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Gun Recoil
Women's Biathlon

Although guns may not be everyday things for many of us, gun recoil is certainly something we're aware of (at least those of us that don't make Hollywood action films with guns whose recoil would instantly kill the person firing them!). Gun recoil is a result of momentum conservation, which is an extremely important fundamental principle. Newton was talking about momentum conservation when he wrote "every action has an equal and opposite reaction".

Momentum Conservation

Momentum characterizes an object's resistance to change in motion. If this is motion along a straight line, we call it linear momentum; if it is rotational motion we call it angular momentum. The basic idea is the same: moving things like to keep moving, and to change their motion we have to apply a force. If no force is present, then momentum doesn't change, ie. it is conserved.

Now, you might point out that a bullet coming out of a gun has a huge force on it from the exploding gunpowder. True enough, and that force is what propels the bullet forward. However, if you look at a bullet and gun together (say while the bullet is still in the barrel but already heading out at full speed), you can say there is no net force on the bullet-gun system. So the momentum of the bullet plus gun should be conserved.

If the bullet has mass mb and speed vb out of the gun, it has momentum pb given simply by

pb = mbvb

in the forward direction. To balance this momentum (and keep the net momentum of the bullet-gun system zero), the gun recoils with momentum in the opposite direction: pg = -pb, or

mgvg = -mbvb

Although the bullet's mass is small, its speed is quite large, so it released with large momentum. The gun has much larger mass, so the recoil speed is much smaller, but still large enough to give a serious kick against the shooter's shoulder.

Example: Winchester .308

Let's look at an example. A Winchester .308 cartridge launches a bullet of mass 150 grains (1 grain = 64.8 mg) with a speed of 2820 ft/s (1 ft = 30.5 cm). In MKS units, then, pb = 8.4 kg m/s. This rifle has a weight of about 8 lbs, or a mass of mg = 3.8 kg. That means the recoil speed of the rifle will be

vg = - pb/mg = -2.2 m/s

This primary recoil is noticeable, but not the main recoil that one feels.

Secondary Recoil

There are actually two distinct recoils from a gun: the first, primary recoil, which I've described above, conserves momentum of the gun-bullet system. However, a larger secondary recoil comes slightly later, when the bullet leaves the muzzle: then the hot expanding gas behind the bullet shoots out of the muzzle, and the muzzle recoils further like a rocket. This is, again, conservation of momentum, but in this case is is the gas momentum out of the barrel that makes the secondary recoil. Gun manufacturers make baffles that reduce the flow of gas out of the muzzle to reduce secondary recoil. Primary recoil cannot be reduced, since it is simply associated with the forward momentum of the bullet.

Equations

• linear momentum: p = mv

Summary

• The total momentum of a system is conserved if there are no outside forces acting on it.
• Gun recoil results from conservation of total momentum of the bullet-gun system: the backward recoil gun momentum balances the forward bullet momentum to maintain zero total momentum.
• Gun recoil actually has two parts: primary recoil from the escaping bullet and secondary recoil from the escaping gas behind the bullet.