A bomb of mass 16kg at rest explodes into two pieces of masses 4 kg and 12 kg. The velolcity of the 12 kg mass is `4 ms^-1`. The kinetic energy of the other mass is

Options

A. `144J`
B. `288J`
C. `192J`
D. `96J`

Answer

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

### Step 1: Understand the problem
We have a bomb of mass 16 kg that is initially at rest. It explodes into two pieces: one piece has a mass of 4 kg and the other has a mass of 12 kg. The velocity of the 12 kg piece after the explosion is given as 4 m/s. We need to find the kinetic energy of the 4 kg piece.

### Step 2: Apply the conservation of momentum
Since the bomb is initially at rest, the total initial momentum is zero. According to the conservation of momentum:
\[ \text{Initial momentum} = \text{Final momentum} \]
\[ 0 = (m_1 \cdot v_1) + (m_2 \cdot v_2) \]
Where:
- \( m_1 = 12 \, \text{kg} \) (mass of the first piece)
- \( v_1 = 4 \, \text{m/s} \) (velocity of the 12 kg piece)
- \( m_2 = 4 \, \text{kg} \) (mass of the second piece)
- \( v_2 \) = velocity of the 4 kg piece (unknown)

Substituting the known values:
\[ 0 = (12 \cdot 4) + (4 \cdot v_2) \]
\[ 0 = 48 + 4v_2 \]

### Step 3: Solve for \( v_2 \)
Rearranging the equation gives:
\[ 4v_2 = -48 \]
\[ v_2 = -12 \, \text{m/s} \]

The negative sign indicates that the 4 kg piece moves in the opposite direction to the 12 kg piece.

### Step 4: Calculate the kinetic energy of the 4 kg piece
The formula for kinetic energy (KE) is:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the values for the 4 kg piece:
\[ KE = \frac{1}{2} \cdot 4 \cdot (-12)^2 \]
\[ KE = \frac{1}{2} \cdot 4 \cdot 144 \]
\[ KE = 2 \cdot 144 \]
\[ KE = 288 \, \text{Joules} \]

### Final Answer
The kinetic energy of the 4 kg mass is **288 Joules**.

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