Consider the following statements A and B. Identify the correct choice in the given answers
A. In a one dimensional perfectly elastic collision between two moving bodies of equal masses the bodies merely exchange their velocities after collision.
B. If a lighter body at rest suffers perfectly elastic collision with a very heavy body moving with a certain velocity, then after collision both travel with same velocity.

Options

A. A and B are correct
B. Both A and B are wrong
C. A is correct, B is wrong
D. A is wrong, B is correct

Explanation

To solve the question, we need to analyze both statements A and B regarding perfectly elastic collisions.

### Step 1: Analyze Statement A
Statement A states that in a one-dimensional perfectly elastic collision between two moving bodies of equal masses, the bodies merely exchange their velocities after the collision.

- **Explanation**: In a perfectly elastic collision, both momentum and kinetic energy are conserved. For two bodies with equal masses (let's denote them as \( m_1 \) and \( m_2 \), where \( m_1 = m_2 \)), if we denote their initial velocities as \( u_1 \) and \( u_2 \), the final velocities \( v_1 \) and \( v_2 \) can be derived using the conservation laws.

Using the equations for perfectly elastic collisions:
1. \( v_1 = \frac{(m_1 - m_2)u_1 + 2m_2u_2}{m_1 + m_2} \)
2. \( v_2 = \frac{(m_2 - m_1)u_2 + 2m_1u_1}{m_1 + m_2} \)

When \( m_1 = m_2 \):
- The equations simplify to:
- \( v_1 = u_2 \)
- \( v_2 = u_1 \)

This shows that the two bodies exchange their velocities after the collision. Thus, **Statement A is correct**.

### Step 2: Analyze Statement B
Statement B states that if a lighter body at rest suffers a perfectly elastic collision with a very heavy body moving with a certain velocity, then after the collision, both travel with the same velocity.

- **Explanation**: Let’s denote the lighter body as \( m_1 \) (with \( m_1 < m_2 \)) and the heavy body as \( m_2 \) (with \( m_2 \) moving with velocity \( u_2 \) and \( m_1 \) initially at rest, \( u_1 = 0 \)).

Using the same equations for perfectly elastic collisions:
1. \( v_1 = \frac{(m_1 - m_2)u_1 + 2m_2u_2}{m_1 + m_2} \)
2. \( v_2 = \frac{(m_2 - m_1)u_2 + 2m_1u_1}{m_1 + m_2} \)

Substituting \( u_1 = 0 \):
- \( v_1 = \frac{2m_2u_2}{m_1 + m_2} \)
- \( v_2 = \frac{(m_2)u_2}{m_1 + m_2} \)

Since \( m_1 \) is much smaller than \( m_2 \), after the collision, \( v_1 \) (the velocity of the lighter body) will not equal \( v_2 \) (the velocity of the heavier body). In fact, the lighter body will move away with a velocity that is less than the velocity of the heavy body. Thus, **Statement B is incorrect**.

### Conclusion
- **Statement A is correct**.
- **Statement B is incorrect**.

### Final Answer
The correct choice is that **Statement A is correct and Statement B is incorrect**.

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