A massive wooden plate of unknown mass M remains is equilibrium in vacuum when n bullets are fired per second on it. The mass of each bullet is `m ( M gt gt m )` e, then M is equal to

Options

A. `( mvn )/( g)`
B. `( mevn)/(g)`
C. `( m v ( 1+e ) n )/(g)`
D. nm

Answer

To solve the problem, we need to analyze the situation where a massive wooden plate of unknown mass \( M \) is in equilibrium while \( n \) bullets, each of mass \( m \), are fired at it per second. The mass of the plate is much greater than the mass of a single bullet (\( M \gg m \)).

### Step-by-Step Solution:

1. **Understand the System**:
- We have a wooden plate with mass \( M \).
- Bullets of mass \( m \) are fired at the plate at a rate of \( n \) bullets per second.

2. **Momentum Consideration**:
- When bullets hit the plate, they impart momentum to the plate.
- The change in momentum per second due to \( n \) bullets can be expressed as:
\[ \text{Change in momentum} = n \cdot m \cdot (V_f - V_i) \]
- Here, \( V_f \) is the final velocity of the plate after bullets hit it, and \( V_i \) is the initial velocity (which is 0 since the plate is initially at rest).

3. **Coefficient of Restitution**:
- The coefficient of restitution \( e \) relates the velocities before and after the collision:
\[ e = \frac{V_f}{V_i} \]
- Since \( V_i = 0 \), we can simplify this to:
\[ V_f = e \cdot V_i \]

4. **Force Acting on the Plate**:
- The force acting on the plate due to the bullets can be expressed as:
\[ F = n \cdot m \cdot V_f \]
- Since \( V_f \) is the velocity after the collision, we can substitute \( V_f \) into the force equation.

5. **Equilibrium Condition**:
- For the plate to remain in equilibrium, the force due to the bullets must equal the weight of the plate:
\[ n \cdot m \cdot V_f = M \cdot g \]

6. **Substituting for \( V_f \)**:
- From the earlier steps, we have:
\[ V_f = e \cdot V_i \]
- Substituting this into the force equation gives:
\[ n \cdot m \cdot (e \cdot V_i) = M \cdot g \]

7. **Solving for \( M \)**:
- Rearranging the equation to solve for \( M \):
\[ M = \frac{n \cdot m \cdot e \cdot V_i}{g} \]

8. **Final Expression**:
- Since \( V_i \) is not defined in the problem, we can express \( M \) in terms of the known quantities:
\[ M = \frac{n \cdot m \cdot (1 + e)}{g} \]

### Final Answer:
Thus, the mass \( M \) of the wooden plate is given by:
\[ M = \frac{n \cdot m \cdot (1 + e)}{g} \]