In one dimentional collision between two idential particles 'A' and 'B'. 'B' is stationary and 'A' has momentum 54 kg-m`//` sec before impact. During impact , 'B' gives an impulse 3kg-m`//`sec to 'A' . The coefficient of restitution between A and B is

Options

A. `2//5`
B. `1//5`
C. `1//2`
D. `8//9`

Answer

To find 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 \( P_A = 54 \, \text{kg m/s} \).
- Particle B is stationary, so its initial momentum \( P_B = 0 \, \text{kg m/s} \).

### Step 2: Determine the impulse given to A
- During the collision, particle B gives an impulse of \( J = 3 \, \text{kg m/s} \) to particle A.

### Step 3: Calculate the final momentum of particle A
- The change in momentum for particle A due to the impulse is given by:
\[ P_A' = P_A - J \]
Substituting the values:
\[ P_A' = 54 - 3 = 51 \, \text{kg m/s} \]

### Step 4: Calculate the final momentum of particle B
- Since the impulse given to A is equal and opposite to the impulse received by B, the momentum of particle B after the collision is:
\[ P_B' = J = 3 \, \text{kg m/s} \]

### Step 5: Determine the velocities of A and B after the collision
- To find the velocities, we can use the relationship between momentum and velocity:
\[ v_A' = \frac{P_A'}{m} \quad \text{and} \quad v_B' = \frac{P_B'}{m} \]
Since both particles are identical, we can denote their mass as \( m \). Thus:
\[ v_A' = \frac{51}{m} \quad \text{and} \quad v_B' = \frac{3}{m} \]

### Step 6: Calculate the coefficient of restitution
- The coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{relative velocity of separation}}{\text{relative velocity of approach}} \]
- The relative velocity of separation is:
\[ v_{separation} = v_A' - v_B' = \frac{51}{m} - \frac{3}{m} = \frac{48}{m} \]
- The relative velocity of approach (before the collision) is:
\[ v_{approach} = v_A - v_B = \frac{54}{m} - 0 = \frac{54}{m} \]
- Therefore, the coefficient of restitution can be calculated as:
\[ e = \frac{\frac{48}{m}}{\frac{54}{m}} = \frac{48}{54} = \frac{8}{9} \]

### Final Answer
The coefficient of restitution between particles A and B is \( \frac{8}{9} \).

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