To analyze the motion of the ball thrown upwards and then held before being released downwards, we need to break the problem into two parts: the upward motion and the downward motion after it is released. Let's go through the steps to find the maximum height reached by the ball and the total distance it covers.
Understanding the Motion of the Ball
When the ball is thrown upwards, it will rise until it reaches its maximum height, where its velocity becomes zero before it starts descending. The height it reaches can be calculated using the initial velocity and the acceleration due to gravity.
Calculating Maximum Height
Let’s denote:
- u = initial velocity of the ball (when thrown upwards)
- g = acceleration due to gravity (approximately 9.81 m/s²)
- h = height of the building
- t = time the ball is held before being released downwards
The maximum height (H) reached by the ball can be calculated using the formula:
H = (u²) / (2g)
This formula derives from the kinematic equations of motion, where the final velocity at the maximum height is zero.
Considering the Time Held
After reaching its maximum height, the ball is held for time t. During this time, it does not gain any additional height. When it is released, it will start falling from the maximum height H.
Calculating Total Distance Covered
The total distance covered by the ball consists of two parts:
- The distance it traveled upwards to reach the maximum height (H)
- The distance it falls back down to the ground after being released
Distance Falling Down
When the ball is released from height H, it will fall down to the ground. The distance it falls is equal to H. Therefore, the total distance covered by the ball is:
Total Distance = H + H = 2H
Final Formulas
To summarize, the maximum height reached by the ball is:
H = (u²) / (2g)
And the total distance covered by the ball is:
Total Distance = 2H
Example Calculation
Let’s say the initial velocity (u) of the ball is 20 m/s. Plugging this into our formula for maximum height:
H = (20²) / (2 * 9.81) ≈ 20.4 m
Thus, the total distance covered by the ball would be:
Total Distance = 2 * 20.4 m = 40.8 m
This example illustrates how to approach the problem step-by-step, ensuring clarity in understanding the motion of the ball in both upward and downward directions.
