A ball of mass `2m` impinges directly on a ball of mass m, which is at rest. If the velocity with which the larger ball impringes be equal to the velocity of the smaller mass after impact then the coefficient of restitution :-

Options

A. `(1)/(3)`
B. `(3)/(4)`
C. `(1)/(2)`
D. `(2)/(5)`

Explanation

To solve the problem, we will use the principles of conservation of momentum and the definition of the coefficient of restitution.

### Step-by-Step Solution:

1. **Identify the masses and initial velocities:**
- Let the mass of the first ball (larger ball) be \(2m\) and its initial velocity be \(V\).
- Let the mass of the second ball (smaller ball) be \(m\) and its initial velocity be \(0\) (at rest).

2. **Set up the conservation of momentum equation:**
- The total initial momentum before the collision is given by:
\[ \text{Initial Momentum} = (2m)V + (m)(0) = 2mV \]
- Let \(V'\) be the final velocity of the larger ball after the collision, and we know that the final velocity of the smaller ball is \(V\) (as given in the problem).
- The total final momentum after the collision is:
\[ \text{Final Momentum} = (2m)V' + (m)V \]

3. **Apply conservation of momentum:**
- According to the conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
- Therefore, we have:
\[ 2mV = (2m)V' + (m)V \]

4. **Simplify the equation:**
- Dividing the entire equation by \(m\) (assuming \(m \neq 0\)):
\[ 2V = 2V' + V \]
- Rearranging gives:
\[ 2V' = 2V - V \]
\[ 2V' = V \]
- Thus, we find:
\[ V' = \frac{V}{2} \]

5. **Determine the coefficient of restitution (e):**
- The coefficient of restitution \(e\) is defined as the ratio of the relative velocity of separation to the relative velocity of approach.
- The relative velocity of separation after the collision is:
\[ \text{Relative Velocity of Separation} = V - V' = V - \frac{V}{2} = \frac{V}{2} \]
- The relative velocity of approach before the collision is:
\[ \text{Relative Velocity of Approach} = V - 0 = V \]
- Therefore, the coefficient of restitution \(e\) is given by:
\[ e = \frac{\text{Relative Velocity of Separation}}{\text{Relative Velocity of Approach}} = \frac{\frac{V}{2}}{V} = \frac{1}{2} \]

### Final Answer:
The coefficient of restitution \(e\) is \(\frac{1}{2}\).

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