Explanation:

When a ball is thrown with an initial velocity at an angle θ with the horizontal, it follows a projectile motion. The path of the ball is a parabolic trajectory and it reaches its maximum height when its vertical velocity becomes zero.

Step 1: Analyzing the Motion
To find the power when the ball is at the maximum height, we first need to analyze the motion of the ball.

- The initial velocity of the ball can be resolved into horizontal (ux) and vertical (uy) components as follows:
- ux = u * cos(θ)
- uy = u * sin(θ)

- The time taken by the ball to reach the maximum height can be found using the vertical component of velocity:
- uy = 0 (at maximum height)
- 0 = u * sin(θ) - g * t (where g is the acceleration due to gravity and t is the time)
- t = u * sin(θ) / g

- The maximum height (h) reached by the ball can be found using the vertical component of displacement:
- h = (uy^2) / (2 * g)
- h = (u^2 * sin^2(θ)) / (2 * g)

Step 2: Calculating Power
Power is defined as the rate at which work is done or energy is transferred. In this case, we need to find the power when the ball is at the maximum height.

- The work done on the ball is equal to the change in its potential energy:
- Work = m * g * h (where m is the mass of the ball and g is the acceleration due to gravity)

- The time taken to reach the maximum height can be used to calculate the power:
- Power = Work / time

Step 3: Substituting Values and Simplifying
Let's substitute the values into the equations and simplify to find the final expression for power.

- Substituting the value of h:
- h = (u^2 * sin^2(θ)) / (2 * g)

- Substituting the value of Work:
- Work = m * g * h
- Work = m * g * (u^2 * sin^2(θ)) / (2 * g)
- Work = m * u^2 * sin^2(θ) / 2

- Substituting the value of time:
- t = u * sin(θ) / g

- Substituting the values into the power equation:
- Power = Work / time
- Power = (m * u^2 * sin^2(θ) / 2) / (u * sin(θ) / g)
- Power = (m * u * sin(θ) * g) / 2

Therefore, the power when the ball is at the maximum height of the trajectory is (m * u * sin(θ) * g) / 2.