A particle moving with kinetic energy E makes a head on elastic collision with an identical particle at rest. During the collision

Options

A. elastic potential energy of the system is always zero
B. maximum elastic potential energy of the systyem is E/2
C. minimum kinetic energy of the system is E/2
D. kinetic energy of the system is constant

Answer

To solve the problem of a particle moving with kinetic energy \( E \) making a head-on elastic collision with an identical particle at rest, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- We have two identical particles, each with mass \( m \).
- The first particle is moving with kinetic energy \( E \).
- The second particle is at rest.

2. **Relate Kinetic Energy to Velocity**:
- The kinetic energy \( E \) of the first particle can be expressed as:
\[ E = \frac{1}{2} m v^2 \]
- From this, we can find the velocity \( v \) of the first particle:
\[ v = \sqrt{\frac{2E}{m}} \]

3. **Apply Conservation of Momentum**:
- Before the collision, the momentum of the system is:
\[ p_{\text{initial}} = mv + 0 = mv \]
- After the collision, let the first particle come to rest and the second particle move with velocity \( v' \):
\[ p_{\text{final}} = 0 + mv' = mv' \]
- By conservation of momentum:
\[ mv = mv' \]
- Thus, we have:
\[ v' = v \]

4. **Apply Conservation of Kinetic Energy**:
- Since the collision is elastic, the total kinetic energy before and after the collision must be equal:
\[ E_{\text{initial}} = E_{\text{final}} \]
- The initial kinetic energy is \( E \) and the final kinetic energy is:
\[ E_{\text{final}} = \frac{1}{2} m (0)^2 + \frac{1}{2} m (v')^2 = \frac{1}{2} m v^2 \]
- Therefore, we have:
\[ E = \frac{1}{2} m v^2 \]

5. **Determine the Minimum Kinetic Energy During Collision**:
- During the collision, the kinetic energy is not constant. The minimum kinetic energy occurs when the two particles are in contact.
- Since the maximum potential energy during the collision is equal to the decrease in kinetic energy, we can find the minimum kinetic energy:
\[ E_{\text{min}} = \frac{E}{2} \]

6. **Calculate the Change in Energy**:
- The change in energy, which corresponds to the potential energy during the collision, is:
\[ \Delta E = E - E_{\text{min}} = E - \frac{E}{2} = \frac{E}{2} \]

### Conclusion:
- The minimum kinetic energy of the system during the collision is \( \frac{E}{2} \).
- The change in energy during the collision (which is the maximum potential energy) is also \( \frac{E}{2} \).