A gun of mass \( M \) fires a bullet of mass \( m \) with a muzzle velocity \( v \) relative to the gun barrel. Assuming the system is initially at rest on a smooth horizontal surface, the recoil speed of the gun relative to the ground is:

Let \( V \) be the recoil speed of the gun relative to the ground in the backward direction. Since the muzzle velocity of the bullet relative to the gun is \( v \), the velocity of the bullet relative to the ground is \( v - V \) in the forward direction. Applying the principle of conservation of linear momentum for the isolated system (initial momentum = 0): \( m(v - V) - MV = 0 \implies mv - mV - MV = 0 \implies mv = (M + m)V \implies V = \frac{mv}{M + m} \). Thus, the correct option is 4.