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 step by step, we will follow the principles of conservation of momentum and the definition of the coefficient of restitution.

### Step 1: Understand the scenario
We have two balls:
- Ball A (mass = 2m) is moving with velocity \( v \).
- Ball B (mass = m) is at rest (velocity = 0).

### Step 2: Apply the conservation of momentum
According to the law of conservation of momentum, the total momentum before the collision must equal the total momentum after the collision.

**Before the collision:**
- Momentum of Ball A = \( 2m \cdot v \)
- Momentum of Ball B = \( m \cdot 0 = 0 \)

Total initial momentum = \( 2m \cdot v + 0 = 2mv \)

**After the collision:**
Let the velocity of Ball A after the collision be \( v' \) and the velocity of Ball B after the collision be \( v \) (as given in the problem).

- Momentum of Ball A after collision = \( 2m \cdot v' \)
- Momentum of Ball B after collision = \( m \cdot v \)

Total final momentum = \( 2m \cdot v' + m \cdot v \)

Setting the initial momentum equal to the final momentum:

\[ 2mv = 2mv' + mv \]

### Step 3: Simplify the equation
We can simplify the equation by dividing everything by \( m \):

\[ 2v = 2v' + v \]

Rearranging gives:

\[ 2v - v = 2v' \]
\[ v = 2v' \]

### Step 4: Solve for \( v' \)
From the equation \( v = 2v' \), we can express \( v' \):

\[ v' = \frac{v}{2} \]

### Step 5: Calculate the coefficient of restitution
The coefficient of restitution \( e \) is defined as the ratio of the relative velocity of separation to the relative velocity of approach.

**Relative velocity of approach:**
Before the collision, the relative velocity of approach is simply the velocity of Ball A since Ball B is at rest:

\[ \text{Relative velocity of approach} = v - 0 = v \]

**Relative velocity of separation:**
After the collision, the relative velocity of separation is:

\[ \text{Relative velocity of separation} = v' - 0 = v' \]

Substituting \( v' = \frac{v}{2} \):

\[ \text{Relative velocity of separation} = \frac{v}{2} \]

### Step 6: Calculate the coefficient of restitution
Now, substituting these values into the formula for the coefficient of restitution:

\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} = \frac{\frac{v}{2}}{v} \]

Simplifying this gives:

\[ e = \frac{1}{2} \]

### Conclusion
The coefficient of restitution is \( \frac{1}{2} \).