A particle is projected at ` 60(@)` to the horizontal with a kinetic energy `K` . The kinetic energy at the highest point is

Options

A. ` K//2`
B. `K`
C. `Zero`
D. `K//4`

Explanation

To find the kinetic energy of a particle at its highest point after being projected at an angle of 60 degrees to the horizontal, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Initial Kinetic Energy**:
The initial kinetic energy (K.E) of the particle when it is projected is given as \( K \).

2. **Break Down the Velocity Components**:
When the particle is projected at an angle of \( 60^\circ \), its initial velocity \( V \) can be broken down into two components:
- Horizontal component: \( V_x = V \cos(60^\circ) \)
- Vertical component: \( V_y = V \sin(60^\circ) \)

Using the values of cosine and sine:
- \( \cos(60^\circ) = \frac{1}{2} \)
- \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)

Therefore:
- \( V_x = V \cdot \frac{1}{2} = \frac{V}{2} \)
- \( V_y = V \cdot \frac{\sqrt{3}}{2} \)

3. **Velocity at the Highest Point**:
At the highest point of the projectile's motion, the vertical component of the velocity becomes zero (as the particle stops rising and is about to fall). Thus, the only component of velocity at the highest point is the horizontal component:
- \( V_{highest} = V_x = \frac{V}{2} \)

4. **Calculate Kinetic Energy at the Highest Point**:
The kinetic energy at the highest point can be calculated using the formula:
\[ K.E_{highest} = \frac{1}{2} m V_{highest}^2 \]
Substituting \( V_{highest} = \frac{V}{2} \):
\[ K.E_{highest} = \frac{1}{2} m \left(\frac{V}{2}\right)^2 = \frac{1}{2} m \cdot \frac{V^2}{4} = \frac{1}{8} m V^2 \]

5. **Relate to Initial Kinetic Energy**:
The initial kinetic energy is given by:
\[ K = \frac{1}{2} m V^2 \]
Therefore, we can express \( K.E_{highest} \) in terms of \( K \):
\[ K.E_{highest} = \frac{1}{8} m V^2 = \frac{1}{4} \cdot \frac{1}{2} m V^2 = \frac{1}{4} K \]

6. **Final Answer**:
Thus, the kinetic energy at the highest point is:
\[ K.E_{highest} = \frac{K}{4} \]

### Conclusion:
The kinetic energy at the highest point is \( \frac{K}{4} \).