A uniform rod lies at rest on a frictionless horizontal surface. A particle, having a mass equal to that of the rod, moves on the surface perpendicular to the rod collides with it, and finally sticks to it. The minimum loss of KE in the collision , under the given conditions, is

Options

A. `5%`
B. `10%`
C. `20%`
D. `30%`

Explanation

To solve the problem of a uniform rod colliding with a particle of equal mass and finding the minimum loss of kinetic energy (KE) in the collision, we can follow these steps:

### Step 1: Define the system
Let the mass of the rod be \( m \) and the mass of the particle also be \( m \). The rod is initially at rest, and the particle is moving with an initial velocity \( v \) perpendicular to the rod.

### Step 2: Calculate the initial kinetic energy
The initial kinetic energy (KE_initial) of the system is solely due to the moving particle since the rod is at rest. Therefore, we can express the initial kinetic energy as:
\[ KE_{\text{initial}} = \frac{1}{2} m v^2 \]

### Step 3: Analyze the collision
Since the particle collides with the rod and sticks to it, this is an inelastic collision. After the collision, the combined mass of the rod and the particle will be \( 2m \).

### Step 4: Calculate the final velocity after collision
Using conservation of momentum, the momentum before the collision must equal the momentum after the collision. The initial momentum of the system is:
\[ p_{\text{initial}} = mv \]
After the collision, the total momentum is:
\[ p_{\text{final}} = (2m) V \]
where \( V \) is the final velocity of the combined system. Setting the initial momentum equal to the final momentum gives:
\[ mv = (2m)V \]
Solving for \( V \):
\[ V = \frac{v}{2} \]

### Step 5: Calculate the final kinetic energy
The final kinetic energy (KE_final) of the combined system is:
\[ KE_{\text{final}} = \frac{1}{2} (2m) V^2 = \frac{1}{2} (2m) \left( \frac{v}{2} \right)^2 = \frac{1}{2} (2m) \left( \frac{v^2}{4} \right) = \frac{mv^2}{4} \]

### Step 6: Calculate the loss of kinetic energy
The loss of kinetic energy during the collision can be calculated as:
\[ \text{Loss of KE} = KE_{\text{initial}} - KE_{\text{final}} = \frac{1}{2} mv^2 - \frac{mv^2}{4} \]
Simplifying this expression:
\[ \text{Loss of KE} = \frac{2mv^2}{4} - \frac{mv^2}{4} = \frac{mv^2}{4} \]

### Step 7: Find the minimum loss of KE
Thus, the minimum loss of kinetic energy in the collision is:
\[ \text{Minimum Loss of KE} = \frac{mv^2}{4} \]

### Final Answer
The minimum loss of kinetic energy in the collision is \( \frac{mv^2}{4} \).

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