Recoil is the 'kick' given by a gun when it is fired. In technical terms, this kick is caused by the gun's backward momentum, which exactly balances the forward momentum of the projectile. In most small arms, the momentum is transferred to the ground through the body of the shooter; while in heavier guns such as mounted machine guns or cannons, the momentum is transferred to the ground through a mounting system.
The change in momentum results in a force which, according to Newton's second law, is equal to the time derivative of the backward momentum of the gun. The backward momentum is equal to the mass of the gun multiplied by its reverse velocity. This backward momentum is equal, by the law of conservation of momentum, to the forward momentum of the ejecta of the gun (the projectile(s), wad, sabot, propellant gases, and so on). Provided that the mass and velocity of the ejecta are known, it is possible to calculate its momentum and thus the recoil. In practice, however, it is often easier simply to measure the recoil force directly, as with a ballistic pendulum.
There are two conservation laws at work when a gun is fired: conservation of momentum and conservation of energy. Recoil is explained by the law of conservation of momentum, and so it is easier to discuss it separately from energy.
The recoil of a firearm, whether large or small, is a result of the law of conservation of momentum. Assuming that the firearm and projectile are both at rest before firing, then their total momentum is zero. Immediately after firing, conservation of momentum requires that the total momentum of the firearm and projectile is the same as before, namely zero. Stating this mathematically:
where pf is the momentum of the firearm and pp is the momentum of the projectile. In other words, immediately after firing, the momentum of the firearm is equal and opposite to the momentum of the projectile.
Since momentum of a body is defined as its mass multiplied by its velocity, we can rewrite the above equation as:
A consideration of energy leads to a different equation. From Newton's second law, the energy of a moving body due to its motion can be stated mathematically as:
This equation is known as the "classic statement" and yields a measurement of energy in joules (or foot-pound force in non-SI units). Et is the amount of work that can be done by the recoiling firearm, firearm system, or projectile because of its motion, and is also called the translational kinetic energy. In the firearms lexicon, the energy of a recoiling firearm is called felt recoil, free recoil, and recoil energy. This same energy from a projectile in motion is called: muzzle energy, bullet energy, remaining energy, down range energy, and impact energy.
There is a difference between these two equations and events. The momentum equations describe conditions immediately after firing, before the projectile has left the barrel, while the energy equation describes conditions after the projectile has left the barrel.
The recoil impulse Ir of a small arm can be roughly described as:
This equation is an approximation. The constant of 1.75 varies for differing propellants.
See physics of firearms for a more detailed discussion.
For small arms, the way in which the shooter perceives the recoil, or kick, can have a significant impact on the shooter's experience and performance. For example, a gun that "kicks like a mule" is going to be approached with trepidation, and the shooter will anticipate the recoil and flinch in anticipation as the shot is released. This leads to the shooter jerking the trigger, rather than pulling it smoothly, and the jerking motion is almost certain to disturb the alignment of the gun and result in a miss.
This perception of recoil is related to the acceleration associated with a particular gun. The actual recoil is associated with the momentum of a gun, the momentum being the product of the mass of the gun times the reverse velocity of the gun. A heavier gun, that is a gun with more mass, will manifest the momentum by exhibiting a lessened acceleration, and, generally, result in a lessened perception of recoil.
One of the common ways of describing the felt recoil of a particular gun/cartridge combination is as "soft" or "sharp" recoiling; soft recoil is recoil spread over a longer period of time, that is at a lower acceleration, and sharp recoil is spread over a shorter period of time, that is with a higher acceleration. With the same gun and two loads with different bullet masses but the same recoil force, the load firing the heavier bullet will have the softer recoil, because the product of mass times acceleration must remain constant, and if mass goes up then acceleration must go down, to keep the product constant.
Keeping the above in mind, you can generally base the relative recoil of firearms by factoring in a number of figures such as bullet weight, powder charge, the weight of the actual firearm etc. The following are base examples calculated through the Handloads.com free online calculator, and bullet and firearm data from respective reloading manuals (of medium/common loads) and manufacturer specs:
In addition to the overall mass of the gun, reciprocating parts of the gun will effect how the shooter perceives recoil. While these parts are not part of the ejecta, and do not alter the overall momentum of the system, they do involve moving masses during the operation of firing. For example, gas operated shotguns are widely held to have a "softer" recoil than fixed breech or recoil operated guns. In a gas operated gun, the bolt is accelerated rearwards by propellant gases during firing, which results in a forward force on the body of the gun. This is countered by a rearward force as the bolt reaches the limit of travel and moves forwards, resulting in a zero sum, but to the shooter, the recoil has been spread out over a longer period of time, resulting in the "softer" feel.
A recoil system absorbs momentum, for example, by the barrel moving backwards. Cannons and such weapons without a recoil system roll several meters backwards when fired.
In a soft-recoil system, a gun's barrel is moved forward prior to shooting. As the barrel is forced backwards by the recoil force, the energy is reduced by friction, resulting in less of an overall "kick". One of the early guns to use this was the French 65 mm mle.1906; however, this method did not receive much attention until the 1970s.
In a mounted gun, a down-recoil system is when the barrel is forced down on a spring mechanism, and immediately springs back up to its original position. This modern system, since around 2003, can work so rapidly that it works with machine guns.
Hollywood depictions of firearm victims being thrown through several feet backwards are inaccurate, although not for the often-cited reason of conservation of energy. Although energy must be conserved, this does not mean that the kinetic energy of the bullet must be equal to the recoil energy of the gun: in fact, it is many times greater. For example, a bullet fired from an M16 rifle has approximately 1300 foot-pounds of kinetic energy as it leaves the muzzle, but the recoil energy of the gun is less than 5 foot-pounds. Despite this imbalance, energy is still conserved because the total energy in the system before firing (the chemical energy stored in the explosive) is equal to the total energy after firing (the kinetic energy of the recoiling firearm, plus the kinetic energy of the bullet and other ejecta, plus the heat energy from the explosion). In order to work out the distribution of kinetic energy between the firearm and the bullet, it is necessary to use the law of conservation of momentum in combination with the law of conservation of energy.
Thus, when a bullet strikes a target, it may have a kinetic energy in the hundreds or even thousands of foot-pounds, which in theory is enough to lift a person well off the ground. (A foot-pound is the energy required to lift a one-pound weight to a height one foot off the ground.) This energy, however, is largely spent in the deformation or shattering of the bullet (depending on bullet construction), damage to the target (depending on target construction), and heat dissipation. In other words, because the bullet strike on the target is an inelastic collision, a minority of the bullet energy is used to actually impart momentum to the target. This is why a ballistic pendulum relies on conservation of bullet momentum and pendulum energy rather than conservation of bullet energy to determine bullet velocity; a bullet fired into a hanging block of wood or other material will spend much of its kinetic energy to create a hole in the wood and dissipate heat as friction as it slows to a stop.
Gunshot victims frequently do collapse when shot, which is usually due to psychological motives, a direct hit to the central nervous system, and/or massive blood loss (see stopping power), and is not the result of the momentum of the bullet pushing them over.