Laws Of Motion

For a rolling ring, the total mechanical energy is the sum of translational and rotational kinetic energies.
Moment of inertia of a ring about its center is , and angular velocity .

This is a perfectly inelastic collision, where both bodies stick together after collision.
Using conservation of linear momentum: So, the velocity of the compound body after the collision is zero.

Using the principle of conservation of momentum, the total momentum before and after firing should remain zero (assuming the system was initially at rest).
Let (mass of gun), (mass of projectile), (velocity of projectile), and be the recoil velocity of the gun.