A body of mass `4 kg` moving with velocity `12 m//s` collides with another body of mass `6 kg` at rest. If two bodies stick together after collision , then the loss of kinetic energy of system is

Options

A. Zero
B. 288J
C. 172.8J
D. 144J

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given data
- Mass of body 1 (m1) = 4 kg
- Velocity of body 1 (u1) = 12 m/s
- Mass of body 2 (m2) = 6 kg
- Velocity of body 2 (u2) = 0 m/s (at rest)

### Step 2: Calculate the initial momentum
The initial momentum (p_initial) of the system can be calculated using the formula:
\[ p_{initial} = m_1 \cdot u_1 + m_2 \cdot u_2 \]
Substituting the values:
\[ p_{initial} = (4 \, \text{kg} \cdot 12 \, \text{m/s}) + (6 \, \text{kg} \cdot 0 \, \text{m/s}) = 48 \, \text{kg m/s} \]

### Step 3: Calculate the final mass after collision
After the collision, the two bodies stick together, so the final mass (m_final) is:
\[ m_{final} = m_1 + m_2 = 4 \, \text{kg} + 6 \, \text{kg} = 10 \, \text{kg} \]

### Step 4: Use conservation of momentum to find the final velocity
According to the conservation of momentum:
\[ p_{initial} = p_{final} \]
\[ p_{final} = m_{final} \cdot v_{final} \]
Setting the initial momentum equal to the final momentum:
\[ 48 \, \text{kg m/s} = 10 \, \text{kg} \cdot v_{final} \]
Solving for \( v_{final} \):
\[ v_{final} = \frac{48 \, \text{kg m/s}}{10 \, \text{kg}} = 4.8 \, \text{m/s} \]

### Step 5: Calculate the initial kinetic energy
The initial kinetic energy (KE_initial) of the system is given by:
\[ KE_{initial} = \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 \]
Substituting the values:
\[ KE_{initial} = \frac{1}{2} (4 \, \text{kg}) (12 \, \text{m/s})^2 + \frac{1}{2} (6 \, \text{kg}) (0 \, \text{m/s})^2 \]
\[ KE_{initial} = \frac{1}{2} (4) (144) + 0 = 288 \, \text{J} \]

### Step 6: Calculate the final kinetic energy
The final kinetic energy (KE_final) after the collision is given by:
\[ KE_{final} = \frac{1}{2} m_{final} v_{final}^2 \]
Substituting the values:
\[ KE_{final} = \frac{1}{2} (10 \, \text{kg}) (4.8 \, \text{m/s})^2 \]
\[ KE_{final} = \frac{1}{2} (10) (23.04) = 115.2 \, \text{J} \]

### Step 7: Calculate the loss of kinetic energy
The loss of kinetic energy (ΔKE) is given by:
\[ \Delta KE = KE_{initial} - KE_{final} \]
Substituting the values:
\[ \Delta KE = 288 \, \text{J} - 115.2 \, \text{J} = 172.8 \, \text{J} \]

### Final Answer
The loss of kinetic energy of the system is **172.8 Joules**.

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