Question

When you fire a shot gun, you feel that you are pushed back by the gun. Can you explain why this happens? is the recoil velocity of the gun equal to the velocity of the bullet? If not is there any other quantity that is equal for the bullet and the gun?

Answer 1

This is a prime example of newtons 3rd law, for every action there is an equal and opposite reaction. This means the shotgun hits your shoulder with the same energy as the bullet hits the target. The stock has enough mass though that the energy is distributed evenly to the surface area of your shoulder touching the gun.

Answer 2

The recoils is caused by the gun trying to balance out the momentum of the bullet and the gas being discharged. The gas helps balance out the momentum of the bullet a little, but the recoil is still needed for the complete balance. This need for balance is proved by Newtons third law. 

The formula for finding the velocity of recoil is as follows:
(Weight of Bullet * Velocity of Bullet + Weight of Gases * Velocity of Gases) / (7000 * Weight of Gun)

Hope this helped!