3. Derive the expression for the range of the ball as a function of its height.

Question

3. Derive the expression for the range of the ball as a function of its height. To do this you should
use the following steps. In the first part of the motion, the ball undergoes vertical free fall until it
hits the bounce plate. At this point, its final velocity in the y-direction becomes the initial
velocity in the x-direction after hitting the plate. Now calculate the time it takes the ball to hit the
ground after hitting the bounce plate. Lastly, find the distance that the ball will travel in the
horizontal direction in this time. This is the range of the ball.

Show steps, thanks.

Explanation / Answer

let the free fall height is h, so h=v^2/2g => v= sqrt(2gh). Now since the intial velocity in the x direction is known which is sqrt(2gh). and now the initial velocity in y direction is 0. Now if the remaining height is H, so the time taken to reach ground = H=1/2 * g * t^2

t= sqrt(2gH)

so the total distance travelled in x direction which is the range is given by sqrt(2gh)*sqrt(2gH) => Range = 2g* sqrt(hH)

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