Question

A Super Ball has a coefficient of restitution of \(0.9115 .\) From what height should the ball be dropped so that its maximum height on its third bounce is \(2.234 \mathrm{~m} ?\)

Short answer

Answer: The Super Ball should be dropped from an initial height of approximately 3.238 meters.

Step-by-step solution

Understand the formula relating coefficient of restitution and height

The coefficient of restitution relates the ratio of the final kinetic energy to the initial kinetic energy (or final height to initial height) in a bounce. Mathematically, it can be written as: \(e = \sqrt{\frac{h_f}{h_i}}\) In this problem, the coefficient of restitution (e) is \(0.9115\) and we need to find the initial height (\(h_i\)) so that the height after the third bounce (\(h_{f3}\)) is \(2.234\) meters.

Calculate the height after two bounces

Since we know the height after the third bounce, we can use the coefficient of restitution to find the height after the second bounce: \(h_{f2} = \frac{h_{f3}}{e^2}\) Plug in the values: \(h_{f2} = \frac{2.234}{(0.9115)^2}\) Calculate the result: \(h_{f2} \approx 2.6944\) meters

Calculate the initial height

Now that we have the height after the second bounce, we can use the coefficient of restitution once more to find the initial height: \(h_i = \frac{h_{f2}}{e^2}\) Plug in the values: \(h_i = \frac{2.6944}{(0.9115)^2}\) Calculate the result: \(h_i \approx 3.238\) meters So, the Super Ball should be dropped from an initial height of approximately \(\boxed{3.238}\) meters so that its maximum height on its third bounce is \(2.234\) meters.