## Recoil

### Recoil Velocity

The forward momentum (sometimes written as impulse), that of the bullet and powder charge are given by

MF = MB*VB + MC*VC

Because momentum is conserved, this is equal to the rearward momentum of the firearm:

MR = MF*VF

We can set these two equations equal to each other and solve for the velocity of the firearm, yielding

VF = (MB*VB + MC*VC)/MF

### Recoil Energy

Once the velocity is found, the recoil energy can be found by solving

EF = 1/2 MF VF2

### Example

Assume a seven pound rifle using 40 grains of powder to shoot a 168 grain bullet at 2700 feet/sec. First find the mass of the bullet, charge and rifle

MB = 168 gr/(32.2 f/s2*7000 gr/lb) = 0.000745 slugs (unit of mass)
MC = 40 gr/(32.2 f/s2*7000 gr/lb) = 0.000177 slugs
MF = 7 lb/(32.2 f/s2) = 0.217 slugs

The constant 32.2 is the acceleration due to gravity, g, and is used to convert weight to mass:

Mass = Weight/g

The free recoil velocity can now be found

VF = (0.000745 slugs * 2700 f/s + 0.000177 slugs * 4700 f/s)/(0.217 slugs) = 13.1 f/s

And the free recoil energy is given by

EF = 0.5 * 0.217 slugs * (13.1 f/s)2 = 18.6 ft•lbs

### Variables

 MB Bullet mass VB Bullet velocity MC Charge mass VC Charge velocity MF Firearm mass VF Firearm velocity EF Firearm energy g Acceleration due to gravity