A body `X` with a momentum `p` collides with another identical stationary body `Y` one dimensionally. During the collision, `Y` gives an impulse `J` to body `X`. Then coefficient of restitution is

Options

A. `(2J)/p-1`
B. `J/P+1`
C. `J/P-1`
D. `J/(2P)-1`

Explanation

To solve the problem, we need to find the coefficient of restitution (e) after a collision between two identical bodies, where one body (X) has an initial momentum (p) and the other body (Y) is stationary. During the collision, body Y imparts an impulse (J) to body X.

### Step-by-Step Solution:

1. **Understanding Initial Conditions:**
- Body X has an initial momentum \( p \).
- Body Y is stationary, so its initial momentum is \( 0 \).
- The masses of both bodies are identical, let's denote the mass as \( m \).

2. **Momentum Before Collision:**
- Total initial momentum of the system:
\[ p_{\text{initial}} = p + 0 = p \]

3. **Momentum After Collision:**
- After the collision, body X will lose momentum due to the impulse \( J \) imparted by body Y. Therefore, the momentum of body X after the collision is:
\[ p_X' = p - J \]
- Body Y gains the same amount of momentum \( J \) due to the impulse, so its momentum after the collision is:
\[ p_Y' = 0 + J = J \]

4. **Total Momentum After Collision:**
- Total momentum after collision:
\[ p_{\text{final}} = (p - J) + J = p \]
- This confirms that momentum is conserved.

5. **Finding Velocities After Collision:**
- The velocity of body X after the collision:
\[ v_X' = \frac{p - J}{m} \]
- The velocity of body Y after the collision:
\[ v_Y' = \frac{J}{m} \]

6. **Finding Velocity of Approach:**
- Before the collision, the velocity of body X was:
\[ v_X = \frac{p}{m} \]
- Since body Y was stationary, its velocity was \( 0 \).

7. **Finding Velocity of Separation:**
- The velocities after the collision indicate that body X is moving in the opposite direction (since it loses momentum) and body Y is moving in the positive direction. Thus, the velocity of separation is:
\[ v_{\text{separation}} = v_Y' - v_X' = \frac{J}{m} - \frac{p - J}{m} = \frac{J - (p - J)}{m} = \frac{2J - p}{m} \]

8. **Coefficient of Restitution (e):**
- The coefficient of restitution is defined as the ratio of the velocity of separation to the velocity of approach:
\[ e = \frac{v_{\text{separation}}}{v_{\text{approach}}} \]
- Substituting the values:
\[ e = \frac{\frac{2J - p}{m}}{\frac{p}{m}} = \frac{2J - p}{p} \]
- Rearranging gives:
\[ e = \frac{2J}{p} - 1 \]

### Final Answer:
Thus, the coefficient of restitution \( e \) is given by:
\[ e = \frac{2J}{p} - 1 \]