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 exchanging their velocities, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**:
- 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 \( P_i \) before the collision is given by:
\[ P_i = m_A V_A + m_B V_B \]

3. **Write the Final Momentum**:
- After the collision, the particles exchange their velocities. Thus, the final momentum \( P_f \) after the collision is:
\[ P_f = m_A V_B + m_B V_A \]

4. **Apply Conservation of Momentum**:
- According to the law of conservation of momentum, the total initial momentum is equal to the total final momentum:
\[ P_i = P_f \]
- Therefore, we can write:
\[ m_A V_A + m_B V_B = m_A V_B + m_B V_A \]

5. **Rearrange the Equation**:
- Rearranging the equation gives:
\[ m_A V_A - m_A V_B = m_B V_A - m_B V_B \]
- This can be rewritten as:
\[ m_A (V_A - V_B) = m_B (V_A - V_B) \]

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

7. **Find the Ratio of Masses**:
- Thus, the ratio of the masses \( \frac{m_A}{m_B} \) is:
\[ \frac{m_A}{m_B} = 1 \]

### Conclusion:
The ratio of the masses \( \frac{m_A}{m_B} \) is 1.

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