Understanding the Physics

* Symmetry: The motion of the ball going up and coming down is symmetrical. This means the time it takes to go up is equal to the time it takes to come down (4 seconds each in this case).

* Acceleration due to gravity: The ball is accelerating downwards due to gravity (approximately 9.8 m/s²).

Using the Equations of Motion

We can use the following equation of motion to find the initial velocity (u):

* s = ut + (1/2)at²

Where:

* s = displacement (0 in this case, as the ball returns to its starting point)

* u = initial velocity (what we want to find)

* t = time (4 seconds, the time it takes to go up)

* a = acceleration due to gravity (-9.8 m/s² - negative because it acts downwards)

Solving for Initial Velocity

1. Plug in the values: 0 = u(4) + (1/2)(-9.8)(4)²

2. Simplify: 0 = 4u - 78.4

3. Solve for u: 4u = 78.4

4. Calculate: u = 19.6 m/s

Therefore, the initial velocity of the ball is 19.6 m/s.