Recoils of gun:

         Consider a system of gun and a bullet. Before firing the gun, both the gun and build are at rest, so the total momentum of the system is zero. As the gun is fired, a bullet shoots out the gun and acquires momentum. To conserve the momentum of the system, the gun recoils.

            According to the law of conservation of momentum, the total momentum of the gun and the bullet will also be zero after the gun is fired. Let m be the mass of the gun and v be its velocity on firing the gun, M be the mass of the gun and V be the velocity with which it recoils. Thus the total momentum of the gun and the bullet after the gun is fired will be.

[total momentum of the gun and the bullet after the un is fired]= MV + mv……(i)

According to the law of conservation of momentum,

 (total momentum of the gun and the bullet=  (total momentum of the gun and the after the gun is fired)  bullet before the gun is fired)

       MV+ mv=0

Or            MV = -mv

Hence V=$\frac{m}{M}v$ ......(ii)

Equation (ii) gives the velocity V of the gun. The negative sign indicates that the velocity of the gun is opposite to the velocity of the bullet i.e, the gun recoils. Since the mass of the gun is much larger than the bullet, therefore the recoil is much smaller than the velocity of the bullet.