Measuring the Coefficient of Restitution

Question

The coefficient of restitution, $C_\mathrm{R}$, of a ball dropped vertically onto a surface is the ratio of the ball's speed just before it hits the surface to its speed immediately after it bounces back from it. If the ball is dropped from a height $h_0$ above the surface, it is easy to show that it will be travelling with speed $v_0 = \sqrt{2g/h_0}$ when it hits; its speed as it bounces back up will then be $v_1 = C_\mathrm{R}v_0$.

More generally, the upward speed of the ball after the $n$th bounce is: $$ v_{n+1} = v_n C_\mathrm{R}^n. $$

An experiment to measure $C_\mathrm{R}$, assuming it is independent of the ball's speed, may be conducted as follows. Drop the ball from a height $h_0$ at time $t=0$ and record the sound of the ball hitting the surface over a number of bounces. The time between the bounce events is then $$ T_n = T_0 C_\mathrm{R}^n, $$ (show this), where $T_0 = 2v_0/g$. Then $$ \ln T_n = n\ln C_\mathrm{R} + \ln T_0, $$ and so a plot of $\ln T_n$ against $n$ should be a straight line with gradient $\ln C_\mathrm{R}$.

The data file bounce-data.txt is a text-version of the recording of an experiment carried out as described above, with a sampling frequency of $4410\;\mathrm{s^{-1}}$ (i.e. data points are separated by $1/4410\;\mathrm{s}$). Analyse these data to find a value for $C_\mathrm{R}$. If $h_0 = 57.5\;\mathrm{cm}$, what is the value of $g$ inferred from the intercept?


Solution

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