A ball of mass `m` moving at a speed `v` collides with another ball of mass `3m` at rest. The lighter block comes to rest after collisoin. The coefficient of restitution is

Options

A. (a) `1/2`
B. (b) `2/3`
C. (c) `1/4`
D. (d) None of these

Explanation

To solve the problem, we need to find the coefficient of restitution for the collision between a ball of mass `m` moving at speed `v` and another ball of mass `3m` at rest. After the collision, the lighter ball comes to rest.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Mass of the first ball (lighter ball) = `m`
- Initial velocity of the first ball = `v`
- Mass of the second ball (heavier ball) = `3m`
- Initial velocity of the second ball = `0` (at rest)

2. **Determine the Final Conditions:**
- After the collision, the lighter ball comes to rest, so its final velocity = `0`.
- Let the final velocity of the heavier ball be `v'`.

3. **Apply the Conservation of Linear Momentum:**
- According to the law of conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
- Initial momentum = \( m \cdot v + 3m \cdot 0 = mv \)
- Final momentum = \( m \cdot 0 + 3m \cdot v' = 3mv' \)
- Setting the initial momentum equal to the final momentum:
\[ mv = 3mv' \]
- Dividing both sides by `m` (assuming `m` is not zero):
\[ v = 3v' \]
- Therefore, we can express the final velocity of the heavier ball as:
\[ v' = \frac{v}{3} \]

4. **Calculate the Coefficient of Restitution (e):**
- The coefficient of restitution is defined as:
\[ e = \frac{\text{Velocity of separation}}{\text{Velocity of approach}} \]
- The velocity of approach (before collision) is the relative speed of the lighter ball towards the heavier ball:
\[ \text{Velocity of approach} = v - 0 = v \]
- The velocity of separation (after collision) is the relative speed of the heavier ball moving away from the lighter ball:
\[ \text{Velocity of separation} = v' - 0 = v' = \frac{v}{3} \]
- Substituting these values into the coefficient of restitution formula:
\[ e = \frac{\frac{v}{3}}{v} = \frac{1}{3} \]

5. **Final Answer:**
- The coefficient of restitution \( e \) is \( \frac{1}{3} \).