A bomb of mass `30 kg` at rest explodes into two pieces of mass `18 kg` and `12 kg`. The velocity of mass `18 kg` is `6 m//s`. The kinetic energy of the other mass is

Options

A. 524 J
B. 256J
C. 486 J
D. 324J

Explanation

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

### Step 1: Understand the problem
We have a bomb of mass 30 kg that explodes into two pieces: one piece of mass 18 kg moving at a velocity of 6 m/s, and another piece of mass 12 kg. We need to find the kinetic energy of the 12 kg piece.

### Step 2: Apply the conservation of momentum
According to the law of conservation of momentum, the total momentum before the explosion must equal the total momentum after the explosion.

- Initial momentum (before explosion) = 0 (since the bomb is at rest)
- Final momentum = momentum of 18 kg piece + momentum of 12 kg piece

Let:
- \( m_1 = 18 \, \text{kg} \) (mass of the first piece)
- \( v_1 = 6 \, \text{m/s} \) (velocity of the first piece)
- \( m_2 = 12 \, \text{kg} \) (mass of the second piece)
- \( v_2 \) = velocity of the second piece (unknown)

Using the conservation of momentum:
\[ 0 = m_1 v_1 + m_2 v_2 \]
Substituting the known values:
\[ 0 = 18 \times 6 + 12 \times v_2 \]
\[ 0 = 108 + 12 v_2 \]

### Step 3: Solve for \( v_2 \)
Rearranging the equation to solve for \( v_2 \):
\[ 12 v_2 = -108 \]
\[ v_2 = \frac{-108}{12} = -9 \, \text{m/s} \]

The negative sign indicates that the 12 kg piece is moving in the opposite direction to the 18 kg piece.

### Step 4: Calculate the kinetic energy of the 12 kg piece
The kinetic energy (KE) is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the values for the 12 kg piece:
\[ KE = \frac{1}{2} \times 12 \times (-9)^2 \]
Calculating the square of velocity:
\[ KE = \frac{1}{2} \times 12 \times 81 \]
\[ KE = 6 \times 81 = 486 \, \text{Joules} \]

### Final Answer
The kinetic energy of the 12 kg piece is **486 Joules**.

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