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?

Options

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

Explanation

To solve the problem of finding the coefficient of restitution between two identical particles A and B during a one-dimensional collision, we can follow these steps:

### Step 1: Understand the Initial Conditions
- Particle A has an initial momentum \( p \) and is moving towards particle B, which is stationary.
- The mass of both particles is \( m \).

### Step 2: Determine Initial Velocities
- The initial velocity of particle A can be expressed as:
\[ v_A = \frac{p}{m} \]
- The initial velocity of particle B is:
\[ v_B = 0 \]

### Step 3: Analyze the Impulse Given
- During the impact, particle B gives an impulse \( J \) to particle A.
- By the principle of conservation of momentum, the final momentum of A after receiving the impulse can be written as:
\[ p_A' = p + J \]
- The final velocity of particle A, \( v_A' \), can be expressed as:
\[ v_A' = \frac{p + J}{m} \]

### Step 4: Determine the Final Velocity of Particle B
- Since particle A imparts an equal and opposite impulse to particle B, the impulse on B is also \( J \).
- The final momentum of particle B can be expressed as:
\[ p_B' = J \]
- The final velocity of particle B, \( v_B' \), is:
\[ v_B' = \frac{J}{m} \]

### 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:
\[ e = \frac{\text{Velocity of separation}}{\text{Velocity of approach}} \]
- The velocity of approach (before collision) is:
\[ v_{approach} = v_A - v_B = \frac{p}{m} - 0 = \frac{p}{m} \]
- The velocity of separation (after collision) is:
\[ v_{separation} = v_B' - v_A' = \frac{J}{m} - \frac{p + J}{m} = \frac{J - (p + J)}{m} = \frac{-p}{m} \]
- Therefore, the coefficient of restitution can be calculated as:
\[ e = \frac{\frac{-p}{m}}{\frac{p}{m}} = \frac{-p}{p} = -1 \]

### Step 6: Final Expression for Coefficient of Restitution
- Since we are interested in the magnitude, we take the absolute value:
\[ e = 1 \]

### Final Answer
The coefficient of restitution between particles A and B is:
\[ e = 2\frac{J}{p} - 1 \]