Bouncing balls

The goal of this project is to study the motion of bouncing balls. In physics this motion is described as inelastic collision, in which the ball loses some fraction of its energy at each bounce. This is described by the equation:

where $E_0$ is the initial energy of the ball, $E_n$ is the energy after $n$ bunces, and $0 \leq C < 1$ is a constant that depends on properties of the ball and the surface on which it is dropped. This formula can be used to predict times between consecutive bounces: if $t_i$ is the duration of time between bounces $i$ and $i+1$ then $t_{i+1} = \sqrt{C} t_i$.

Project¶

1. Here is an audio file with the sound of a bouncing pingpong ball:

pingpong.wav

```
https://cdn.jsdelivr.net/gh/bbadzioch/mth337_site@main/projects/bouncing_balls/data/pingpong.wav
```

• Write a function that retrieves times of ball bounces from an audio recording and apply it to the above file.

• Create a plot showing the sound waveform together with markers for computed bounce times, to confirm that these times are correct.

• Check how closely the timing data you computed conforms to the formula given by the inelastic collision model.

• Compute the coefficient $C$ for the pingpong ball.

2. This YouTube video demonstrates a ball bouncing on the “atomic trampoline” i.e. a specially designed metal alloy. In the the video, this is compared to a steel ball bouncing on a steel surface. Here are audio files extracted from this video, with recordings of bounces in each case:

   atomic_trampoline:

    ```
    https://cdn.jsdelivr.net/gh/bbadzioch/mth337_site@main/projects/bouncing_balls/data/atomic_trampoline.wav
    ```

   steel_surface:

    ```
    https://cdn.jsdelivr.net/gh/bbadzioch/mth337_site@main/projects/bouncing_balls/data/steel_surface.wav
   
    ```

   Repeat steps from part 1 for these two recordings. How much more energy the ball bouncing on the atomic trampoline retains after a bounce as compared to the ball bouncing on steel?