A bomb at rest has mass 60 kg. It explodes and a fragment of 40 kg has kinetic energy 96 joule. Then kinetic energy of other fragment is-

Options

A. 180 J
B. 190 J
C. 182 J
D. 192 J

Answer

To solve the problem, we will use the principles of conservation of momentum and kinetic energy.

### Step-by-Step Solution:

1. **Identify the masses and the initial conditions:**
- The total mass of the bomb before the explosion is \( M = 60 \, \text{kg} \).
- After the explosion, one fragment has a mass of \( m_1 = 40 \, \text{kg} \) and the other fragment has a mass of \( m_2 = 20 \, \text{kg} \) (since \( 60 \, \text{kg} - 40 \, \text{kg} = 20 \, \text{kg} \)).
- The initial velocity of the bomb is \( 0 \, \text{m/s} \).

2. **Apply the conservation of momentum:**
- Since the bomb is initially at rest, the total momentum before the explosion is \( 0 \).
- After the explosion, the momentum must also equal \( 0 \):
\[ m_1 v_1 + m_2 v_2 = 0 \]
- Here, \( v_1 \) is the velocity of the \( 40 \, \text{kg} \) fragment and \( v_2 \) is the velocity of the \( 20 \, \text{kg} \) fragment. We can express \( v_2 \) in terms of \( v_1 \):
\[ 40 v_1 + 20 v_2 = 0 \implies v_2 = -2 v_1 \]

3. **Use the kinetic energy of the first fragment:**
- The kinetic energy of the \( 40 \, \text{kg} \) fragment is given as \( K_1 = 96 \, \text{J} \).
- The formula for kinetic energy is:
\[ K = \frac{1}{2} m v^2 \]
- For the \( 40 \, \text{kg} \) fragment:
\[ 96 = \frac{1}{2} \times 40 \times v_1^2 \]
- Simplifying gives:
\[ 96 = 20 v_1^2 \implies v_1^2 = \frac{96}{20} = 4.8 \implies v_1 = \sqrt{4.8} \]

4. **Calculate the kinetic energy of the second fragment:**
- Now, substitute \( v_2 = -2 v_1 \) into the kinetic energy formula for the \( 20 \, \text{kg} \) fragment:
\[ K_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} \times 20 \times (2 v_1)^2 \]
- This simplifies to:
\[ K_2 = \frac{1}{2} \times 20 \times 4 v_1^2 = 10 \times 4 v_1^2 = 40 v_1^2 \]
- Now substitute \( v_1^2 = 4.8 \):
\[ K_2 = 40 \times 4.8 = 192 \, \text{J} \]

### Final Answer:
The kinetic energy of the other fragment is \( 192 \, \text{J} \).