A body of mass of 10 kg. and velocity 9 m/sec strikes another body of mass 20 kg. and velocity 6 m/sec moving in the same direction. If the velocity of the first body becomes 6 m/sec after impact, then the coefficient of restitution and the loss of kinetic energy are .... and ...

Answer

To solve the problem, we need to find the coefficient of restitution and the loss of kinetic energy after the collision between two bodies. Let's break it down step by step.

### Given:
- Mass of body 1, \( m_1 = 10 \, \text{kg} \)
- Initial velocity of body 1, \( u_1 = 9 \, \text{m/s} \)
- Mass of body 2, \( m_2 = 20 \, \text{kg} \)
- Initial velocity of body 2, \( u_2 = 6 \, \text{m/s} \)
- Final velocity of body 1 after impact, \( v_1 = 6 \, \text{m/s} \)

### Step 1: Find the final velocity of body 2 after the collision (v2)
Using the conservation of momentum:
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]
Substituting the known values:
\[ 10 \cdot 9 + 20 \cdot 6 = 10 \cdot 6 + 20 v_2 \]
Calculating the left side:
\[ 90 + 120 = 60 + 20 v_2 \]
\[ 210 = 60 + 20 v_2 \]
Now, isolate \( v_2 \):
\[ 210 - 60 = 20 v_2 \]
\[ 150 = 20 v_2 \]
\[ v_2 = \frac{150}{20} = 7.5 \, \text{m/s} \]

### Step 2: Calculate the coefficient of restitution (e)
The coefficient of restitution is given by the formula:
\[ e = \frac{v_2 - v_1}{u_1 - u_2} \]
Substituting the values:
\[ e = \frac{7.5 - 6}{9 - 6} \]
Calculating the numerator:
\[ e = \frac{1.5}{3} \]
\[ e = 0.5 \]

### Step 3: Calculate the initial and final kinetic energy
The initial kinetic energy (KE_initial) is given by:
\[ KE_{\text{initial}} = \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 \]
Calculating:
\[ KE_{\text{initial}} = \frac{1}{2} \cdot 10 \cdot 9^2 + \frac{1}{2} \cdot 20 \cdot 6^2 \]
\[ = \frac{1}{2} \cdot 10 \cdot 81 + \frac{1}{2} \cdot 20 \cdot 36 \]
\[ = 405 + 360 = 765 \, \text{J} \]

The final kinetic energy (KE_final) is given by:
\[ KE_{\text{final}} = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \]
Calculating:
\[ KE_{\text{final}} = \frac{1}{2} \cdot 10 \cdot 6^2 + \frac{1}{2} \cdot 20 \cdot 7.5^2 \]
\[ = \frac{1}{2} \cdot 10 \cdot 36 + \frac{1}{2} \cdot 20 \cdot 56.25 \]
\[ = 180 + 562.5 = 742.5 \, \text{J} \]

### Step 4: Calculate the loss of kinetic energy
The loss of kinetic energy is given by:
\[ \text{Loss of KE} = KE_{\text{initial}} - KE_{\text{final}} \]
Calculating:
\[ \text{Loss of KE} = 765 - 742.5 = 22.5 \, \text{J} \]

### Final Answers:
- Coefficient of restitution, \( e = 0.5 \)
- Loss of kinetic energy = 22.5 J