# Ballistics

**BALLISTICS** (from the Gr. *"ballein"*, to throw), the science of throwing warlike missiles or projectiles. It is now divided into two parts: -
*Exterior Ballistics*, in which the motion of the projectile is considered after it has received its initial impulse, when the projectile is moving freely under the
influence of gravity and the resistance of the air, and it is required to determine the circumstances so as to hit a certain object, with a view to its destruction or
perforation; and *Interior Ballistics*, in which the pressure of the powder-gas is analysed in the bore 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*, etc., 1865-1870; *Final Report*, etc., 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 *d*2, 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 =
*d*2*p*, 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 , then R = τ*d*2*p*, and τ is called the
*coefficient of tenuity*.

The factor 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 = 0.975; on the other hand, for a flat-headed projectile, as required at proof-butts, = 1.8, say 2 on the average.

For spherical shot 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.

In the long range high angle fire the shot ascends to such a height that the correction for the tenuity of the air becomes important, and the curvature - θ of an arc should be so chosen that φ*y*θ
the height ascended, should be limited to about 1000 ft., equivalent to a fall of 1 inch in the barometer or 3% diminution in the tenuity factor .

A convenient rule has been given by Captain James M. Ingalls, U.S.A., for approximating to a high angle trajectory in a single arc, which assumes that the mean density of the air may be taken as the density at two-thirds of the estimated height of the vertex; the rule is founded on the fact that in an unresisted parabolic trajectory the average height of the shot is two-thirds the height of the vertex, as illustrated in a jet of water, or in a stream of bullets from a Maxim gun.

The longest recorded range is that given in 1888 by the 9.2-in. gun to a shot weighing 380 lb fired with velocity 2375 f/s at elevation 40°; the range was about 12 m., with a time for flight of about 64 sec., shown in fig. 2.

A calculation of this trajectory is given by Lieutenant A. H. Wolley-Dod, R.A., in the *Proceedings R.A. Institution*, 1888, employing Siacci's method and about
twenty arcs; and Captain Ingalls, by assuming a mean tenuity-factor =0.68, corresponding to a height of about 2 m., on the estimate that the shot would reach a height of
3 m., was able to obtain a very accurate result, working in two arcs over the whole trajectory, up to the vertex and down again (Ingalls, *Handbook of Ballistic
Problems*).

Siacci's altitude-function is useful in direct fire, for giving immediately the angle of elevation required for a given range of R yds. or X ft., between limits V and
*v* of the velocity, and also the angle of descent β.

In direct fire the pseudo-velocities U and *u*, and the real velocities V and *v*, are undistinguishable, and sec η may be replaced by
unity so that, putting y = 0 in (79),

(88) tan = C [ I(V) - A ] .

S

Also

(89) tan - tan = C [I(V) - L(*v*)]

so that

(90) tan = C [ A - I(*v*) ] ,

S

or, as (88) and (90) may be written for small angles,

(91) sin 2 = 2C [ I(V) - A ] ,

S

(92) sin 2 = 2C [ A - I(*v*) ] .

S

To simplify the work, so as to look out the value of sin 2φ without the intermediate calculation of the remaining velocity *v*, a
double-entry table has been devised by Captain Braccialini Scipione (*Problemi del Tiro*, Roma, 1883), and adapted to yd., ft., in. and lb units by A. G. Hadcock,
late R.A., and published in the *Proc. R.A. Institution*, 1898, and in *Gunnery Tables*, 1898.

In this table

(93) sin 2 = C*a*,

where *a* is a function tabulated for the two arguments, V the initial velocity, and R/C the reduced range in yards.

The table is too long for insertion here. The results for φ and , as calculated for the range tables above, are also given there for comparison.

*Drift*. - An elongated shot fired from a rifled gun does not move in a vertical plane, but as if the mean plane of the trajectory was inclined to the true
vertical at a small angle, 2° or 3°; so that the shot will hit the mark aimed at if the back sight is tilted to the vertical at this angle , called the permanent angle of
deflection (see *Sights*).

This effect is called *drift* and the reason of it is not yet understood very clearly.

It is evidently a gyroscopic effect, being reversed in direction by a change from a right to a left-handed twist of rifling, and being increased by an increase of rotation of the shot.

The axis of an elongated shot would move parallel to itself only if fired in a vacuum; but in air the couple due to a sidelong motion tends to place the axis at right
angles to the tangent of the trajectory, and acting on a rotating body causes the axis to precess about the tangent. At the same time the frictional drag damps the
nutation and causes the axis of the shot to follow the tangent of the trajectory very closely, the point of the shot being seen to be slightly above and to the right of
the tangent, with a right-handed twist. The effect is as if there was a mean sidelong thrust *w* tan on the shot from left to right in order to deflect the plane of
the trajectory at angle to the vertical. But no formula has yet been invented, derived on theoretical principles from the physical data, which will assign by calculation
a definite magnitude to .

An effect similar to drift is observable at tennis, golf, base-ball and cricket; but this effect is explainable by the inequality of pressure due to a vortex of air carried along by the rotating ball, and the deviation is in the opposite direction of the drift observed in artillery practice, so artillerists are still awaiting theory and crucial experiment.

After all care has been taken in laying and pointing, in accordance with the rules of theory and practice, absolute certainty of hitting the same spot every time is unattainable, as causes of error exist which cannot be eliminated, such as variations in the air and in the muzzle-velocity, and also in the steadiness of the shot in flight.

To obtain an estimate of the accuracy of a gun, as much actual practice as is available must be utilized for the calculation in accordance with the laws of probability
of the 50% zones shown in the range table (see *Probability*.)

II. *Interior Ballistics*

The investigation of the relations connecting the pressure, volume and temperature of the powder-gas inside the bore of the gun, of the work realized by the expansion of the powder, of the dynamics of the movement of the shot up the bore, and of the stress set up in the material of the gun, constitutes the branch of interior ballistics.

A gun may be considered a simple thermo-dynamic machine or heat-engine which does its work in a single stroke, and does not act in a series of periodic cycles as an ordinary steam or gas-engine.

An indicator diagram can be drawn for a gun (fig. 3) as for a steam-engine, representing graphically by a curve CPD the relation between the volume
and pressure of the powder-gas; and in addition the curves AQE of energy *e*, A*v*V of velocity *v*, and A*t*T of time *t* can be plotted or
derived, the velocity and energy at the muzzle B being denoted by V and E.

After a certain discount for friction and the recoil of the gun, the net work realized by the powder-gas as the shot advances AM is represented by the area ACPM, and this is equated to the kinetic energy e of the shot, in foot-tons,

(1) *e* = *w* ( 1 + 4*k*2 tan2 ) *v*2 ,

2240
*d*2 2*g*

in which the factor 4(*k*2/*d*2)tan2δ represents the fraction due to the rotation of the shot, of diameter *d* and axial radius of gyration *k*, and represents the angle of the
rifling; this factor may be ignored in the subsequent calculations as small, less than 1%.

The mean effective pressure (M.E.P.) in tons per sq. in. is represented in fig. 3 by the height AH, such that the rectangle AHKB is equal to the area APDB; and the
M.E.P. multiplied by π*d*2, the cross-section of the bore in square inches, gives in tons the mean effective
thrust of the powder on the base of the shot; and multiplied again by *l*, the length in inches of the travel AB of the shot up the bore, gives the work realized in
inch-tons; which work is thus equal to the M.E.P. multiplied by *d*2*l* = B - C, the volume in cubic inches of the rifled part
AB of the bore, the difference between B the total volume of the bore and C the volume of the powder-chamber.

Equating the muzzle-energy and the work in foot-tons

(2) E = *w* V2 = B - C M.E.P.

2240 2*g* 12

(3) M.E.P. = *w* V2 12 .

2240 2*g* B - C

Working this out for the 6-in. gun of the range table, taking L = 216 in., we find B - C = 6100 cub. in., and the M.E.P. is about 6.4 tons per sq. in.

But the maximum pressure may exceed the mean in the ratio of 2 or 3 to 1, as shown in fig. 4, representing graphically the result of Sir Andrew Noble's experiments
with a 6-in. gun, capable of being lengthened to 100 calibres or 50 ft. (*Proc. R.S.*, June 1894).

On the assumption of uniform pressure up the bore, practically realizable in a Zalinski pneumatic dynamite gun, the pressure-curve would be the straight line HK of
fig. 3 parallel to AM; the energy-curve AQE would be another straight line through A; the velocity-curve A*v*V, of which the ordinate *v* is as the square root
of the energy, would be a parabola; and the acceleration of the shot being constant, the time-curve A*t*T will also be a similar parabola.

If the pressure falls off uniformly, so that the pressure-curve is a straight line PDF sloping downwards and cutting AM in F, then the energy-curve will be a parabola curving downwards, and the velocity-curve can be represented by an ellipse, or circle with centre F and radius FA; while the time-curve will be a sinusoid.

But if the pressure-curve is a straight line F'CP sloping upwards, cutting AM behind A in F', the energy-curve will be a parabola curving upwards, and the velocity-curve a hyperbola with center at F'.

These theorems may prove useful in preliminary calculations where the pressure-curve is nearly straight; but, in the absence of any observable law, the area of the pressure-curve must be read off by a planimeter, or calculated by Simpson's rule, as an indicator diagram.

To measure the pressure experimentally in the bore of a gun, the crusher-gauge is used as shown in fig. 6, nearly full size; it records the maximum pressure by the compression of a copper cylinder in its interior; it may be placed in the powder-chamber, or fastened in the base of the shot.

In Sir Andrew Noble's researches a number of plugs were inserted in the side of the experimental gun, reaching to the bore and carrying crusher-gauges, and also chronographic appliances which registered the passage of the shot in the same manner as the electric screens in Bashforth's experiments; thence the velocity and energy of the shot was inferred, to serve as an independent control of the crusher-gauge records (figs. 4 and 5).

As a preliminary step to the determination of the pressure in the bore of a gun, it is desirable to measure the pressure obtained by exploding a charge of powder in a closed vessel, varying the weight of the charge and thereby the density of the powder-gas.

The earliest experiments of this nature are due to Benjamin Robins in 1743 and Count Rumford in 1792; and their method has been revived by Dr Kellner, War Department
chemist, who employed the steel spheres of bicycle ball-bearings as safety-valves, loaded to register the pressure at which the powder-gas will blow off, and thereby
check the indications of the crusher-gauge (*Proc. R.S.*, March 1895).

Chevalier d'Arcy, 1760. also experimented on the pressure of powder and the velocity of the bullet in a musket barrel; this he accomplished by shortening the barrel successively, and measuring the velocity obtained by the ballistic pendulum; thus reversing Noble's procedure of gradually lengthening the gun.

But the most modern results employed with gunpowder are based on the experiments of Noble and Abel (*Phil. Trans.*, 1875-1880-1892-1894 and following years).

A charge of powder, or other explosive, of varying weight P lb, is fired in an explosion-chamber (fig. 7, scale about 1/5) of which the volume C, cub. in., is known
accurately, and the pressure *p*, tons per sq. in., was recorded by a crusher-gauge (fig. 6).

The result is plotted in figs. 8 and 9, in a curve showing the relation between *p* and D the *gravimetric density*, which is the specific gravity of the P
lb of powder when filling the volume C, cub. in., in a state of gas; or between *p* and *v*, the reciprocal of D, which may be called the *gravimetric
volume* (G.V.), being the ratio of the volume of the gas to the volume of an equal weight of water.

The term *gravimetric density* (G.D.) is peculiar to artillerists; it is required to distinguish between the specific gravity (S.G.) of the powder filling a given
volume in a state of gas, and the specific gravity of the separate solid grain or cord of powder.

Thus, for instance, a lump of solid lead of given S.G., when formed into a charge of lead shot composed of equal spherules closely packed, will have a G.D. such that

(4) G.D. of charge of lead shot = 1 √2 = 0.7403;

S.G. of lump of solid lead 6

while in the case of a bundle of cylindrical sticks of cordite,

(5) G.D. of charge of cordite = 1 √3 = 0.9067.

S.G. of stick of cordite 6

At the standard temperature of 62° F. the volume of the gallon of 10 lb of water is 277.3 cub. in.; or otherwise, 1 cub. ft. or 1728 cub. in. of water at this temperature weighs 62.35 lb, and therefore 1 lb of water bulks 1728 ÷ 62.35 = 27.73 cub. in.

Thus if a charge of P lb of powder is placed in a chamber of volume C cub. in., the

(6) G.D.= 27.73P/C, G.V. = C/27.73 P.

Sometimes the factor 27.68 is employed, corresponding to a density of water of about 62.4 lb per cub. ft., and a temperature 12° C., or 54° F.

With metric units, measuring P in kg., and C in litres, the G.D. = P/C, G.V. = C/P, no factor being required.

From the Table I., or by quadrature of the curve in fig. 9, the work E in foot-tons realized by the expansion of 1 lb of the powder from one gravimetric volume to
another is inferred; for if the average pressure is *p* tons per sq. in., while the gravimetric volume changes from *v* - *v* to *v* + Δ*v*, a change of volume of 27.73Δ*v* cub. in., the work done is 27.73*p**v* inch-tons, or

(7) E = 2.31 *p**v* foot-tons;

and the differences E being calculated from the observed values of *p*, a summation, as in the ballistic tables, would give E in a tabular form, and conversely
from a table of E in terms of *v*, we can infer the value of *p*.

On drawing off a little of the gas from the explosion vessel it was found that a gramme of cordite-gas at 0° C. and standard atmospheric pressure occupied 700 ccs., while the same gas compressed into 5 ccs. at the temperature of explosion had a pressure of 16 tons per sq. in., or 16 2240 / 14.7 = 2440 atmospheres, of 14.7 lb per sq. in.; one ton per sq. in. being in round numbers 150 atmospheres.

The absolute centigrade temperature T is thence inferred from the gas equation

(8) R = *pv* / *T* = *p*0*v*0/273,

which, with *p* = 2440, *v* = 5, *p*0 = 1, *v*0 = 700, makes T = 4758, a temperature
of 4485° C. or 8105° F.

In the heading of the 6-in. range table we find the description of the charge.

Charge: weight 13 lb 4 oz.; gravimetric density 55.01/0.504; nature, cordite, size 30.

So that P = 13.25, the G.D. = 0.504, the upper figure 55.01 denoting the specific volume of the charge measured in cubic inches per lb, filling the chamber in a state of gas, the product of the two numbers 55.01 and 0.504 being 27.73; and the chamber capacity C = 13.25 55.01 = 730 cub. in., equivalent to 25.8 in. or 2.15 ft. length of bore, now called the equivalent length of the chamber (E.L.C.).

If the shot was not free to move, the closed chamber pressure due to the explosion of the charge at this G.D. (= 0.5) would be nearly 49 tons per sq. in., much too great to be safe.

But the shot advances during the combustion of the cordite, and the chief problem in interior ballistics is to adjust the G.D. of the charge to the weight of the shot so that the advance of the shot during the combustion of the charge should prevent the maximum pressure from exceeding a safe limit, as shown by the maximum ordinate of the pressure curve CPD in fig. 3.

Suppose this limit is fixed at 16 tons per sq. in., corresponding in Table 1. to a G.D., 0.2; the powder-gas will now occupy a volume b = 3/2 C = 1825 cub. in., corresponding to an advance of the shot 3/2 2.15 = 3.225 ft.

Assuming an average pressure of 8 tons per sq. in., the shot will have acquired energy 8 *d*2 3.225 = 730 foot-tons, and a
velocity about *v* = 1020 f/s, so that the time over the 3.225 ft. at an average velocity 510 f/s is about 0.0063 sec.

Comparing this time with the experimental value of the time occupied by the cordite in burning, a start is made for a fresh estimate and a closer approximation.

Assuming, however, that the agreement is close enough for practical requirement, the combustion of the cordite may be considered complete at this stage P, and in the
subsequent expansion it is assumed that the gas obeys an adiabatic law in which the pressure varies inversely as some *m**th*
power of the volume.

The work done in expanding to infinity from *p* tons per sq. in. at volume *b* cub. in. is then *pb*/(*m* - 1) inch-tons, or to any volume B cub.
in. is

(9) *pb* [ 1 - ( *b* ) *m*-1

] .

*m* - 1 B

It is found experimentally that *m* = 1.2 is a good average value to take for cordite; so now supposing the combustion of the charge of the 6-in. is complete in
0.0063 sec., when *p* = 16 tons per sq. in., *b* = 1825 cub. in., and that the gas expands adiabatically up to the muzzle, where

(10) B = 216 + 25.8 = 3.75

*b* 2.5 25.8

we find the work realized by expansion is 2826 foot-tons, sufficient to increase the velocity from 1020 to 2250 f/s at the muzzle.

This muzzle velocity is about 5% greater than the 2150 f/s of the range table, so on these considerations we may suppose about 10% of work is lost by friction in the
bore: this is expressed by saying that the *factor of effect* is *f* = 0.9.

The experimental determination of the time of burning under the influence of the varying pressure and density, and the size of the grain, is thus of great practical importance, as thereby it is possible to estimate close limits to the maximum pressure that will be reached in the bore of a gun, and to design the chamber so that the G.D. of the charge may be suitable for the weight and acceleration of the shot. Empirical formulas based on practical experience are employed for an approximation to the result.

A great change has come over interior ballistics in recent years, as the old black gunpowder has been abandoned in artillery after holding the field for six hundred years. It is replaced by modern explosives such as those indicated on fig. 4, capable of giving off a very much larger volume of gas at a greater temperature and pressure, more than threefold as seen on fig. 8, so that the charge may be reduced in proportion, and possessing the military advantage of being nearly smokeless. (See Explosives.)

The explosive cordite is adopted in the British service; it derives the name from its appearance as cord in short lengths, the composition being squeezed in a viscous state through the hole in a die, and the cordite is designated in size by the number of hundredths of an inch in the diameter of the hole. Thus the cordite, size 30, of the range table has been squeezed through a hole 0.30 in. diameter.

The thermochemical properties of the constituents of an explosive will assign an upper limit to the volume, temperature and pressure of the gas produced by the
combustion; but much experiment is required in addition. Sir Andrew Noble has published some of his results in the *Phil. Trans.*, 1905-1906 and following years.

*Authorities*. - Tartaglia, *Nova Scientia* (1537); Galileo (1638); Robins, *New Principles of Gunnery* (1743); Euler (trans. by Hugh Brown), *The
True Principles of Gunnery* (1777); Didion, Hélie, Hugoniot, Vallier, Baills, etc., *Balistique* (French); Siacci, *Balistica* (Italian); Mayevski,
Zabudski, *Balistique* (Russian); La Llave, Ollero, Mata, etc., *Balistica* (Spanish); Bashforth, *The Motion of Projectiles* (1872); *The Bashforth
Chronograph* (1890); Ingalls, *Exterior and Interior Ballistics, Handbook of Problems in Direct and Indirect Fire*; Bruff, *Ordnance and Gunnery*; Cranz,
*Compendium der Ballistik* (1898); *The Official Text-Book of Gunnery* (1902); Charbonnier, *Balistique* (1905); Lissak, *Ordnance and Gunnery*
(1907).

(A. G. G.)

*Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)*