A moving block having mass `m` , collides with another stationary block having mass `4m`. The lighter block comes to rest after collision. When the initial velocity of the block is `v`, then the value of coefficient of restitution `( e)` will be

Options

A. 0.8
B. 0.25
C. 0.5
D. 0.4

Answer

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:**
- Mass of the moving block (m) = m
- Mass of the stationary block (4m) = 4m
- Initial velocity of the moving block (u1) = v
- Initial velocity of the stationary block (u2) = 0

2. **Apply the conservation of linear momentum:**
The total initial momentum before the collision must equal the total final momentum after the collision.
\[ \text{Initial momentum} = \text{Final momentum} \]
\[ mu_1 + 4mu_2 = mv_1' + 4mv_2' \]
Substituting the values:
\[ mv + 4m(0) = mv' + 4m(v_2') \]
Simplifying:
\[ mv = mv' + 4mv_2' \]

3. **Determine the final velocities:**
Since the lighter block (mass m) comes to rest after the collision, its final velocity \(v_1' = 0\).
Substitute \(v_1' = 0\) into the momentum equation:
\[ mv = 0 + 4mv_2' \]
This simplifies to:
\[ mv = 4mv_2' \]
Dividing both sides by \(m\) (assuming \(m \neq 0\)):
\[ v = 4v_2' \]
Thus, we find:
\[ v_2' = \frac{v}{4} \]

4. **Calculate the coefficient of restitution (e):**
The coefficient of restitution is defined as:
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \]
- **Relative velocity of separation:** After the collision, the lighter block is at rest (0), and the heavier block moves with velocity \(v_2' = \frac{v}{4}\). Thus, the relative velocity of separation is:
\[ \text{Relative velocity of separation} = v_2' - 0 = \frac{v}{4} \]
- **Relative velocity of approach:** Before the collision, the relative velocity is:
\[ \text{Relative velocity of approach} = v - 0 = v \]

5. **Substituting values into the coefficient of restitution formula:**
\[ e = \frac{\frac{v}{4}}{v} = \frac{1}{4} \]

### Final Answer:
The value of the coefficient of restitution \(e\) is \(\frac{1}{4}\).