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

Explanation

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 explodes into two pieces: one piece has a mass of 4 kg and the other has a mass of 12 kg. The 12 kg piece moves with a velocity of 4 m/s. We need to find the kinetic energy of the 4 kg piece.

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

### Step 3: Set Up the Equation
Substituting the known values into the momentum equation:
\[ 0 = 4 v_1 + 12 \times 4 \]
This simplifies to:
\[ 0 = 4 v_1 + 48 \]

### Step 4: Solve for \( v_1 \)
Rearranging the equation to solve for \( v_1 \):
\[ 4 v_1 = -48 \]
\[ v_1 = -12 \, \text{m/s} \]
The negative sign indicates that the 4 kg piece moves in the opposite direction to the 12 kg piece.

### Step 5: Calculate the Kinetic Energy of the 4 kg Piece
The kinetic energy (KE) is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the values for the 4 kg piece:
\[ KE = \frac{1}{2} \times 4 \times (-12)^2 \]
Calculating:
\[ KE = \frac{1}{2} \times 4 \times 144 \]
\[ KE = 2 \times 144 = 288 \, \text{Joules} \]

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

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