A ball of mass `m_(1)` moving with a certain speed collides elastically with another ball of mass `m_(2)` kept at rest . If the first ball starts moving in reversed direction after the impact then

Options

A. `m_(1)=m_(2)`
B. `m_(1) gt m_(2)`
C. `m_(1)lt m_(2)`
D. None of these

Explanation

To solve the problem, we need to analyze the elastic collision between two balls, where one ball (mass \( m_1 \)) is moving and the other ball (mass \( m_2 \)) is at rest. After the collision, the first ball moves in the reverse direction. We will derive the conditions under which this scenario occurs.

### Step-by-Step Solution:

1. **Define Initial Conditions**:
- Let the mass of the first ball be \( m_1 \) and its initial velocity be \( u_1 = u \).
- Let the mass of the second ball be \( m_2 \) and its initial velocity be \( u_2 = 0 \) (since it is at rest).

2. **Apply Conservation of Momentum**:
- According to the law of conservation of momentum, the total momentum before the collision must equal the total momentum after the collision.
- Initial momentum: \( m_1 u + m_2 \cdot 0 = m_1 u \)
- Final momentum: \( m_1 v_1 + m_2 v_2 \)
- Therefore, we have the equation:
\[ m_1 u = m_1 v_1 + m_2 v_2 \quad \text{(1)} \]

3. **Apply Conservation of Kinetic Energy**:
- Since the collision is elastic, kinetic energy is also conserved.
- Initial kinetic energy: \( \frac{1}{2} m_1 u^2 + 0 = \frac{1}{2} m_1 u^2 \)
- Final kinetic energy: \( \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \)
- Therefore, we have the equation:
\[ \frac{1}{2} m_1 u^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \quad \text{(2)} \]

4. **Use the Coefficient of Restitution**:
- The coefficient of restitution \( e \) for elastic collisions is 1. It is defined as:
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} = 1 \]
- This gives us:
\[ v_2 - v_1 = u_1 - u_2 \quad \Rightarrow \quad v_2 - v_1 = u \quad \text{(3)} \]

5. **Substitute Equation (3) into Equation (1)**:
- From equation (3), we can express \( v_2 \) in terms of \( v_1 \):
\[ v_2 = v_1 + u \]
- Substitute this into equation (1):
\[ m_1 u = m_1 v_1 + m_2 (v_1 + u) \]
- Simplifying this gives:
\[ m_1 u = m_1 v_1 + m_2 v_1 + m_2 u \]
\[ m_1 u - m_2 u = (m_1 + m_2) v_1 \]
\[ (m_1 - m_2) u = (m_1 + m_2) v_1 \quad \text{(4)} \]

6. **Determine the Condition for Reversed Direction**:
- For the first ball to move in the reverse direction, \( v_1 \) must be negative:
\[ v_1 < 0 \]
- From equation (4), if \( u > 0 \) (initial velocity of \( m_1 \) is positive), then for \( v_1 \) to be negative, we need:
\[ m_1 - m_2 < 0 \quad \Rightarrow \quad m_1 < m_2 \]

### Conclusion:
The condition that must be fulfilled for the first ball to start moving in the reverse direction after the elastic collision is:
\[ m_1 < m_2 \]