Answer:

The variation of kinetic energy (K) with vertical displacement (y) of a particle projected with kinetic energy (Ko) and an angle of projection (θ) can be explained by considering the basic principles of projectile motion.

Projectile Motion:
When a particle is projected into the air at an angle, it follows a curved path called a projectile motion. This motion can be divided into two independent components: horizontal motion and vertical motion.

Horizontal Motion:
The horizontal motion of the particle remains constant throughout its path. There is no external force acting horizontally on the particle, so its velocity in the horizontal direction remains constant. Therefore, the kinetic energy in the horizontal direction remains constant.

Vertical Motion:
The vertical motion of the particle is influenced by the force of gravity. As the particle moves upward, the force of gravity acts against its motion, slowing it down. As the particle reaches its maximum height, its velocity becomes zero. Then, as it falls back to the ground, the force of gravity accelerates it, increasing its velocity.

Variation of Kinetic Energy (K) with Vertical Displacement (y):
1. Linear Variation:
When the particle is projected vertically upwards or downwards (θ = 90° or 270°), the variation of kinetic energy with vertical displacement is linear. As the particle moves upward, its kinetic energy decreases linearly until it reaches its maximum height, where the kinetic energy becomes zero. Similarly, as the particle falls downward, its kinetic energy increases linearly.

2. Parabolic Variation:
When the particle is projected at an angle other than 90° or 270°, the variation of kinetic energy with vertical displacement is parabolic. The kinetic energy initially decreases as the particle moves upward, reaches its minimum value at the maximum height, and then increases symmetrically as the particle falls back to the ground.

3. Hyperbolic Variation:
Hyperbolic variation of kinetic energy with vertical displacement is not observed in projectile motion.

4. Periodic Variation:
Periodic variation of kinetic energy with vertical displacement is not observed in projectile motion.

In conclusion, the variation of kinetic energy (K) with vertical displacement (y) in projectile motion can be either linear or parabolic, depending on the angle of projection.