Two particles of mass `m_(A)` and `m_(B)` and their velocities are `V_(A)` and `V_(B)` respectively collides. After collision they interchanges their velocities, then ratio of `m_(A)/m_(B)` is

Options

A. `v_(A)/v_(B)`
B. `v_(B)/v_(A)`
C. `(v_(A) + v_(B))/(v_(B) - v_(A))`
D. 1

Explanation

To solve the problem of two particles colliding and interchanging their velocities, we will use the principle of conservation of momentum. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Define the Initial Conditions**:
- Let the mass of particle A be \( m_A \) and its initial velocity be \( V_A \).
- Let the mass of particle B be \( m_B \) and its initial velocity be \( V_B \).

2. **Write the Initial Momentum**:
- The total initial momentum \( P_{initial} \) before the collision is given by:
\[ P_{initial} = m_A V_A + m_B V_B \]

3. **Define the Final Conditions**:
- After the collision, the particles interchange their velocities.
- Therefore, the final velocity of particle A becomes \( V_B \) and the final velocity of particle B becomes \( V_A \).

4. **Write the Final Momentum**:
- The total final momentum \( P_{final} \) after the collision is given by:
\[ P_{final} = m_A V_B + m_B V_A \]

5. **Apply Conservation of Momentum**:
- According to the conservation of momentum, the initial momentum must equal the final momentum:
\[ P_{initial} = P_{final} \]
- This gives us the equation:
\[ m_A V_A + m_B V_B = m_A V_B + m_B V_A \]

6. **Rearranging the Equation**:
- Rearranging the equation, we get:
\[ m_A V_A - m_A V_B = m_B V_A - m_B V_B \]
- This simplifies to:
\[ m_A (V_A - V_B) = m_B (V_A - V_B) \]

7. **Factor Out the Common Term**:
- If \( V_A \neq V_B \), we can divide both sides by \( (V_A - V_B) \):
\[ m_A = m_B \]

8. **Calculate the Ratio**:
- Therefore, the ratio of the masses \( \frac{m_A}{m_B} \) is:
\[ \frac{m_A}{m_B} = 1 \]

### Final Answer:
The ratio of \( \frac{m_A}{m_B} \) is \( 1 \).