A ball of mass m is moving with velocity u and collides head on with another identical body at rest . What is the maximum possible loss of kinetic energy due to collision ?

Options

A. `1/2"mu"^(2)`
B. `1/3"mu"^(2)`
C. `1/4"mu"^(2)`
D. `1/5"mu"^(2)`

Explanation

To solve the problem of finding the maximum possible loss of kinetic energy due to a head-on collision between two identical balls, we can follow these steps:

### Step 1: Understand the scenario
We have two identical balls, each with mass \( m \). One ball is moving with an initial velocity \( u \) while the other is at rest.

### Step 2: Determine the type of collision
For maximum loss of kinetic energy, we consider a perfectly inelastic collision, where the two balls stick together after the collision. In this case, the coefficient of restitution is zero.

### Step 3: Apply conservation of momentum
The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision.

Before the collision:
- Momentum of the moving ball = \( mu \)
- Momentum of the stationary ball = \( 0 \)

Total initial momentum = \( mu + 0 = mu \)

After the collision, let \( v \) be the common velocity of the two balls (since they stick together):
- Total mass after collision = \( m + m = 2m \)
- Total momentum after collision = \( (2m)v \)

Setting the initial momentum equal to the final momentum:
\[ mu = 2mv \]

### Step 4: Solve for the final velocity \( v \)
From the equation \( mu = 2mv \), we can simplify it:
\[ v = \frac{u}{2} \]

### Step 5: Calculate initial and final kinetic energy
1. **Initial kinetic energy (KE1)** of the moving ball:
\[ KE_1 = \frac{1}{2}mu^2 \]

2. **Final kinetic energy (KE2)** after the collision:
\[ KE_2 = \frac{1}{2}(2m)v^2 = \frac{1}{2}(2m)\left(\frac{u}{2}\right)^2 = \frac{1}{2}(2m)\left(\frac{u^2}{4}\right) = \frac{mu^2}{4} \]

### Step 6: Calculate the loss of kinetic energy
The loss of kinetic energy due to the collision is given by:
\[ \text{Loss} = KE_1 - KE_2 \]
Substituting the values we found:
\[ \text{Loss} = \frac{1}{2}mu^2 - \frac{mu^2}{4} \]

To simplify:
\[ \text{Loss} = \frac{2}{4}mu^2 - \frac{1}{4}mu^2 = \frac{1}{4}mu^2 \]

### Conclusion
Thus, the maximum possible loss of kinetic energy due to the collision is:
\[ \text{Loss} = \frac{1}{4}mu^2 \]