Consider a gun of mass m (when unloaded) that fires a shell of mass m with muzzle speed v. (that is, the shell's speed relative to the gun is v.) assuming that the gun is completely free to recoil (no external forces on gun or shell), use conservation of momentum to show that the shell's speed relative to the ground is v /(1 m/m).

M = - v m / M

Explanation:

Let's use the law of conservation for now, the system is formed by the cannon plus the bullet, at all times

Moment before shooting

     po = 0

It is zero because the elements have no speed

Moment after shooting

     pf = Mvg + m v

the moment is conserved, by all internal forces    

     po = pf

       0 = M + m v

      M = - v m / M

The negative sign indicates that the cannon recedes