From the viewpoint of physics (dynamics, to be exact), a firearm, as for most weapons, is a system for delivering maximum destructive energy to the target with minimum delivery of energy on the shooter. The force and momentum delivered to the target however cannot be any more than that (due to recoil) on the shooter. This is because the force and momentum on the bullet is equal to that on the gun-shooter system.

Force

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Assuming the gun and shooter are at rest, the force on the bullet is equal to that on the gun-shooter. This is due to the law of conservation of momentum. Consider a system where the gun and shooter have a combined mass M and the bullet has a mass m. When the gun is fired, the two systems move away from one another with new velocities V and v respectively. But the law of conservation of momentum states that their momenta must be equal:

$MV=mv\; \backslash qquad\; (1)$

Since force equals the change in momentum and the initial momenta are zero, the force on the bullet must therefore be the same as the force on the gun/shooter.

Hollywood depictions of firearm victims being thrown through plate-glass windows are inaccurate. Were this to be the case, the shooter would also be thrown backwards with equal force. Gunshot victims frequently fall or collapse when shot; this is less a result of the momentum of the bullet pushing them over, but is primarily caused by physical damage or psychological effects, perhaps combined with being off-balance. This is not the case if the victim is hit by heavier projectiles such as 20 mm cannon shell, where the momentum effects can be enormous; this is why very few such weapons can be fired without being mounted on a weapons platform or involve a recoilless system e.g.Recoilless rifle.)

Velocity

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From Eq. 1 we can write for the velocity of the gun/shooter: V = mv/M. This shows that despite the high velocity of the bullet, the small bullet-mass to shooter-mass ratio results in a low recoil velocity (V) although the force and momentum are equal.

Kinetic Energy

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However, the smaller mass of the bullet, compared that of the gun-shooter system, allows significantly more energy to be imparted to the bullet than to the shooter. The kinetic energies for the two systems are $\backslash begin\{matrix\}\backslash frac\{1\}\{2\}\backslash end\{matrix\}MV^2$ for the gun-shooter system and $\backslash begin\{matrix\}\backslash frac\{1\}\{2\}\backslash end\{matrix\}mv^2$ for the bullet. The energy imparted to the shooter can then be written as:

$\backslash frac\{1\}\{2\}MV^2=\backslash frac\{1\}\{2\}M\backslash left(\backslash frac\{mv\}\{M\}\backslash right)^2=\backslash frac\{m\}\{M\}\backslash frac\{1\}\{2\}mv^2$

If we now write for the ratio of the energies we have:

$\backslash frac\{\backslash frac\{1\}\{2\}MV^2\}\{\backslash frac\{1\}\{2\}mv^2\}\; =\; \backslash frac\{m\}\{M\}\; \backslash qquad\; (2)$

The ratio of the energies is the same as the ratio of the masses (and is independent of velocity). Since the mass of the bullet is much less than that of the shooter there is more kinetic energy transferred to the bullet than to the shooter. The larger kinetic energy of the bullet is then dissipated in the target.

Transfer of Energy

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When the bullet strikes, its high velocity and small area means that it will exert large stresses in any object it hits. This usually results in it penetrating any soft object, such as flesh. The energy is then dissipated in the wound tract formed by the passage of the bullet. See terminal ballistics for a fuller discussion of these effects.

Bulletproof vests work by dissipating the bullet's energy in another way; the vest's material, usually Kevlar, works by presenting a series of material layers which catch the bullet and spread its imparted force over a larger area, hopefully bringing the round to a stop before it can penetrate into the body. While the vest can prevent a bullet from penetrating, the wearer will still be affected by the kinetic energy of the bullet, which can produce serious internal injuries.

Other information

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The television show MythBusters undertook to study the behavior of projectiles fired into water; comparing several firearms including a muzzle-loader, a Beretta 9mm automatic pistol, a .357 magnum, an M1 Garand, a Browning .50 caliber, and a shotgun with a deer slug, _Season 2#Bulletproof Water that the lower velocity weapons would still provide lethal penetration at a distance of 8-10 feet, while the projectiles from the higher velocity weapons would fragment upon contact with the water.

They followed this with Season 3#Guns Fired Underwater, using a Beretta 9mm automatic pistol, a .357 magnum, a Garand M1, and a shotgun. While all the guns successfully fired underwater, the shotgun was completely destroyed, with the barrel splitting. The bullets were able to provide lethal penetration after traveling no more than two feet.

See also

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• ballistics

Firearms | applied and interdisciplinary physics | classical mechanics