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 perfectly inelastic, then loss in kinetic energ is

Options

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

Answer

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

### Step 1: Identify the masses and initial velocities
- Mass of ball 1 (m1) = 5 kg, initial velocity (u1) = 8 m/s
- Mass of ball 2 (m2) = 15 kg, initial velocity (u2) = 0 m/s (stationary)

### Step 2: Calculate the initial kinetic energy (KE_initial)
The formula for kinetic energy is:
\[ KE = \frac{1}{2} m v^2 \]
For ball 1:
\[ KE_{1} = \frac{1}{2} m_1 u_1^2 = \frac{1}{2} \times 5 \times (8)^2 = \frac{1}{2} \times 5 \times 64 = 160 \, \text{J} \]

For ball 2 (which is stationary):
\[ KE_{2} = \frac{1}{2} m_2 u_2^2 = \frac{1}{2} \times 15 \times (0)^2 = 0 \, \text{J} \]

Thus, the total initial kinetic energy is:
\[ KE_{initial} = KE_{1} + KE_{2} = 160 + 0 = 160 \, \text{J} \]

### Step 3: Apply conservation of momentum to find the final velocity (V)
In a perfectly inelastic collision, the two objects stick together after the collision. The total momentum before the collision equals the total momentum after the collision.

Initial momentum:
\[ p_{initial} = m_1 u_1 + m_2 u_2 = 5 \times 8 + 15 \times 0 = 40 \, \text{kg m/s} \]

Final momentum (after collision):
\[ p_{final} = (m_1 + m_2) V = (5 + 15) V = 20V \]

Setting initial momentum equal to final momentum:
\[ 40 = 20V \implies V = \frac{40}{20} = 2 \, \text{m/s} \]

### Step 4: Calculate the final kinetic energy (KE_final)
Now we can calculate the kinetic energy after the collision:
\[ KE_{final} = \frac{1}{2} (m_1 + m_2) V^2 = \frac{1}{2} \times 20 \times (2)^2 = \frac{1}{2} \times 20 \times 4 = 40 \, \text{J} \]

### Step 5: Calculate the loss in kinetic energy
The loss in kinetic energy is given by:
\[ \text{Loss in KE} = KE_{initial} - KE_{final} = 160 - 40 = 120 \, \text{J} \]

### Final Answer
The loss in kinetic energy during the collision is **120 J**.

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