Introduction:
When a bullet is fired from a gun, the gun experiences a recoil due to the conservation of momentum. This recoil can be used to compress a spring. In this scenario, we will calculate the velocity of the bullet after it is fired based on the mass of the bullet, the mass of the gun, the force constant of the spring, and the distance the spring is compressed.

Given:
Mass of the bullet (m)
Mass of the gun (M)
Force constant of the spring (K)
Distance the spring is compressed (d)

Conservation of momentum:
When the bullet is fired, the momentum of the bullet and the gun before firing is zero since they are initially at rest. Therefore, the total momentum after firing should also be zero to conserve momentum.

Recoil velocity of the gun:
Let the recoil velocity of the gun be v.
According to the conservation of momentum, we have:
m * v = -M * v
Since the masses are opposite in sign, the negative sign is included to indicate opposite directions of motion.

Energy stored in the spring:
The energy stored in the spring is given by the formula:
Potential energy (PE) = (1/2) * K * d^2
This energy comes from the kinetic energy of the recoiling gun.

Equating the energy:
The kinetic energy of the recoiling gun is equal to the potential energy stored in the spring:
(1/2) * M * v^2 = (1/2) * K * d^2
Simplifying the equation, we have:
M * v^2 = K * d^2

Calculating the velocity of the bullet:
The final velocity of the bullet can be calculated using the equation:
m * v_bullet = -M * v
Rearranging the equation, we get:
v_bullet = -(M / m) * v

Summary:
To calculate the velocity of the bullet, we first determine the recoil velocity of the gun using the conservation of momentum. Then, we equate the kinetic energy of the recoiling gun to the potential energy stored in the spring. By rearranging the equation, we can calculate the velocity of the bullet using the mass of the bullet, the mass of the gun, and the recoil velocity of the gun.