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

Explanation

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

### Step 1: Understand the situation
We have two blocks:
- Block A with mass \( m \) is moving with an initial velocity \( v \).
- Block B with mass \( 4m \) is stationary (initial velocity = 0).

After the collision, Block A comes to rest, meaning its final velocity is \( 0 \).

### Step 2: Apply the conservation of momentum
The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision.

**Initial momentum:**
\[ \text{Initial momentum} = m \cdot v + 4m \cdot 0 = mv \]

**Final momentum:**
\[ \text{Final momentum} = m \cdot 0 + 4m \cdot v_f = 4m \cdot v_f \]

Setting initial momentum equal to final momentum:
\[ mv = 4m \cdot v_f \]

### Step 3: Simplify the equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ v = 4v_f \]

From this, we can solve for \( v_f \):
\[ v_f = \frac{v}{4} \]

### Step 4: Calculate the coefficient of restitution
The coefficient of restitution \( e \) is defined as:
\[ e = \frac{v_2 - v_1}{u_1 - u_2} \]
where:
- \( u_1 \) is the initial velocity of Block A (which is \( v \)),
- \( u_2 \) is the initial velocity of Block B (which is \( 0 \)),
- \( v_1 \) is the final velocity of Block A (which is \( 0 \)),
- \( v_2 \) is the final velocity of Block B (which we found to be \( \frac{v}{4} \)).

Substituting these values into the equation:
\[ e = \frac{\frac{v}{4} - 0}{v - 0} = \frac{\frac{v}{4}}{v} \]

### Step 5: Simplify the expression for \( e \)
\[ e = \frac{1}{4} \]

### Final Answer
The coefficient of restitution \( e \) is \( \frac{1}{4} \) or \( 0.25 \).

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