A bullet of mass m leaves a gun of mass M kept on a smooth horizontal surface . If the speed of the bullet relative to the gun is v , the magnitude of recoil speed of the gun will be

Options

A. `m/Mv`
B. `m/(M+m)v`
C. `m/(M-m)v`
D. `M/mv`

Answer

To solve the problem, we will use the principle of conservation of momentum. Here are the steps to find the magnitude of the recoil speed of the gun:

### Step 1: Understand the system
We have a bullet of mass \( m \) leaving a gun of mass \( M \) on a smooth horizontal surface. The bullet moves with a speed \( v \) relative to the gun.

### Step 2: Define the velocities
Let:
- \( V \) be the recoil speed of the gun (which we need to find).
- The speed of the bullet with respect to the ground will be \( v + V \) (since the bullet moves forward while the gun recoils backward).

### Step 3: Apply the conservation of momentum
Since there are no external forces acting on the system, the total momentum before and after the bullet is fired must be conserved. Initially, both the gun and bullet are at rest, so the initial momentum is zero.

The momentum after the bullet is fired can be expressed as:
\[ \text{Final Momentum} = \text{Momentum of bullet} + \text{Momentum of gun} \]
\[ 0 = mv + MV \]
Here, the momentum of the bullet is \( m(v + V) \) and the momentum of the gun is \( -MV \) (negative because it is in the opposite direction).

### Step 4: Set up the equation
From the conservation of momentum:
\[ m(v + V) - MV = 0 \]
This simplifies to:
\[ mv + mV - MV = 0 \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ mV - MV = -mv \]
\[ (m - M)V = -mv \]

### Step 6: Solve for \( V \)
Now, isolate \( V \):
\[ V = \frac{mv}{m + M} \]

### Conclusion
Thus, the magnitude of the recoil speed of the gun is:
\[ V = \frac{mv}{M + m} \]