Free Recoil Energy

We will assume that we can ignore the forces the shooter exerts on the gun between the time the primer is ignited and the shot leaves the barrel. If this is the case, then the total linear momentum of the shot, wad, powder and gun is conserved (that is, it doesn't change), so

(M_s + M_W)V_s + M_pV_p = M_gV_g

or

(W_s + W_W)V_s + W_pV_p = W_gV_g

where

• W = M × g, or weight = mass × g (gravitational constant)
• M_s = mass of shot
• W_s = weight of shot in grains (7000 gr = 1 lb) = M_sg / 7000
• M_w = mass of wad
• W_w = weight of wad in grains = M_wg / 7000
• V_s = muzzle speed of wad and shot
• M_p = mass of powder
• W_p = weight of powder in grains = M_pg / 7000
• V_p = expansion velocity of the now-unconstrained gas produced by the powder, commonly taken to be 4700 fps, the speed of gas expansion under standard conditions
• M_g = mass of gun
• W_g = weight of gun in pounds = M_gg
• V_g = speed at which gun moves backward
• g = gravitational constant, 32.174 ft/s²

solving for the gun speed:

V_g = [(W_s + W_W)V_s + W_pV_p] / (7000 W_g)

This is the speed at which the gun begins to push back against your shoulder. A better measure of how much work the gun does on your shoulder is the kinetic energy of the gun immediately after firing. The kinetic energy of the gun is ½ M_gV_g², also called the free recoil energy ("free" since we ignored the shooter standing behind the gun), is

Free recoil energy	= ½ MgVg2
	= WgVg2 / (2 g)
	= [(Ws + WW)Vs + WpVp]2 / (2 × 70002 × Wg × g)

This is the amount of work that the gun exerts on the shooter because of recoil.