In a one-dimensional collision between two identical particles. A and B, B is stationary and A has momentum `p` before impact. During impact, B gives an impulse J to A. Find the coefficient of restitution between A and B?

Explanation

To find the coefficient of restitution (e) between two identical particles A and B during a one-dimensional collision where A has momentum p before impact and B is stationary, we can follow these steps:

### Step 1: Understand the Initial Conditions
- Particle A has momentum \( p \) before the collision.
- Particle B is stationary, so its initial momentum is \( 0 \).

### Step 2: Define the Impulse Given
- During the collision, B gives an impulse \( J \) to A. This means that A's momentum changes due to the impulse received from B.

### Step 3: Write the Momentum After Collision
- Let \( P_1 \) be the momentum of particle A after the collision.
- Let \( P_2 \) be the momentum of particle B after the collision.
- The impulse \( J \) can be expressed in terms of the change in momentum of A:
\[ J = P_1 - p \]

### Step 4: Apply Conservation of Momentum
- The total momentum before the collision is equal to the total momentum after the collision:
\[ p + 0 = P_1 + P_2 \]
Therefore,
\[ P_1 + P_2 = p \]

### Step 5: Express \( P_2 \) in Terms of \( J \)
- From the impulse equation, we can express \( P_1 \):
\[ P_1 = p + J \]
- Substituting \( P_1 \) into the momentum conservation equation:
\[ (p + J) + P_2 = p \]
This simplifies to:
\[ P_2 = -J \]

### Step 6: 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:
\[ e = \frac{V_{B, \text{after}} - V_{A, \text{after}}}{V_{A, \text{before}} - V_{B, \text{before}}} \]
- Since B is stationary before the collision, its initial velocity is \( 0 \), and the initial velocity of A can be expressed as:
\[ V_{A, \text{before}} = \frac{p}{m} \quad \text{(where m is the mass of A)} \]
- After the collision, the velocities can be expressed as:
\[ V_{A, \text{after}} = \frac{P_1}{m} = \frac{p + J}{m} \]
\[ V_{B, \text{after}} = \frac{P_2}{m} = \frac{-J}{m} \]
- Substituting these into the coefficient of restitution formula:
\[ e = \frac{\left(\frac{-J}{m}\right) - \left(\frac{p + J}{m}\right)}{ \frac{p}{m} - 0} \]
Simplifying gives:
\[ e = \frac{-J - p - J}{p} = \frac{-2J - p}{p} = -2\frac{J}{p} - 1 \]

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