If a body of mass m collides head on, elastically with velocity u with another identical boday at rest. After collision velocty of the second body will be

Options

A. zero
B. u
C. 2u
D. data insufficient.

Explanation

To solve the problem of finding the velocity of the second body after an elastic collision with the first body, we can use the principles of conservation of momentum and conservation of kinetic energy.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Mass of the first body, \( m_1 = m \)
- Initial velocity of the first body, \( u_1 = u \)
- Mass of the second body, \( m_2 = m \)
- Initial velocity of the second body, \( u_2 = 0 \) (at rest)

2. **Apply Conservation of Momentum:**
The total momentum before the collision must equal the total momentum after the collision.
\[ m u + m \cdot 0 = m v_1 + m v_2 \]
Simplifying this gives:
\[ mu = mv_1 + mv_2 \]
Dividing through by \( m \) (since \( m \neq 0 \)):
\[ u = v_1 + v_2 \quad \text{(1)} \]

3. **Apply Conservation of Kinetic Energy:**
In an elastic collision, the total kinetic energy before the collision equals the total kinetic energy after the collision.
\[ \frac{1}{2} m u^2 + \frac{1}{2} m \cdot 0^2 = \frac{1}{2} m v_1^2 + \frac{1}{2} m v_2^2 \]
Simplifying this gives:
\[ \frac{1}{2} mu^2 = \frac{1}{2} mv_1^2 + \frac{1}{2} mv_2^2 \]
Dividing through by \( \frac{1}{2} m \):
\[ u^2 = v_1^2 + v_2^2 \quad \text{(2)} \]

4. **Solve the Equations:**
From equation (1), we can express \( v_1 \) in terms of \( v_2 \):
\[ v_1 = u - v_2 \]
Substitute \( v_1 \) into equation (2):
\[ u^2 = (u - v_2)^2 + v_2^2 \]
Expanding the equation:
\[ u^2 = (u^2 - 2uv_2 + v_2^2) + v_2^2 \]
Simplifying:
\[ u^2 = u^2 - 2uv_2 + 2v_2^2 \]
Rearranging gives:
\[ 0 = -2uv_2 + 2v_2^2 \]
Factoring out \( 2v_2 \):
\[ 0 = 2v_2(v_2 - u) \]
This gives us two solutions:
\[ v_2 = 0 \quad \text{or} \quad v_2 = u \]

5. **Conclusion:**
Since \( v_2 = 0 \) corresponds to the initial state (before collision), the relevant solution is:
\[ v_2 = u \]
Therefore, the velocity of the second body after the collision is \( u \).