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 and the relationship between the velocities of the bullet and the gun. Here are the steps to derive the magnitude of the recoil speed of the gun:

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Mass of the bullet = \( m \)
- Mass of the gun = \( M \)
- Speed of the bullet relative to the gun = \( v \)

2. **Define the Directions:**
- Assume the positive direction is the direction in which the bullet moves when fired.
- The gun will recoil in the opposite direction, which we will consider as negative.

3. **Establish the Relationship of Velocities:**
- Let \( V_g \) be the recoil speed of the gun (which will be negative).
- Let \( V_b \) be the speed of the bullet after it leaves the gun.
- According to the relative speed, we have:
\[ v = V_b - V_g \]
- Since the gun recoils backward, we can express \( V_g \) as negative:
\[ v = V_b - (-V_g) \implies v = V_b + V_g \]

4. **Use Conservation of Momentum:**
- Initially, the system (gun + bullet) is at rest, so the initial momentum is 0.
- After firing, the final momentum must also equal 0:
\[ \text{Final Momentum} = \text{Momentum of the bullet} + \text{Momentum of the gun} \]
\[ 0 = m V_b - M V_g \]
- Rearranging gives:
\[ m V_b = M V_g \implies V_g = \frac{m}{M} V_b \]

5. **Substitute for \( V_b \):**
- From the earlier relationship \( v = V_b + V_g \), substitute \( V_g \):
\[ v = V_b + \frac{m}{M} V_b \]
- Factor out \( V_b \):
\[ v = V_b \left(1 + \frac{m}{M}\right) \]
- Solve for \( V_b \):
\[ V_b = \frac{v}{1 + \frac{m}{M}} = \frac{v M}{M + m} \]

6. **Find the Recoil Speed of the Gun:**
- Substitute \( V_b \) back into the equation for \( V_g \):
\[ V_g = \frac{m}{M} V_b = \frac{m}{M} \left(\frac{v M}{M + m}\right) \]
- Simplifying gives:
\[ V_g = \frac{m v}{M + m} \]

7. **Magnitude of the Recoil Speed:**
- The magnitude of the recoil speed of the gun is:
\[ |V_g| = \frac{m v}{M + m} \]

### Final Answer:
The magnitude of the recoil speed of the gun is:
\[ \boxed{\frac{m v}{M + m}} \]