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. 172.8 J
D. 144 J

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the masses and initial velocities
- Mass of body A (m₁) = 4 kg (moving with velocity u₁ = 12 m/s)
- Mass of body B (m₂) = 6 kg (at rest, so u₂ = 0 m/s)

### Step 2: Calculate the initial momentum
The initial momentum (p_initial) of the system can be calculated using the formula:
\[ p_{\text{initial}} = m_1 \cdot u_1 + m_2 \cdot u_2 \]
Since body B is at rest, its velocity is 0, so:
\[ p_{\text{initial}} = (4 \, \text{kg} \cdot 12 \, \text{m/s}) + (6 \, \text{kg} \cdot 0 \, \text{m/s}) \]
\[ p_{\text{initial}} = 48 \, \text{kg m/s} \]

### Step 3: Calculate the final velocity after collision
After the collision, the two bodies stick together, so we can use the conservation of momentum:
\[ p_{\text{final}} = (m_1 + m_2) \cdot V \]
Setting the initial momentum equal to the final momentum:
\[ 48 \, \text{kg m/s} = (4 \, \text{kg} + 6 \, \text{kg}) \cdot V \]
\[ 48 \, \text{kg m/s} = 10 \, \text{kg} \cdot V \]
Solving for V:
\[ V = \frac{48 \, \text{kg m/s}}{10 \, \text{kg}} = 4.8 \, \text{m/s} \]

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

### Step 5: Calculate the final kinetic energy (KE_final)
After the collision, the kinetic energy of the combined mass is:
\[ KE_{\text{final}} = \frac{1}{2} (m_1 + m_2) V^2 \]
\[ KE_{\text{final}} = \frac{1}{2} (10 \, \text{kg}) (4.8 \, \text{m/s})^2 \]
\[ KE_{\text{final}} = 5 \cdot 23.04 \]
\[ KE_{\text{final}} = 115.2 \, \text{J} \]

### Step 6: Calculate the loss of kinetic energy
The loss of kinetic energy (ΔKE) is given by:
\[ \Delta KE = KE_{\text{initial}} - KE_{\text{final}} \]
\[ \Delta KE = 288 \, \text{J} - 115.2 \, \text{J} \]
\[ \Delta KE = 172.8 \, \text{J} \]

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

---