ so that}}}$
(13) ${\displaystyle \Delta s=wv\Delta v/nd^{2}pg=C\Delta S,{\mbox{ where}}\,}$
(14) ${\displaystyle \Delta S=v\Delta v/gp=v\Delta T,\,}$

and ΔS is the advance in feet of a shot for which C=1, while the velocity falls Δv in passing through the average velocity v.

Denoting by S(v) the sum of all the values of ΔS up to any assigned velocity v,

(15) ${\displaystyle S(v)=\Sigma (\Delta S)+{\mbox{ a constant}}\,}$, by which S(v) is calculated from ΔS, and then between two assigned velocities V and v,
(16) ${\displaystyle S(V)-S(v)={\begin{matrix}\sum _{v}^{V}\Delta T\end{matrix}}=\sum {\frac {v\Delta v}{gp}}{\mbox{or}}\int _{v}^{V}{\frac {vdv}{gp}},}$

and if s feet is the advance of a shot whose ballistic coefficient is C,

(17) ${\displaystyle s=C\left\lbrack S(V)-S(v)\right\rbrack .\,}$

In an extended table of S, the value is interpolated for unit increment of velocity.

A third table, due to Sir W.D. Niven, F.R.S., called the degree table, determines the change of direction of motion of the shot while the velocity changes from V to v, the shot flying nearly horizontally.

To explain the theory of this table, suppose the tangent at the point of the trajectory, where the velocity is v, to make an angle i radians with the horizon.

Resolving normally in the trajectory, and supposing the resistance of the air to act tangentially,

(18) ${\displaystyle v(di/dt)=g\cos i\,}$, where di denotes the infinitesimal decrement of i in the infinitesimal increment of time dt.