Introduction:
When a particle is projected from the ground at an angle theta with the horizontal, it follows a curved trajectory. At the highest point of this trajectory, the particle's velocity becomes zero. We need to find the value of theta for which the kinetic energy of the particle is equal to its potential energy.

Understanding the Problem:
To solve this problem, we need to consider the conservation of energy. At any point along the trajectory, the total mechanical energy (sum of kinetic energy and potential energy) remains constant. At the highest point, the kinetic energy is zero, so the total mechanical energy is equal to the potential energy.

Derivation:
Let's break down the problem into two components: horizontal and vertical.

Horizontal Component:
- The horizontal component of the velocity remains constant throughout the motion.
- Therefore, the kinetic energy in the horizontal direction remains constant and is equal to 1/2 * m * v^2, where m is the mass of the particle and v is the initial velocity.

Vertical Component:
- The vertical component of the velocity changes as the particle moves up and then comes down.
- At the highest point, the vertical velocity becomes zero, so the kinetic energy in the vertical direction is zero.
- The potential energy at the highest point is given by m * g * h, where g is the acceleration due to gravity and h is the maximum height reached by the particle.

Equating Kinetic and Potential Energy:
Since the total mechanical energy is conserved, we can equate the kinetic energy in the horizontal direction to the potential energy at the highest point:
1/2 * m * v^2 = m * g * h

Simplification:
Simplifying the equation, we get:
v^2 = 2 * g * h

Conclusion:
From the above equation, we can see that the angle theta does not appear in the equation. Therefore, the value of theta does not affect the equality between kinetic and potential energy at the highest point of the trajectory. The kinetic energy at the highest point will always be equal to the potential energy regardless of the angle theta.

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