Recoil, in common everyday language, is considered the backward kick or force produced by a gun when it is fired. In more precise scientific terms, this force is equal to the time derivative of the backward momentum resulting when a gun is fired.
The backward momentum is equal to the mass of the gun times 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 gasses, etc.). Provided that enough information is known about the mass and velocity of the ejecta, 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, such as with a ballistic pendulum.
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.
In mathematics, the recoil and energy of a recoiling firearm are different as well. The recoil of a firearm, both large and small, is a function of the law conservation of momentum and can be stated mathematically as mf·vf = mp·vp from Newton's third law. The equation describes the recoil of a firearm and the user or gun-mount system in its entirety. The action of firing includes the ejecta at any position along the barrel. The equal sign (=) that represents the total thermodynamic energy held within the powder charge. "mf·vf" found on the left side of the equation represents the backwards motion of the firearm or firearm system. "mp·vp" found on the right of the equation represents the forward motion of the ejecta moving down the barrel, before either has left the barrel.
The energy of a recoiling firearm can be stated mathematically as Et=0.5·m·v² from Newton's second law. This equation is known as the "classic statement" and yields a measurement of force in joules (foot-pound force in non-SI units). The calculated energy is the amount of work that can be done by the recoiling firearm, firearm system, or a projectile because of their motion. When looking at the classic statement (Et=0.5·m·v²): the "Et" in energy represents the translational kinetic energy of a recoiling firearm, firearm system, or ejacta's motion after it exits the barrel. The "m" in the equation represents the mass of the firearm, firearm system, or ejecta. The "v²" in the equation represents the square of the velocity for the firearm or firearm system in backwards motion and the ejecta's forward motion after it exits the barrel.
The difference between these two equations and events is that during recoil, m·v = m·v, a projectile has not left the barrel. When calculating the energy of a recoiling firearm, Et=0.5·m·v², the projectile has exited the barrel. Again, recoil is a principle of the law of conservation of momentum and the energy of a recoiling firearm, firearm system, or projectile is a quality of work that can be measured.
It can also be helpful to think of the energy of a recoiling firearm or firearm system as a byproduct of work being done to accelerate a projectile towards its target.
See physics of firearms for a more detailed discussion.
This perception of recoil is related to the momentum associated with a particular gun. The total force of 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.
Of course this does not apply if the victim is hit by heavy weapons fire, such as aircraft cannon, where the momentum effects can be significant. The recoil from such a weapon is equally enormous, and requires it to be mounted on a weapons platform. An example of the latter would be the Handheld Minigun shown in the film Predator. For example, the gun shown in the movie is based on the prototype XM214 Microgun, a six barrel electric Gatling gun which fires the 5.56 × 45 mm NATO cartridge at a rate of 4000 to 6,000 rounds per minute. Each cartridge accelerates a bullet to about ; at a moderate rate of fire of 4,000 rounds per minute, a one second burst will accelerate of mass. Assuming a user, and for the XM214 and ammunition, that one second burst will result in the gun and user being accelerated to .