A tennis ball is hit vertically upward and returns 6 sec later calculate. A) the greatest height reached by the ball. B) the initial velocity of the ball .

To calculate the greatest height reached by the ball and the initial velocity, we need to use the equations of motion for vertical motion. The key equation is:

y = ut + (1/2)at^2,

where:
y is the vertical displacement or height,
u is the initial velocity,
t is the time, and
a is the acceleration due to gravity (-9.8 m/s^2 for objects near the Earth's surface).

A) The greatest height reached by the ball can be determined by finding the highest point of its trajectory. At this point, the vertical velocity is zero, and the total time for the ball to reach this point is half of the total time for the complete motion. In this case, t = 6 seconds, so the time to reach the highest point would be t/2 = 6/2 = 3 seconds.

1. First, calculate the initial velocity (u) using the equation:
0 = u + a(t/2)
Rearranging the equation, we get u = -a(t/2).

Substituting the known values, u = -(9.8)(3/2) = -14.7 m/s.

2. Next, calculate the height (y) using the equation:
y = ut + (1/2)at^2.
Replace u with the initial velocity we found in the previous step:
y = -14.7(3) + (1/2)(-9.8)(3^2).

Simplifying the equation:
y = -44.1 - 44.1 = -88.2 m.

The greatest height reached by the ball is 88.2 meters above its initial position.

B) The initial velocity of the ball can also be determined from the above calculations. The negative sign indicates that the ball was hit in the upward direction:
The initial velocity of the ball is -14.7 m/s.