A particle moving with kinetic energy K makes a head on elastic collision with an identical particle at rest. Find the maximum elastic potential energy of the system during collision.

Explanation

To solve the problem of finding the maximum elastic potential energy of the system during the collision, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- We have two identical particles. One particle is moving with kinetic energy \( K \), and the other is at rest.
- The mass of each particle is denoted as \( m \).

2. **Relating Kinetic Energy to Velocity**:
- The kinetic energy \( K \) of the moving particle can be expressed as:
\[ K = \frac{1}{2} mv^2 \]
- From this, we can solve for the initial velocity \( v_i \):
\[ v_i = \sqrt{\frac{2K}{m}} \]

3. **Applying Conservation of Momentum**:
- In an elastic collision, momentum is conserved. The initial momentum of the system is:
\[ p_{initial} = mv_i + 0 = mv_i \]
- After the collision, let the final velocities of the two particles be \( v_f \) (for the initially moving particle) and \( v_l \) (for the initially stationary particle). Since they are identical, we can use the conservation of momentum:
\[ mv_i = mv_f + mv_l \]
- Since the collision is elastic and the particles are identical, we can also apply the elastic collision formula which gives:
\[ v_f = 0 \quad \text{and} \quad v_l = v_i \]

4. **Maximum Potential Energy During Collision**:
- At the point of maximum compression during the collision, the kinetic energy of the system is converted to potential energy. The total kinetic energy before the collision is \( K \).
- The maximum elastic potential energy \( U_{max} \) can be expressed as:
\[ U_{max} = \frac{1}{2} k_{max} x^2 \]
- However, we can also relate it directly to the initial kinetic energy. In elastic collisions, the maximum potential energy stored during the collision is equal to the initial kinetic energy of the moving particle divided by 4:
\[ U_{max} = \frac{1}{4} K \]

5. **Final Result**:
- Therefore, the maximum elastic potential energy of the system during the collision is:
\[ U_{max} = \frac{1}{4} K \]

### Summary:
The maximum elastic potential energy of the system during the collision is \( \frac{1}{4} K \).