A ball of mass M strikes another smooth ball of mass m at rest. If they separate in mutually perpendicular directions, then the coefficient of impact (e) is :

Options

A. `M/m`
B. `m/M`
C. `m/(2M)`
D. Zero

Explanation

To solve the problem, we need to analyze the collision between two balls, one of mass \( M \) that strikes another ball of mass \( m \) at rest. After the collision, they move in mutually perpendicular directions. We will find the coefficient of restitution \( e \).

### Step-by-Step Solution:

1. **Understanding the Collision**:
- Let the mass \( M \) be moving with an initial velocity \( u_1 \) towards the mass \( m \) which is at rest (initial velocity \( u_2 = 0 \)).
- After the collision, let the velocity of mass \( M \) be \( v_2 \) and the velocity of mass \( m \) be \( v_1 \). They move in perpendicular directions.

2. **Coefficient of Restitution Formula**:
- The coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \]
- Mathematically, this can be expressed as:
\[ e = \frac{v_2 - v_1}{u_1 - u_2} \]

3. **Substituting Values**:
- Since \( u_2 = 0 \), the formula simplifies to:
\[ e = \frac{v_2 - v_1}{u_1} \]
- Since \( v_1 \) is in the perpendicular direction, we can assume \( v_1 = 0 \) for the purpose of calculating the coefficient of restitution in the direction of \( u_1 \).

4. **Conservation of Momentum**:
- Since the collision is elastic and momentum is conserved, we can write the momentum conservation equations:
\[ M \cdot u_1 + m \cdot 0 = M \cdot v_2 + m \cdot v_1 \]
- Rearranging gives:
\[ M \cdot u_1 = M \cdot v_2 + m \cdot v_1 \]

5. **Solving for Velocities**:
- Since \( v_1 \) is perpendicular to \( v_2 \), we can express the velocities in terms of each other. However, we can simplify our analysis by noting that in a perfectly elastic collision, the velocities relate directly to their masses.
- From the conservation of momentum, we can also deduce that:
\[ v_2 = \frac{M}{m} \cdot u_1 \]
- Thus, substituting \( v_2 \) back into the coefficient of restitution formula:
\[ e = \frac{v_2}{u_1} = \frac{M}{m} \]

6. **Final Result**:
- Therefore, the coefficient of restitution \( e \) is:
\[ e = \frac{M}{m} \]

### Conclusion:
The coefficient of impact \( e \) is \( \frac{M}{m} \).