Two identical balls `A and B` having velocity of `0.5 m//s and -0.3 m//s` respectively collide elastically in one dimension. The velocities of `B and A` after the collision respectively will be

Options

A. `-0.5m//s "and" 0.3m//s`
B. `0.5m//s "and" -0.3m//s`
C. `-0.3m//s "and" 0.5m//s`
D. `0.3m//s "and" 0.5m//s`

Explanation

To solve the problem of two identical balls A and B colliding elastically in one dimension, we will follow these steps:

### Step 1: Understand the Initial Conditions
- Ball A has an initial velocity \( u_1 = 0.5 \, \text{m/s} \).
- Ball B has an initial velocity \( u_2 = -0.3 \, \text{m/s} \) (the negative sign indicates that it is moving in the opposite direction).

### Step 2: Apply the Conservation of Momentum
Since the collision is elastic and the balls are identical, we can use the conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.

The equation for conservation of momentum is:
\[ m u_1 + m u_2 = m v_1 + m v_2 \]
Where:
- \( v_1 \) is the final velocity of ball A.
- \( v_2 \) is the final velocity of ball B.

Since the masses \( m \) cancel out, we simplify to:
\[ u_1 + u_2 = v_1 + v_2 \]
Substituting the values:
\[ 0.5 + (-0.3) = v_1 + v_2 \]
\[ 0.2 = v_1 + v_2 \quad \text{(Equation 1)} \]

### Step 3: Apply the Coefficient of Restitution
For elastic collisions, the coefficient of restitution \( e = 1 \). The coefficient of restitution is defined as:
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \]
Thus, we have:
\[ 1 = \frac{v_2 - v_1}{u_1 - u_2} \]
Substituting the values:
\[ 1 = \frac{v_2 - v_1}{0.5 - (-0.3)} \]
\[ 1 = \frac{v_2 - v_1}{0.8} \]
This leads to:
\[ v_2 - v_1 = 0.8 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations Simultaneously
Now we have two equations:
1. \( v_1 + v_2 = 0.2 \) (Equation 1)
2. \( v_2 - v_1 = 0.8 \) (Equation 2)

We can solve these equations by adding them:
\[ (v_1 + v_2) + (v_2 - v_1) = 0.2 + 0.8 \]
This simplifies to:
\[ 2v_2 = 1 \]
Thus, we find:
\[ v_2 = 0.5 \, \text{m/s} \]

### Step 5: Substitute Back to Find \( v_1 \)
Now, substitute \( v_2 \) back into Equation 1:
\[ v_1 + 0.5 = 0.2 \]
Solving for \( v_1 \):
\[ v_1 = 0.2 - 0.5 = -0.3 \, \text{m/s} \]

### Final Result
The final velocities after the collision are:
- Velocity of ball B (\( v_2 \)): \( 0.5 \, \text{m/s} \)
- Velocity of ball A (\( v_1 \)): \( -0.3 \, \text{m/s} \)

### Summary
Thus, the velocities of balls B and A after the collision are \( 0.5 \, \text{m/s} \) and \( -0.3 \, \text{m/s} \) respectively.

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