A particle of mass `m` and momentum `p` moves on a smooth horizontal table and collides directly and elastically with a similar particle (of mass m) having momentum `-2p`. The loss `(-)` or gain `(+)` in the kinetic energy of the first particle in the collision is

Options

A. (a) `+(p^2)/(2m)`
B. (b) `-(p^2)/(4m)`
C. (c) `+(3p^2)/(2m)`
D. (d) zero

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given data
We have two particles, both with mass \( m \). The first particle has momentum \( p \), and the second particle has momentum \( -2p \).

### Step 2: Calculate the velocities of the particles
Using the formula for momentum \( p = mv \), we can find the velocities of both particles.

- For the first particle:
\[ p = mv_1 \implies v_1 = \frac{p}{m} \]

- For the second particle:
\[ -2p = mv_2 \implies v_2 = \frac{-2p}{m} \]

### Step 3: Calculate the initial kinetic energies
The kinetic energy \( KE \) of a particle is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]

- For the first particle:
\[ KE_1 = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{1}{2} m \cdot \frac{p^2}{m^2} = \frac{p^2}{2m} \]

- For the second particle:
\[ KE_2 = \frac{1}{2} m \left(\frac{-2p}{m}\right)^2 = \frac{1}{2} m \cdot \frac{4p^2}{m^2} = \frac{2p^2}{m} \]

### Step 4: Calculate the total initial kinetic energy
The total initial kinetic energy \( KE_{initial} \) is the sum of the kinetic energies of both particles:
\[ KE_{initial} = KE_1 + KE_2 = \frac{p^2}{2m} + \frac{2p^2}{m} = \frac{p^2}{2m} + \frac{4p^2}{2m} = \frac{5p^2}{2m} \]

### Step 5: Analyze the elastic collision
In an elastic collision between two equal masses, the velocities are exchanged. Thus, after the collision:
- The first particle will have the velocity of the second particle \( v_2 \).
- The second particle will have the velocity of the first particle \( v_1 \).

- After the collision, the velocity of the first particle becomes \( v_2 = \frac{-2p}{m} \).

### Step 6: Calculate the final kinetic energy of the first particle
Now we calculate the kinetic energy of the first particle after the collision:
\[ KE_{final} = \frac{1}{2} m \left(\frac{-2p}{m}\right)^2 = \frac{1}{2} m \cdot \frac{4p^2}{m^2} = \frac{2p^2}{m} \]

### Step 7: Calculate the change in kinetic energy
The change in kinetic energy \( \Delta KE \) for the first particle is given by:
\[ \Delta KE = KE_{final} - KE_{initial} = \frac{2p^2}{m} - \frac{p^2}{2m} \]

To combine these:
\[ \Delta KE = \frac{4p^2}{2m} - \frac{p^2}{2m} = \frac{3p^2}{2m} \]

### Step 8: Determine if it is a gain or loss
Since \( \Delta KE \) is positive, it indicates a gain in kinetic energy.

### Final Answer
The gain in kinetic energy of the first particle in the collision is:
\[ \frac{3p^2}{2m} \]