A ball of mass 5 kg moving with speed 8 m/s collides head on with another stationary ball of mass 15 kg. If collision is perfecty inelastic, then loss in kinetic energy is

Options

A. 160 J
B. 80 J
C. 40 J
D. 120 J

Explanation

To solve the problem of finding the loss in kinetic energy during a perfectly inelastic collision, we can follow these steps:

### Step 1: Calculate the Initial Kinetic Energy
The initial kinetic energy (KE_initial) is only due to the moving ball (mass = 5 kg, speed = 8 m/s). The formula for kinetic energy is:
\[ KE = \frac{1}{2}mv^2 \]
For the moving ball:
\[ KE_{\text{initial}} = \frac{1}{2} \times 5 \, \text{kg} \times (8 \, \text{m/s})^2 \]
Calculating this gives:
\[ KE_{\text{initial}} = \frac{1}{2} \times 5 \times 64 = \frac{320}{2} = 160 \, \text{J} \]

### Step 2: Calculate the Final Velocity After Collision
In a perfectly inelastic collision, the two balls stick together and move with a common velocity after the collision. We can use the conservation of momentum to find this final velocity (V_final).

The initial momentum (P_initial) is given by:
\[ P_{\text{initial}} = m_1v_1 + m_2v_2 \]
Where:
- \(m_1 = 5 \, \text{kg}\) (mass of the moving ball)
- \(v_1 = 8 \, \text{m/s}\) (initial speed of the moving ball)
- \(m_2 = 15 \, \text{kg}\) (mass of the stationary ball)
- \(v_2 = 0 \, \text{m/s}\) (initial speed of the stationary ball)

Calculating the initial momentum:
\[ P_{\text{initial}} = (5 \times 8) + (15 \times 0) = 40 \, \text{kg m/s} \]

After the collision, the total mass is:
\[ m_{\text{total}} = m_1 + m_2 = 5 + 15 = 20 \, \text{kg} \]

Using conservation of momentum:
\[ P_{\text{final}} = m_{\text{total}} \times V_{\text{final}} \]
Setting initial momentum equal to final momentum:
\[ 40 = 20 \times V_{\text{final}} \]
Solving for \(V_{\text{final}}\):
\[ V_{\text{final}} = \frac{40}{20} = 2 \, \text{m/s} \]

### Step 3: Calculate the Final Kinetic Energy
Now we can calculate the final kinetic energy (KE_final) using the combined mass and the final velocity:
\[ KE_{\text{final}} = \frac{1}{2} m_{\text{total}} V_{\text{final}}^2 \]
Substituting the values:
\[ KE_{\text{final}} = \frac{1}{2} \times 20 \, \text{kg} \times (2 \, \text{m/s})^2 \]
Calculating this gives:
\[ KE_{\text{final}} = \frac{1}{2} \times 20 \times 4 = 40 \, \text{J} \]

### Step 4: Calculate the Loss in Kinetic Energy
The loss in kinetic energy (\(\Delta KE\)) is given by:
\[ \Delta KE = KE_{\text{initial}} - KE_{\text{final}} \]
Substituting the values:
\[ \Delta KE = 160 \, \text{J} - 40 \, \text{J} = 120 \, \text{J} \]

### Final Answer
The loss in kinetic energy during the perfectly inelastic collision is:
\[ \Delta KE = 120 \, \text{J} \]

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