BASIC BALLISTICS

© Anthony G Williams

Version 7 (with thanks to Bryden Ritchie):  January 2013

I am often asked questions about the basic principles governing ballistics and related issues, so this is an attempt to provide some understanding of the most popular topics without getting too technical. I hasten to say that it has to be a basic guide as I am neither a physicist nor a mathematician, and I dislike complicated formulae. There are computer programmes available for working out advanced problems but I hope this article will at least point people in the right direction.

The study of ballistics is usually divided into three: internal, external and terminal. Internal (or interior) ballistics concerns what happens between the cartridge being fired and the projectile leaving the muzzle (I will deal with recoil under this heading as well). External (or exterior) ballistics is concerned with the flight of the projectile from the muzzle to the target. Terminal ballistics describes what happens when the target is hit.

INTERNAL BALLISTICS

As soon as the primer ignites the propellant, gas is generated which rapidly builds up a considerable pressure. This pushes the projectile out of the case and up the barrel. The characteristics of propellant powders are such that the peak gas pressures are generated almost immediately, as the projectile begins its trip up the barrel. That is why the gun steel is thickest at this point. As the projectile accelerates up the barrel, it makes space for the gas to expand so gas pressure declines. It is still significant when the projectile leaves the muzzle, resulting in a rapid expansion into the open air causing the characteristic 'report' of a gun firing. This final expansion, coupled with the end of the friction between the projectile and the barrel, results in a final boost to the projectile so its maximum velocity is attained just beyond the muzzle (although "muzzle velocity" is usually measured at several metres past the muzzle anyway).

Different weapons operate at different gas pressures; pistols and shotguns generally work at much lower pressures than rifles and automatic cannon. Magnum pistol ammunition, plus some military pistol ammunition intended for sub-machine guns, is loaded to higher pressures than normal in order to generate higher velocities. It can be dangerous to use this ammunition in guns not designed for it, unless they are very strong. Rifle and cannon ammunition is generally loaded up to the highest practical pressure level, taking into account barrel wear, the risk of a case being stuck to the chamber and other potential problems. Chamber pressure is measured in various different ways: the Imperial/US system still uses "pounds per square inch" (psi), while metric systems may use "kilograms per square centimetre" (kg/cm2), bar (a measurement of atmospheric pressure) or the more scientific megapascals (MPa). The conversion factors are:

1 MPa = 10 bar = 10.2 kg/cm2 = 145 psi

Typical chamber pressures for shotguns are 75-90 MPa (11,000-13,000 psi); for pistol rounds 100-240 MPa (15,000-35,000 psi), although a few magnums are loaded to rifle pressures; and for rifle rounds 310-380 MPa (45,000-55,000 psi), with some extreme loadings reaching 450 MPa (65,000 psi). By the time a rifle bullet reaches the muzzle, the pressure acting on it has dropped to around 50-70 MPa (7,000-10,000 psi).

One cartridge issue affecting gun design is bolt thrust. This is the rearwards push on the gun's bolt or breechblock caused by the cartridge firing. This depends on two factors: the chamber pressure, and the inside diameter of the base of the cartridge case. Put simply, the wider the cartridge case, the greater the surface area the pressure has to work on, so the greater the bolt thrust for any given chamber pressure. So a long thin cartridge case may develop the same chamber pressure as a short fat one with double the internal base area, but the latter will develop double the bolt thrust. The greater the bolt thrust, the stronger the gun's locking mechanism has to be to withstand it.

Silenced (or, more accurately, suppressed) weapons are fitted with a suppressor; a container attached to the muzzle (or, in specially-designed weapons, integrated with the barrel). This allows the gas to expand within it in a controlled way, during which time it is cooled down and drops in pressure, to be released slowly afterwards instead of bursting violently out of the muzzle. This is why suppressors have to be bulky; the more propellant, the bigger they have to be. The muzzle report is by far the major source of noise from an unsuppressed weapon, but supersonic projectiles (with a muzzle velocity in excess of around 340 m/s - 1,115 fps - at sea level: it drops with altitude) also generate a supersonic "crack" audible as they go past. Probably about half of pistol ammunition is subsonic, half supersonic; all rifle ammunition is supersonic, except for .22 rimfire target loadings and some special-purpose ammunition intended for suppressed weapons. Suppressors work best with subsonic ammunition, but are also of use with sniper rifles firing supersonic ammunition; the enemy will still hear the "crack" of a bullet passing but will be unable to determine from where it was fired without some sophisticated detection equipment.

Note that a suppressor is different from a flash suppressor, which is a slotted or pronged device fitted to the muzzle to reduce the size of the muzzle flash. Older MGs sometimes had flash hiders (also known as flash eliminators) which were cone-shaped muzzle attachments primarily designed to shield the flash from the gunner, in order to preserve his night vision.

MUZZLE ENERGY

The cartridge develops a "muzzle energy", measured either in joules (metric) or foot-pounds (ft lbs). This is calculated as follows (please note that although the correct term is "mass", I have used "weight" instead for easier comprehension. Mass is a constant regardless of gravitational pull, whereas weight depends on the gravity. However, on the Earth's surface the two are effectively the same):

Joules: multiply the projectile weight in grams by the square of the muzzle velocity in metres per second (m/s), then divide the result by 2,000. So a 40g projectile fired at 800 m/s will generate (40 x 800 x 800)/2,000 = 12,800j

Foot-pounds: multiply the projectile weight in pounds by the square of the muzzle velocity in feet per second (fps), then divide the result by 64. Note that there are 7,000 grains in a pound, so for bullet calculations you can enter the weight in grains then divide the resulting calculation by 7,000.

To convert foot-pounds to joules, multiply by 1.348.

To convert joules to foot-pounds, multiply by 0.742.

15.432 grains = 1 gram, 2.205 pounds = 1 kg and 3.281 feet = 1 metre

Note that in developing muzzle energy, muzzle velocity is much more important than projectile weight. Doubling the muzzle velocity of a projectile quadruples its energy, whereas doubling the projectile weight only doubles its energy.

The muzzle energy which is generated by a given amount of propellant will depend on the calibre (spelled "caliber" in the USA) of the gun. Think of the barrel as the cylinder of an engine, and the bullet as a piston. In a small-calibre weapon, the gas has a very small piston area - the base of the bullet - on which to work. As the pressure it can generate is limited, it can only apply a limited amount of force to the bullet. In a larger calibre weapon, the piston area is greater so the same amount of propellant can do more work. This explains why, in the case of a rifle cartridge made in several different calibres (e.g. the .30-06, also made in .25, .27 and .35 calibres), there is usually a direct relationship between the calibre and the muzzle energy generated; the bigger the calibre, the higher the muzzle energy from a given quantity of propellant. This calculation is similar to that for bolt thrust (see above) except that it is projectile base area rather than case internal base area which is involved.

For a given calibre, there is a practical limit to the amount of propellant which can be used. The law of diminishing returns applies, and using bigger cartridge cases holding more propellant will achieve ever-smaller increases in velocity from the extra propellant. A cartridge which is so big as to be unable to use all its propellant efficiently is described as "over bore". Such cartridges have very unpleasant firing characteristics, with high levels of flash and blast, and usually wear out barrels quickly. They also need long barrels to give the necessarily slow-burning propellant time to generate a high velocity, which can be inconvenient.

Incidentally, in any given cartridge different projectile weights may produce different energy levels; typically, an "average" weight for the calibre produces the highest energy, with unusually light or heavy projectiles doing less well. This may in part reflect the characteristics of the propellant, although these are adjustable; heavier projectiles need slower-burning powders to keep the pressure peak down, whereas light projectiles need faster-burning powders to accelerate them quickly enough to reach a high velocity. Very heavy projectiles may protrude deeper into the case, reducing the space for propellant.

What is the maximum velocity which a projectile can be pushed to? This is ultimately limited by the expansion rate of the gas from the burning propellant. In rifles, the practical limit is around 1,200 m/s ( nearly 4,000 fps) achieved in small-calibre guns which only need light bullets (plus a couple of WW2 7.92mm anti-tank rifles). This is also about the maximum velocity for cannon firing conventional full-calibre HE shells. The highest velocities currently achieved are in tank guns firing APFSDS shot, which is extremely light for the calibre and allows velocities to be pushed up to 1,800 m/s (nearly 6,000 fps), which is close to the theoretical limit for conventional powder propellants. To go much faster would require a different technology. There is more on this subject in In Search of High Velocity on this website.

The barrel length in comparison with the calibre is obviously an important factor in muzzle velocity. In cannon calibres, this is expressed as the "calibre length", which is simply the length of the barrel divided by the calibre. For example, the current Bofors 40mm gun has a barrel 2.8m long, and therefore has a calibre length of 70, expressed as L/70. The WW2 Bofors had a less powerful cartridge and needed a calibre length of only L/56.

Any loading of any cartridge has a ballistic optimum barrel length, at which the highest velocity is generated. Shorter barrels will lose velocity because the propellant gas has had insufficient time to accelerate the projectile, and some of the energy will therefore be lost in excess muzzle blast and flash.  Longer barrels will lose velocity because the gas pressure begins to drop too low to overcome the friction of the projectile travelling up the barrel (a projectile will eventually grind to a halt given a barrel of sufficient length). In practice, most guns are built with a barrel shorter than the ballistic maximum, because the fall-off of velocity with reductions in length is initially quite small, and a shorter barrel provides a lighter and handier gun.

RECOIL

Two factors determine recoil: the cartridge ballistics and the characteristics of the gun (most especially, its weight).

The recoil impulse generated by firing a cartridge has two components; the momentum of the projectile, and the "rocket effect" of the escaping gas. The simple formula for calculating this is as follows:

(bullet weight x muzzle velocity) + (propellant weight x 1,200 m/s / 4,000 fps)

Therefore a cartridge firing a 10g bullet at 1,000 m/s should have the same bullet momentum as one firing a 20g bullet at 500 m/s. Note that this is a different calculation from muzzle energy, as bullet weight and muzzle velocity are of equal value. This explains why in different bullet-weight loadings of the same cartridge which generate the same muzzle energy, the heavy bullet loading will produce heavier recoil.

The recoil caused by the escaping gas is much more difficult to calculate because it depends on the relationship between the burning characteristics of the propellant and the length of the barrel. If you assume two rifles firing the same cartridge, one with a barrel of optimum length and the other with a much shorter barrel, the optimum length one will produce the higher muzzle velocity and therefore the greater recoil through bullet momentum. However, in the short-barrel gun the gas will be at a higher pressure when the bullet leaves the muzzle, and will therefore expand more violently, causing more muzzle blast and flash and generating a stronger "rocket effect". In this case, a higher proportion of the recoil will be generated by the expanding gas than with the optimum barrel.

For this reason, there is no simple ratio which will tell you exactly what proportion of the recoil is generated by the escaping gas as opposed to the projectile. However, a good approximation can be made, based on the weight multiplied by the velocity of the propellant compared with the weight multiplied by the velocity of the projectile. In a large number of empirical tests, the velocity of the gas escaping from the muzzle of a rifle has been determined to be 1,200 m/s (4,000 fps) plus or minus 10%. In larger high-velocity military weapons, which can operate at very high pressures and velocities, the escaping gas velocity may be significantly higher.

It is therefore fairly simple to work out what proportion of the recoil impulse is generated by the escaping gas. Take for example the 7.62x51 NATO military rifle/MG cartridge in M80 ball loading. This uses 3.0g (46 grains) of propellant to fire a 9.5g (146 grain) bullet at a muzzle velocity of 840 m/s (2,750 fps). The calculation goes like this (the units of measurement don't matter as long as they are used consistently):

Bullet momentum: 9.5 x 840 = 8,000 (rounded). Propellant gas momentum: 3.0 x 1,200 = 3,600. So the total recoil momentum is 8,000 + 3,600 = 11,600, of which the gas produces 3,600 / 11,600 x 100 = 31%

This figure of around 30% is typical for a medium-velocity rifle cartridge. In a higher-velocity rifle like the 5.56mm NATO it is in the region of 35-40%. In handguns it is much lower, in the region of 10-15%, although in the big Magnums it can exceed 20%. In powerful military cannon it can be as high as 50%.

The only way of reducing the recoil force generated by a cartridge while maintaining the muzzle energy, is to reduce the effect of the escaping gas by diverting some of it, either to one side or (preferably) to the rear. A device to achieve this is known as a muzzle brake. The extent to which a muzzle brake can reduce recoil obviously depends upon the proportion of the recoil impulse generated by the propellant gas - it gives the greatest benefit in very powerful, high-velocity weapons. One text on military cannon states that an efficient muzzle brake can reduce the recoil impulse by up to 30%. Higher figures are possible, but only by using brakes which are so large that they would be impractical. A disadvantage of a muzzle brake is that the rearwards-deflected gas greatly increases the muzzle blast and noise perceived by the firer, and may also kick up dust, revealing the weapon's position and affecting the user's visibility.

A type of muzzle brake is the compensator. This deflects some of the muzzle blast upwards in order to counteract the tendency for the gun barrel of a hand-held weapon to point upwards as a result of recoil. It is therefore mainly found in powerful handguns, or in automatic weapons like sub-machine guns.

Of course, a recoilless gun deflects most of the gas directly behind the weapon, so in this case the "rocket effect" more or less balances the projectile momentum. However, this requires the use of several times as much propellant as with a conventional gun of the same muzzle energy, so the ammunition is bulky and expensive.

So far I have only discussed the "recoil impulse" generated by the cartridge as opposed to the recoil experienced. This is because the recoil experienced depends on the weapon, and on the mounting. In the case of a rifle or handgun, the weight of the weapon has the greatest effect. Momentum works both ways, equally (you can't defeat Newton's law of equal and opposite reactions!), so the rearwards momentum of the gun matches the forwards momentum of the bullet plus the expanding gas. It therefore follows that the heavier the weapon, the more slowly it will move backwards under recoil, giving a smooth push rather than the sharper kick of a lighter weapon firing the same ammunition. The recoil momentum experienced by the firer is the same, but it is delivered in a different way; the lighter weapon, recoiling more quickly, has the same momentum but higher energy and is perceived to "kick harder".

For instance, let's take our 7.62x51 NATO cartridge and work out the recoil energy it would generate in different rifles. As we have seen, the cartridge generates a total recoil impulse of 11,600 (grams per metre per second). If a rifle weighs 4.0 kg (8.8.lbs) or 4,000g, it will therefore be pushed back at 11,600 / 4,000 = 2.9 metres per second. 4.0 kg at 2.9 m/s = 17 joules (whereas the cartridge develops 3,350 joules - which shows the importance of velocity in calculating energy). If a lightweight rifle of only 3.0 kg (6.6 lbs) is used to fire the same cartridge, it will be pushed back at 11,600 / 3,000 = 3.9 m/s, producing 23 joules muzzle energy - an increase in recoil energy of 35%, even though the recoil momentum is the same.

To translate this into practical consequences, the 7.62x51 NATO cartridge generates only about double the recoil momentum of the 5.56x45 NATO, but in rifles of the same weight this translates to double the rifle recoil speed, therefore four times the recoil energy. An intermediate military rifle cartridge like the Russian 7.62x39 used in the famous Kalashnikov AK assault rifle fits about half-way between the 7.62x51 and 5.56x45 in recoil generated; regarded as around the maximum for (just about) controllable automatic fire in a military rifle.

Self-loading weapons, whether recoil or gas-operated, tend to reduce the perceived recoil because some of the energy is used to drive the reloading mechanism.

The stock design can also affect the perceived recoil; in rifles, a "straight" stock which directs the recoil impulse into the shoulder causes less barrel jump than a dropped stock, which has the thrust line over the shoulder. A stock with a cheekpiece designed to move away from the face under recoil is also a lot more comfortable to shoot than one which hits the face. And the simple addition of a shock-absorbing butt pad can make a big difference. In handguns, a revolver has a higher thrust line than other types and this may also cause more perceived recoil (although in the old-fashioned type of "western" revolver, the grip tends to rotate in the hand, absorbing some of the kick at the cost of considerable muzzle jump). The type of hand grips fitted can also make a big difference to perceived recoil.

In heavier military guns which are fitted to mountings, the nature of the mounting can make quite a difference. In all but the lightest weapons, the mounting allows the gun to recoil backwards between shots (there is commonly some kind of buffer). Note that this doesn't reduce the recoil force - only a muzzle brake can do that - it merely reduces the peak recoil blow by spreading out the recoil force over a longer period and thereby puts less strain on the mounting (or to put it another way, allows a lighter mounting to be used). Detail design can make a big difference; the US Edgewater mounting used on aircraft HMG and cannon mountings in WW2 substantially reduced the peak recoil blow.

An even more effective way of doing this is with a differential recoil or floating firing mounting, in which the gun is held back in the full recoil position before firing. On firing, the gun runs forwards "into battery" and the weapon is fired just before it gets there. The recoil force therefore has to overcome the forward momentum of the gun before it can start pushing it back again. This idea was first invented over a century ago. It is a highly effective way of smoothing out the recoil pulses and is commonly used in modern automatic AA cannon, plus some automatic grenade launchers (AGLs). The US XM307 25mm AGL, plus the 12.7mm XM312 MG based on this, use this system, which enables both  the guns and their mountings to be very light. The disadvantage is that there is quite a long delay between pressing the trigger and the gun firing the first shot, during which there is considerable gun movement, so it is only suitable for mounted weapons.  It is also important to have reliable ammunition, as a "hang fire" in which a cartridge ignites slightly late, after the barrel has come to a stop, would have unpleasant effects.

Some automatic gun mechanisms have a greater recoil-smoothing effect than others. One of them is the long-recoil type, in which the barrel recoils a considerable distance between shots. This makes for a slow-firing gun and is mainly used in large-calibre weapons. It was commonly used in large aircraft cannon in WW2, such as the US 37mm and the British 40mm and 57mm guns - the big 57mm Molins had a peak recoil blow similar to that of the 20mm Hispano (although the total recoil thrust was obviously much greater). The other is the advanced primer ignition blowback type, as pioneered by Becker in WW1 and much used by Oerlikon and similar 20mm cannon in WW2. In this, the gun fires from an open bolt, which means that when the firing button is pressed, the bolt moves forwards and chambers a cartridge before firing. The key point of the API blowback is that the cartridge is fired while it is still moving forwards, so there is a kind of internal differential recoil effect. This was only possible because the Becker/Oerlikon guns used special ammunition with cartridge cases featuring a rebated rim (of small diameter than the body), enabling the extractor to hook over the rim while remaining within the overall diameter of the cartridge. This allowed the entire bolt to slide inside an extended firing chamber. Some SMGs and AGLs are claimed to use the API blowback system, but without the special ammunition and extended chambers, the effect is much reduced.

There is a popular fallacy that firing a large-calibre cannon in an aircraft (such as the 75mm M4 in the B-25) had a drastic effect on aircraft speed - or even briefly brought it to a stop! A simple comparison of speeds and weights between the shell and the aircraft will show that the aircraft had about 200 times the momentum of the shell. Firing several shots in quick succession would slow the plane a little, but not by more than about 5%.

EXTERNAL BALLISTICS

Just two key factors determine the external ballistics of a projectile; the muzzle velocity and the ballistic coefficient. The ballistic coefficient is significant because it determines the rate at which the projectile slows down, and in conjunction with the muzzle velocity this decides the maximum range (at any given elevation) and the time of flight to any particular distance. The time of flight in turn decides the amount by which the projectile drops downwards as this happens at a constant rate due to gravity. The curved path of the projectile which results from the muzzle velocity, the ballistic coefficient and gravity drop is called the trajectory.

In most types of long-range shooting (whether by rifles or large cannon) a short time of flight is considered desirable because it maximizes the hit probability by reducing the time of flight and flattening the trajectory. It also results in the projectile striking the target at a high velocity and therefore with greater effect. The main exception is when artillery fires in the "upper register" (above 45 degrees elevation) to achieve plunging fire.

The advantages of a high muzzle velocity in reducing the time of flight are self-evident. So are the disadvantages: more propellant is required, the barrel will need to be longer, the gun will be heavier and (in the case of a mounted weapon) so will be the mounting to cope with the greater recoil. In an automatic weapon, the rate of fire is also usually lower. As we have seen, there is also a practical limit to how high the velocity of any given projectile can be pushed. To make the most of the muzzle velocity, we need to achieve a high ballistic coefficient.

There are two elements which decide the ballistic coefficient (BC); the sectional density (SD) and the form factor (FF). The SD is a simple calculation as it is the ratio between calibre and projectile weight. The formula is:

For metric measurements: multiply the projectile weight in grams by 1.422, then divide the result by the square of the calibre in millimetres. So for a 12.7mm bullet weighing 40 grams: (40x1.422)/(12.7x12.7) = an SD of 0.353

For Imperial measurements: divide the projectile weight in pounds by the square of the calibre in inches (if bullet weights are in grains, divide the result by 7,000).

The higher the SD figure, the better the velocity retention (assuming equal form factors).

What the SD measures is the weight (or momentum, when moving) behind every square millimetre of the projectile calibre (i.e. the cross-sectional area of the projectile). If projectiles were solid cylinders then for a given SD figure they would all be the same length regardless of their calibre. In practice, of course, the length varies with the calibre; a 40mm projectile will be about twice the length of a 20mm, and will therefore have about double the SD figure. This explains why artillery shells travel much further than rifle bullets, no matter how fast or streamlined. Other things being equal, the bigger the calibre, the longer the range and the shorter the flight time to any given range.

Other things are of course far from equal, which is where the form factor comes in. The FF measures the aerodynamic efficiency of the projectile's shape, and is much more complicated to calculate; without access to manufacturers' data, only approximate estimates can be made. It is obvious that a projectile with a pointed nose will have much less air resistance than a simple cylinder, and it will therefore have a better FF, but problems arise when you try to become more specific.

The first problem is that the FF is different at subsonic and supersonic velocities, because shapes which work best at subsonic speeds are not the best at supersonic velocities. At subsonic speeds, the drag caused by the low-pressure area created at the back or base of the projectile is significant, and major reductions in drag can be made by tapering this to some extent (streamlining or boat-tailing). At supersonic speeds, it is the nose shape that is critical; finely pointed noses are needed, but the back end doesn't matter so much. Some taper towards the base is useful, but the optimum taper angle is different from that at subsonic velocities. The benefit of boat-tailing at very long range can be demonstrated by two .30-06 bullets, both weighing 180 grains (11.7g) and fired at 2,700 fps (823 m/s). At sea level, the flat-based bullet will travel a maximum of 3,800m, the boat-tail 5,200m.

A further factor affecting military projectiles is the addition of tracer elements. These generate gas which helps to fill the low-pressure area at the base, reducing drag. This gives them a different trajectory by comparison with non-tracer rounds, not helped by the fact that as the tracer burns up the weight of the projectile reduces, thereby worsening its sectional density. Tracers can therefore never achieve a perfect match with other projectiles and can only ever be an approximate guide to their trajectory.

Putting all of this together, the most aerodynamically sophisticated projectiles in use today are the long-range artillery shells known as ERFBBB (extended-range full-bore base-bleed). These have a long, finely pointed nose to work well at their initial supersonic speeds, and a tapered base filled with a "base bleed" burning chemical which essentially does the same aerodynamic job as a tracer. Furthermore, the nose is so pointed that only the base of the shell is in contact with the barrel, so small streamlined stubs are fitted part way up the shell to keep it centred in the bore. It was discovered that these generate some aerodynamic lift, like tiny wings, and extend the range still further. The advantage of all of this can be seen in the range improvement over a conventional 155mm HE shell; in a 39 calibre barrel, the standard M107 shell has a range of 18,100m, the ERFB shell 25,500m and the ERFBBB 32,400m. Furthermore, unlike rocket-assisted or sub-calibre shells, there is no penalty in effectiveness as they carry at least as much HE (in fact, the South African 155mm M57 ERFB shell contains 30% more HE than the standard M107 shell).

It is possible to obtain some idea of typical FFs by comparing manufacturers' BC data with the calculated SDs for the same projectiles. In the case of small arms bullets, this provides the following approximate FFs (this figure should be multiplied by the SD to give the BC):

Flat-nose lead: 0.8
Round-nose lead: 0.9
Round-nose jacketed: 1.0
Semi-pointed soft point: 0.9-1.1
Pointed soft point: 1.2-1.6 (depending on sharpness of point)
Pointed full jacket: 1.5-1.8
Pointed full-jacket boat-tailed: 1.9-2.0

Comparing the BCs with ballistic tables for the ammunition gives the following results. These figures show the approximate percentage velocity loss over 100m for supersonic projectiles (900 m/s) with the following BCs:

BC     0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
V loss  % 25 18 14 11.5 9.5 8 7 6.5

Another type of Form Factor is traditionally used for artillery - and especially naval - shells. This is the "caliber radius head" (CRH) which measures how pointed the nose is. To give an example, if the curve of a shell nose is the same as that of a circle with a radius of 500mm in diameter, and the calibre is 100mm, then the shell has a CRH of 5. The higher the CRH, the better the FF.

In calculating SD and BC, it should be noted that the notional cartridge calibre is not necessarily the same as the actual projectile diameter, particularly with small arms. The bore diameter (ie the inside diameter of the barrel ignoring any rifling grooves) may be used instead, or some notional figure. The following list of bullet diameters is followed by some of the common military and commercial cartridge designations of that calibre (there are many other calibres and cartridges…). For more detail on cartridge designations, see this article.

Bullet diameter                         Cartridge designation
mm/inches
5.60/.221                                     5.45mm Russian
5.69/.224                                     5.56mm, 5.6mm, .218, 219, .22, .220, .221, .222, .223, .224, .225, .226
6.17/.243                                     6mm, .240, .243, .244
6.53/.257                                     6.35mm, .25, .250, .257
6.71/.264                                     6.5mm, .256, .260, .264
7.04/.277                                    .270, .277
7.21/.284                                     7mm, .275, .276, .280, .284
7.82/.308                                     7.5mm, 7.62mm, 7.63mm, 7.65mm, .30, .300, .307, .308
7.90/.311                                     7.62mm (Russian), 7.65mm (Belgian), 7.7mm, 8mm Swedish,.303, .311
8.20/.323                                     7.92mm, 8mm (most - but not all)
8.59/.338                                     8.58mm, .33, .330, .338, .340
12.9/.510                                    12.7mm (Russian), .50
13.1/.514                                    12.7mm (Breda, Ho-103), .5 (Vickers)
13.5/.530                                    13mm (IJN Type 3 and 96), 13.2mm (Hotchkiss)
13.5/.533                                    13mm MG 131, 13mm IJN Type 2
15.6/.614                                    15mm MG 151
16.2/.638                                    15mm ZB vz/60, Besa
19.8/.780                                    20mm ShVAK, B-20
19.9/.785                                    20mm (other)
22.9/.900                                    23mm VYa, ZU
24.9/.980                                    25mm (general)
26.9/1.06                                    27mm Mauser
29.9/1.18                                    30mm (general)
35.0/1.38                                    35mm Oerlikon
36.8/1.45                                    37mm (general), 1 pdr, 1.5 pdr
39.9/1.57                                    40mm (general), 2 pdr
47.0/1.85                                    47mm, 3 pdr
50.0/1.97                                    50mm, 5 cm
56.9/2.24                                    57mm, 6 pdr

An important aspect of external ballistics is the stability of the projectile. If an ordinary bullet or cannon shell were fired from a smoothbored gun, it would probably start to tumble, ruining its aerodynamics and accuracy. Old-fashioned smoothbore muskets fired round balls, so tumbling wasn't a problem, but such a shape has a poor BC. Even round balls fired from smoothbores tend to drift off target as the range increases anyway, as they will not be entirely symmetrical. The answer was found to be to cause the bullet to spin rapidly by cutting spiral grooves into the barrel, called rifling. This evens out any asymmetries and keeps pointed bullets heading point-first.

Initially, bullets and shells were provided with studs to fit into the rifling but these were slow to load. An alternative approach was to make a polygonal-section barrel with shells manufactured to fit. Subsequently, rifles were provided with Miniť type bullets which had a hollow base, designed to expand under the pressure of firing and "take" the rifling. Modern rifle, pistol and heavy machine gun (HMG) bullets are given a metal jacket (usually cupro-nickel) which has a slightly larger diameter than the bore of the gun. It is therefore squeezed into the rifling grooves on firing, which leaves characteristic angled grooves engraved into the bullet. Modern cannon shells are usually made of steel which is too hard for this, so they are given a driving band near the base of the projectile, which is larger in diameter than the shell and is gripped by the rifling. The driving bands are traditionally copper but this is too soft for modern high-velocity cannon which normally use soft steel driving bands instead. The 30mm projectile for the US GAU-8/A cannon are unusual in using plastic driving bands.

With large-calibre, high-velocity cannon there is some risk of the shock of impact with the rifling "stripping" the driving band. To combat this, some weapons have progressive rifling, in which the rifling grooves start out parallel then gradually increase in twist down the barrel. In a return to earlier "studded" shell principles, the WW1 Paris Guns had pre-engraved rifling bands to minimise the friction and the risk of stripping the band.

There is also some concern with cannon that the driving bands, which stick out from the shell body and tend to be rather chewed up by the rifling, have a poor effect on aerodynamics. One approach which helps with this problem is to include a smooth (unrifled)  final section of the barrel tapered to achieve the same diameter as the shell at the muzzle, which squeezes the driving band flat against the projectile. This is known as a "Probertised" barrel (technically "RD Rifling") after Probert, the British inventor, and was used in the high-velocity 3.7" Mk VI AA gun in World War 2. It should be noted that the main purpose of this system was to extend the barrel life by allowing greater wear at the neck of the chamber. The system worked by separating the roles of imparting spin and forward obturation to two separate bands. The forward driving band, up by the bourrelet was designed to engage the rifling and the rearward obturation band to provide the gas seal. This second band was very deep to allow for significant erosion at the case mouth. The leed was very long (about 7” between the end of the chamber and the C of R) on a probertised barrel to allow for this. The 3.7, being fixed QF, could not take up wear by ramming so would be condemned very quickly if rifled in the conventional manner.

Rifling permits a high degree of accuracy over the maximum range of a weapon. There is a relationship between the rifling twist (the angle of the rifling to the barrel) and the length of the projectile. For a given calibre, the longer (ie heavier) the projectile, the steeper the twist has to be in order to stabilise it. Clearly, with a particular rifling twist some light projectiles will be very stable, some medium-weight ones marginally stable and some heavy ones not stabilised at all. This can have consequences for the terminal as well as the external ballistics, as we shall see. As projectiles become steadily longer so rifling can no longer cope, and long, thin projectiles such as APFSDS (armour-piercing fin-stabilised discarding-sabot) require fins, like arrows, at the back to keep them pointing, nose-first, in the right direction. Such projectiles are disturbed by rifling and work better from an (almost) smoothbore barrel (some degree of spin is considered useful in effecting clean sabot separation). HEAT (high-explosive anti-tank, also known as hollow-charge) projectiles also work best when not spun, so these two types of munitions have become associated with smoothbore barrels, almost exclusively fitted to AFVs. Recently, automatic cannon with rifled barrels have taken to using APFSDS, but the spinning causes the projectiles to yaw (fail to point straight ahead) for several hundred metres, so they only become fully effective at 300+m.

The optimum gun elevation for achieving the maximum range depends on the range capability. In a vacuum, the maximum range would be achieved at 45 degrees for all weapons (and this is in reality approximately  true for most artillery) but this varies at the extremes due to the variation in air pressure and resistance at different altitudes. Large calibre, very high-velocity artillery (e.g. the WW1 German Paris Guns which ranged out to 120 km / 75 miles) achieve their maximum range at an elevation of about 55 degrees, because aerodynamic drag reduces along with air pressure so the sooner the shell gets up into the thin upper air the further it will travel. Rifle bullets are restricted to the lower and denser atmosphere and their optimum elevation is about 30-35 degrees. For the same reason, an aircraft gun will have a much longer effective range at high altitude than in the thick air at ground level. While the effective range and maximum range of artillery shells are the same, the effective range of small-arms ammunition is much shorter than its maximum ballistic range, due to aiming problems. Even the little .22LR rimfire round, which is usually credited with an effective range of no more than 100 metres, can reach out to 1,500 m, while typical military rifle bullets will travel to more than 3,500 m.

TERMINAL BALLISTICS

There are two different aspects to this; the effect of projectile strike against soft targets (animals or people) and the effect against armour. The former is described in more detail here.

First, against soft targets (the squeamish have permission to duck this section!). A military (i.e. fully jacketed, pointed, non-expanding) rifle bullet will be destabilised when hitting a soft target and will tumble. This is because its shape means that the centre of gravity of the bullet is towards the rear so it naturally prefers to fly base-first. Spinning the bullet by means of the rifling keeps the bullet flying point-first through the air, but flesh is about 400 times denser than air so spinning is no longer enough; the bullet destabilises and turns over to travel base-first, a process known as tumbling.  In so doing it obviously inflicts a far more serious wound than if it carried on flying straight through the body. Incidentally, bullets designed for penetrating heavy game animals like elephant - which need to penetrate very deeply in a straight line and must therefore not yaw or tumble - have long, parallel sides and blunt round noses, just like early military rifle bullets.

Not all bullets tumble at the same rate. Other things being equal, small bullets will tumble more quickly than large ones, but the design of the bullet is also important; some visually identical bullets will tumble at different speeds, generally depending on the internal construction. For example, the Yugoslavian bullet for the 7.62x39 has a lead core and has been found in tests to tumble much more quickly than the Russian steel-cored bullet in the same cartridge. Various tricks have been used to increase the probability of a bullet tumbling; the British .303 Mk VII bullet had a lightweight tip filler with the weight concentrated towards the rear of the bullet, and the current Russian 5.45mm rifle bullet has a hollow tip.

If a bullet has a relatively weak jacket, the stresses of tumbling may cause it to break apart while it is travelling sideways through flesh - a process known as fragmentation - which further increases the wounding effect. Most 5.56x45 military bullets fragment, although they have to be travelling at high velocity to do so. This limits their maximum effectiveness to fairly short range, particularly from short-barrelled carbines which have a lower muzzle velocity. Most 7.62x51 NATO bullets do not fragment, although the German one does - by accident rather than design. Fragmentation is not an official requirement for any military bullets; if it were, there might be some legal challenge over the international prohibition on bullets designed to cause unnecessary suffering. The noses of hunting rifle bullets (and many commercial handgun bullets) are designed to expand on impact, which greatly increases the size of the wound channel.  Such bullets are illegal for military use.

It is often claimed by hunters that as the striking velocity of the bullet increases beyond about 700 m/s (2,300 fps), so hydrostatic shock begins to appear, with the effect that animals drop dead much more dramatically than if hit in the same place with a low-velocity bullet. However, this effect does not seem to be replicated in people; there are many cases of soldiers continuing to fight for some time despite receiving severe (and ultimately fatal) wounds from high-velocity rifle bullets. Furthermore, serious shock effects are only likely if the bullet exceeds the speed of sound in flesh, which is around 1,500 m/s (4,900 fps), but even this has been disputed.

This brings us onto the vexed question of stopping power, about which it is impossible to make any pronouncements without stimulating fierce arguments. Stopping power may be defined as the ability of a particular weapon to immediately disable an opponent so he can take no further part in the fighting. It is not the same as lethality; quite low-powered weapons can be lethal, but considerably more power is normally required to achieve reliable stopping power. Incidentally, this shows that the notion that modern military rifle bullets are meant to wound rather than kill is a myth; if it is powerful enough to disable, it is more than powerful enough to kill.

Clearly, bullet placement is vital to achieving effective stopping power; it is much more effective to hit an immediately vital area with a low-powered weapon than to inflict a minor wound with a high-powered one. Also, the psychological state of the target has a considerable effect. Someone who is relaxed, or frightened, may be put out of the fight by a minor wound, someone who is highly charged with aggression will require far more power to stop them, and yet another person high on drugs may continue fighting despite suffering the most appalling wounds.

Stopping power is simplest to define with pistols, which have too low a velocity for hydrostatic shock to be a factor. The classic formula, named after the American Julian Hatcher, is calculated by multiplying the bullet weight by the muzzle velocity and then by the square of the calibre. The result is then multiplied by a form factor, similar in principle to that used for calculating the BC, except that in this case, the blunter the bullet shape the more effective it is. It will immediately be seen that calibre is the most important factor, and indeed large calibre pistols such as the .45" have always had a good reputation for stopping power. It should be noted that even the most powerful handgun or rifle will not physically knock someone down; if they were that powerful, Newton's law would require the firer to be thrown backwards with equal force. The recent spread of body armour has changed the perceptions of desirable pistol ballistics to some extent, as a high-velocity small-calibre bullet will punch through body armour which will easily stop a large-calibre, low-velocity bullet.

This brings us onto the subject of penetration. This is not just to do with military armour, but also against tough animals like elephants or water buffalo. As already indicated, early big game hunters found that the most important characteristics of a bullet against such tough game were that it should be round-nosed, strongly built, and have a good SD. A pointed bullet would not follow a straight path through a mass of bone, and one with too high a velocity also often followed an erratic path. Amazingly, one of the most successful early elephant guns (albeit only in very skilled hands) was the little 6.5mm Mannlicher. Why? Because its very long, 160 grain (10.4g), round-nosed bullet and its moderate velocity allowed it to penetrate remarkable thicknesses of bone - but it was only effective with a precise aim.

The subject of the penetration of armour is highly technical and complex. Furthermore, different national definitions of penetration and different types and qualities of armour used to test projectiles against make comparisons difficult. However, certain broad principles still hold generally true. As with elephant guns, a high SD is desirable and so (more surprisingly) is a blunt nose, although this is often concealed by a pointed ballistic cap or windshield. However, one major difference is that the higher the striking velocity the better, at least until the velocities are so high that the projectile is more likely to shatter than penetrate. For hardened steel penetrators, this happens at velocities much over 1,000 m/s.

SUB-CALIBRE PROJECTILES

This term is used to describe projectiles which are smaller than the calibre of the gun they are fired from. Nowadays this normally means APDS or APFSDS, but I will also deal with two related developments; APCR and squeezebore guns.

As we have seen, a large calibre will permit more energy to be generated than a small one. On the other hand, for a given projectile weight a smaller calibre will have a higher SD and therefore better long-range and AP performances. Designers have therefore tried different ways of combining the advantages of the two.

The simplest type was known to the British in WW2 as APCR (armour piercing, composite rigid - I have seen an early document which referred to this as "composite rigid armour piercing" but they presumably thought better of the acronym…), to the Americans as HVAP (high velocity armour piercing) and to the Germans as Hartkernmunition or Pzgr.40. However, it was probably the French who fielded it first, in the M1935 loading for the little 37x94R round still being used in some tank guns (there's a picture of one, plus sub-calibre projectiles, in the photo gallery on this website). It is nowadays commonly known as APHC, for armour piercing hard core, and is mainly used in MGs, HMGs and small-calibre cannon.

As the names suggest, this consists of a lightweight projectile (normally mainly aluminium) with a hard, small calibre core (normally tungsten alloy, which is heavier and harder than steel). The light projectile in a large-calibre gun gives a high muzzle velocity but when it strikes the target, only the hard core penetrates so it can go through much more armour than a full-calibre projectile of the same weight. The only disadvantage is that the light projectile has a low SD and therefore slows down more quickly than a normal projectile, steadily losing its penetration advantage as the range increases. To overcome this problem, later versions tended to be little if any lighter than a standard shell, thereby trading some of their short-range penetration for better long-range effectiveness. A modern example of this is the 30mm API used in the GAU-8/A cannon fitted to the A-10 aircraft; this is also unusual in having a depleted uranium core.

Another approach to achieving the best of both worlds was the squeezebore gun, of which there were two basic types; the Gerlich and the Littlejohn. In both, a projectile fitted with flanges to fit a large caliber barrel was squeezed down to a smaller calibre before it left the muzzle. The difference between them was that the Gerlich guns had tapered barrels whereas the Littlejohns had normal barrels with a tapered attachment fitted to the muzzle, in principle not unlike a shotgun choke. These worked very well and both saw limited service in WW2, the Gerlich in some German AT guns and the Littlejohn (named after the Czech designer, Janecek, which translates as "little John") in some Allied armoured car and light tank guns. Their main problem, apart from the cost of the tungsten-cored ammo (and in the case of the Gerlich, the expensive barrel manufacturing) was that they could only fire this type of ammunition; they could not fire full-calibre HE shells. For this reason, they lost favour as soon as a better solution emerged.

The better solution was APDS, for armour piercing discarding sabot. This was like the APCR shell, except that the light alloy sabot (French for shoe) was designed to fall away from the small-calibre penetrator as soon as the projectile left the muzzle. This therefore combined the advantages of a large calibre for maximum energy with a small calibre for best flight and penetration performance, and allowed conventional ammunition to be fired from the same gun. It was initially designed in France before WW2, but was then developed in Canada and the UK, being issued for British 6pdr and 17pdr guns from mid-1944 onwards.

Apart from the cost and availability of the tungsten (always an issue in WW2) the only problem was that early version were very inaccurate because the flight of the projectile was disturbed by sabot separation. The British carried on using conventional AP tank ammunition into the 1950s, and APDS only really became supreme with the British 105mm tank gun of the late 1950s, which became the NATO standard for many years.

The replacement for APDS in tank guns (it is still used in small calibre cannon and HMGs) was APFSDS, which takes the design principles to their logical conclusion in producing the longest and thinnest practical projectile. The problem, as we have seen, is that achieving stability by spinning doesn't work with such long projectiles so they have to be fin stabilised. Modern manufacturing quality means that a high degree of accuracy can be achieved, and APFSDS seems likely to remain the supreme penetrator until conventional guns are replaced by different technologies.

Further information about high-velocity military AP ammunition, together with illustrations, is in The Search for High Velocity on this website.

SOURCES:

NRA Firearms Fact Book (3rd edition, 1989). A fascinating compendium of assorted data from the American National Rifle Association.

Handbook for Shooters and Reloaders by P. O. Ackley - a classic, which comes with a pack of charts and tables for do-it-yourself ballistics without formulae, let alone computer programmes.

Brassey's Military Ballistics, by Moss, Leeming and Farrar

...and decades of collecting odd snippets of information from all over the place.

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