Recoil

Recoil is the rearward thrust generated when a gun is being discharged. In technical terms, the recoil is a result of conservation of momentum, for according to Newton's third law the force required to accelerate something will evoke an equal but opposite reactional force, which means the forward momentum gained by the projectile and exhaust gases will be mathematically balanced out by an equal and opposite impulse exerted back upon the gun.

Basics

Any launching system generates recoil. However recoil only constitutes a problem in the field of artillery and firearms due to the magnitude of the forces at play. Gun chamber pressures and projectile acceleration forces are tremendous, on the order of tens to hundreds megapascal and tens of thousands of times the acceleration of gravity, both necessary to launch the projectile at useful velocity during the very short time it is travelling inside the barrel. Meanwhile, the same pressures acting on the base of the projectile are acting on the rear face of the gun chamber, accelerating the gun rearward during firing with just the same force it is accelerating the projectile forward.
This moves the gun rearward and generates the recoil momentum. This recoil momentum is the product of the mass and the acceleration of the projectile and propellant gasses combined, reversed: the projectile moves forward, the recoil is rearward. The heavier and the faster the projectile, the more recoil will be generated. The gun acquires a rearward velocity that is ratio of this momentum by the mass of the gun: the heavier the gun, the slower the rearward velocity.
As an example, a 8 g bullet of 9×19mm Parabellum flying forward at 350 m/s muzzle speed generates a momentum to push a 0.8 kg pistol firing it at 3.5 m/s rearward, if unopposed by the shooter.

Countering recoil

In order to bring the rearward moving gun to a halt, the momentum acquired by the gun is dissipated by a forward-acting counter-recoil force applied to the gun over a period of time during and after the projectile exits the muzzle. In hand-held small arms, the shooter will apply this force using their own body, resulting in a noticeable impulse commonly referred to as a "kick". In heavier mounted guns, such as heavy machine guns or artillery pieces, recoil momentum is transferred through the platform on which the weapon is mounted. Practical weight gun mounts are typically not strong enough to withstand the maximum forces accelerating the gun during the short time the projectile is in the barrel. To mitigate these large recoil forces, recoil buffering mechanisms spread out the counter-recoiling force over a longer time, typically ten to a hundred times longer than the duration of the forces accelerating the projectile. This results in the required counter-recoiling force being proportionally lower, and easily absorbed by the gun mount.
To apply this counter-recoiling force, modern mounted guns may employ recoil buffering comprising springs and hydraulic recoil mechanisms, similar to shock-absorbing suspension on automobiles. Early cannons used systems of ropes along with rolling or sliding friction to provide forces to slow the recoiling cannon to a stop. Recoil buffering allows the maximum counter-recoil force to be lowered so that strength limitations of the gun mount are not exceeded.

Contribution of propellant gasses

Modern cannons also employ muzzle brakes very effectively to redirect some of the propellant gasses rearward after projectile exit. This provides a counter-recoiling force to the barrel, allowing the buffering system and gun mount to be more efficiently designed at even lower weight.
Propellant gases are even more tapped in recoilless guns, where much of the high pressure gas remaining in the barrel after projectile exit is vented rearward though a nozzle at the back of the chamber, creating a large counter-recoiling force sufficient to eliminate the need for heavy recoil mitigating buffers on the mount.

Hand-held guns

The same physics principles affecting recoil in mounted guns also applies to hand-held guns. However, the shooter's body assumes the role of gun mount, and must similarly dissipate the gun's recoiling momentum over a longer period of time than the bullet travel-time in the barrel, in order not to injure the shooter. Hands, arms and shoulders have considerable strength and elasticity for this purpose, up to certain practical limits. Nevertheless, "perceived" recoil limits vary from shooter to shooter, depending on body size, the use of recoil padding, individual pain tolerance, the weight of the firearm, and whether recoil buffering systems and muzzle devices are employed. For this reason, establishing recoil safety standards for small arms remains challenging, in spite of the straightforward physics involved.

Physics: momentum, energy and impulse

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.
Momentum is simply mass multiplied by velocity. Velocity is speed in a particular direction. In a very technical sense, speed is a scalar : a magnitude; while velocity is a vector : magnitude and direction. Momentum is conservative: any change in momentum of an object requires an equal and opposite change of some other objects. Hence the recoil: imparting momentum to the projectile requires imparting opposite momentum to the gun.
A change in the momentum of a mass requires the application of a force. Forces within a firearm wildly change, so what matters is impulse: the change of momentum is equal to the impulse. The rapid change of velocity of the gun is a shock and will be countered as if by a shock absorber.
Energy in firing a firearm comes in many forms but for understanding recoil what matters is kinetic energy, which is half mass multiplied by squared speed. For the recoiling gun, this means that for a given rearward momentum, doubling the mass halves the speed and also halves the kinetic energy of the gun, making it easier to dissipate.

Momentum

If all the masses and velocities involved are accounted for, the vector sum, magnitude and direction, of the momentum of all the bodies involved does not change; that is, momentum of the system is conserved. This conservation of momentum is why gun recoil occurs in the opposite direction of bullet projection—the mass times velocity of the projectile in the positive direction equals the mass times velocity of the gun in the negative direction. In summation, the total momentum of the system ) equals zero just as it did before the trigger was pulled.
From a practical engineering perspective, therefore, through the mathematical application of conservation of momentum, it is possible to calculate a first approximation of a gun's recoil momentum and kinetic energy simply based on estimates of the projectile speed coming out the barrel. And then to properly design recoil buffering systems to safely dissipate that momentum and energy. To confirm analytical calculations and estimates, once a prototype gun is manufactured, the projectile and gun recoil energy and momentum can be directly measured using a ballistic pendulum and ballistic chronograph.
The nature of the recoil process is determined by the force of the expanding gases in the barrel upon the gun, which is equal and opposite to the force upon the ejecta. It is also determined by the counter-recoil force applied to the gun. The recoil force only acts during the time that the ejecta are still in the barrel of the gun. The counter-recoil force is generally applied over a longer time period and adds forward momentum to the gun equal to the backward momentum supplied by the recoil force, in order to bring the gun to a halt. There are two special cases of counter recoil force: Free-recoil, in which the time duration of the counter-recoil force is very much larger than the duration of the recoil force, and zero-recoil, in which the counter-recoil force matches the recoil force in magnitude and duration. Except for the case of zero-recoil, the counter-recoil force is smaller than the recoil force but lasts for a longer time. Since the recoil force and the counter-recoil force are not matched, the gun will move rearward, slowing down until it comes to rest. In the zero-recoil case, the two forces are matched and the gun will not move when fired. In most cases, a gun is very close to a free-recoil condition, since the recoil process generally lasts much longer than the time needed to move the ejecta down the barrel. An example of near zero-recoil would be a gun securely clamped to a massive or well-anchored table, or supported from behind by a massive wall. However, employing zero-recoil systems is often neither practical nor safe for the structure of the gun, as the recoil momentum must be absorbed directly through the very small distance of elastic deformation of the materials the gun and mount are made from, perhaps exceeding their strength limits. For example, placing the butt of a large caliber gun directly against a wall and pulling the trigger risks cracking both the gun stock and the surface of the wall.
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. Assuming a near free-recoil condition, and neglecting the gases ejected from the barrel,, then 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 is the momentum of the firearm and 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:
where:

is the mass of the firearm
is the velocity of the firearm immediately after firing
is the mass of the projectile
is the velocity of the projectile immediately after firing

A force integrated over the time period during which it acts will yield the momentum supplied by that force. The counter-recoil force must supply enough momentum to the firearm to bring it to a halt. This means that:
where:

is the counter-recoil force as a function of time
is duration of the counter-recoil force

A similar equation can be written for the recoil force on the firearm:
where:

is the recoil force as a function of time
is duration of the recoil force

Assuming the forces are somewhat evenly spread out over their respective durations, the condition for free-recoil is, while for zero-recoil,