A ball collides impinges directly on a similar ball at rest. The first ball is brought to rest after the impact. If half of the kinetic energy is lost by impact, the value of coefficient of restitution `( e)` is

Explanation

To solve the problem, we need to analyze the collision between two identical balls, where one ball is initially moving and the other is at rest. The first ball comes to rest after the collision, and we are given that half of the kinetic energy is lost during the impact. We need to find the coefficient of restitution \( e \).

### Step-by-Step Solution:

1. **Define the Variables:**
Let:
- Mass of both balls = \( m \)
- Initial velocity of the first ball (moving ball) = \( u \)
- Initial velocity of the second ball (at rest) = \( 0 \)
- Final velocity of the first ball (after collision) = \( 0 \) (it comes to rest)
- Final velocity of the second ball = \( v \)

2. **Calculate Initial Kinetic Energy:**
The initial kinetic energy (KE_initial) of the first ball is given by:
\[ KE_{\text{initial}} = \frac{1}{2} m u^2 \]

3. **Calculate Final Kinetic Energy:**
After the collision, the first ball is at rest, and the second ball moves with velocity \( v \). The final kinetic energy (KE_final) is:
\[ KE_{\text{final}} = \frac{1}{2} m v^2 \]

4. **Determine the Kinetic Energy Lost:**
According to the problem, half of the initial kinetic energy is lost:
\[ KE_{\text{lost}} = \frac{1}{2} KE_{\text{initial}} = \frac{1}{2} \left( \frac{1}{2} m u^2 \right) = \frac{1}{4} m u^2 \]

5. **Relate Initial and Final Kinetic Energies:**
The kinetic energy lost can also be expressed as:
\[ KE_{\text{lost}} = KE_{\text{initial}} - KE_{\text{final}} = \frac{1}{2} m u^2 - \frac{1}{2} m v^2 \]
Setting these equal gives:
\[ \frac{1}{4} m u^2 = \frac{1}{2} m u^2 - \frac{1}{2} m v^2 \]

6. **Simplify the Equation:**
Dividing through by \( \frac{1}{2} m \) (assuming \( m \neq 0 \)):
\[ \frac{1}{2} u^2 = u^2 - v^2 \]
Rearranging gives:
\[ v^2 = u^2 - \frac{1}{2} u^2 = \frac{1}{2} u^2 \]

7. **Find Final Velocity \( v \):**
Taking the square root:
\[ v = \sqrt{\frac{1}{2}} u = \frac{u}{\sqrt{2}} \]

8. **Calculate the Coefficient of Restitution \( e \):**
The coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{relative velocity after collision}}{\text{relative velocity before collision}} \]
Before the collision, the relative velocity is \( u - 0 = u \).
After the collision, the relative velocity is \( 0 - v = -v \). Thus:
\[ e = \frac{-v}{u} = \frac{-\left(-\frac{u}{\sqrt{2}}\right)}{u} = \frac{1}{\sqrt{2}} \]

### Final Answer:
The value of the coefficient of restitution \( e \) is:
\[ e = \frac{1}{\sqrt{2}} \]