To solve the problem of finding the loss in kinetic energy during an inelastic collision, we can follow these steps:
### Step-by-Step Solution:
1. **Identify the masses and initial velocities**:
- Mass of the first body, \( m_1 = 5 \times 10^3 \, \text{kg} \)
- Mass of the second body, \( m_2 = 15 \times 10^3 \, \text{kg} \)
- Initial velocity of the first body, \( u_1 = 2 \, \text{m/s} \)
- Initial velocity of the second body, \( u_2 = 0 \, \text{m/s} \) (since it is at rest)
2. **Calculate the initial kinetic energy (KE_initial)**:
- The kinetic energy of the first body before the collision:
\[
KE_{initial} = \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2
\]
- Substituting the values:
\[
KE_{initial} = \frac{1}{2} (5 \times 10^3) (2^2) + \frac{1}{2} (15 \times 10^3) (0^2)
\]
\[
KE_{initial} = \frac{1}{2} (5 \times 10^3) (4) + 0 = 10 \times 10^3 \, \text{J} = 10,000 \, \text{J}
\]
3. **Use conservation of momentum to find the final velocity (v)**:
- Before the collision, the total momentum is:
\[
p_{initial} = m_1 u_1 + m_2 u_2 = (5 \times 10^3)(2) + (15 \times 10^3)(0) = 10 \times 10^3 \, \text{kg m/s}
\]
- After the collision, both bodies stick together, so:
\[
p_{final} = (m_1 + m_2) v
\]
- Setting initial momentum equal to final momentum:
\[
10 \times 10^3 = (5 \times 10^3 + 15 \times 10^3) v
\]
\[
10 \times 10^3 = 20 \times 10^3 v
\]
\[
v = \frac{10 \times 10^3}{20 \times 10^3} = \frac{1}{2} \, \text{m/s}
\]
4. **Calculate the final kinetic energy (KE_final)**:
- The combined mass after the collision is \( m_1 + m_2 = 20 \times 10^3 \, \text{kg} \).
- The final kinetic energy is:
\[
KE_{final} = \frac{1}{2} (m_1 + m_2) v^2
\]
- Substituting the values:
\[
KE_{final} = \frac{1}{2} (20 \times 10^3) \left(\frac{1}{2}\right)^2
\]
\[
KE_{final} = \frac{1}{2} (20 \times 10^3) \left(\frac{1}{4}\right) = \frac{20 \times 10^3}{8} = 2.5 \times 10^3 \, \text{J} = 2500 \, \text{J}
\]
5. **Calculate the loss in kinetic energy**:
- The loss in kinetic energy is given by:
\[
\text{Loss in KE} = KE_{initial} - KE_{final}
\]
- Substituting the values:
\[
\text{Loss in KE} = 10,000 \, \text{J} - 2500 \, \text{J} = 7500 \, \text{J}
\]
6. **Convert the loss in kinetic energy to kilojoules**:
- Since \( 1 \, \text{kJ} = 1000 \, \text{J} \):
\[
\text{Loss in KE} = \frac{7500}{1000} = 7.5 \, \text{kJ}
\]
### Final Answer:
The loss in kinetic energy of the system is **7.5 kJ**.
