The Answer!

How long would this ball bounce, if you could measure the tiniest micro-bounces? As we just showed, the total time T is given by an infinite series

T = t₁ (1 + r + r² + r³ + r⁴ + ···) = t₁ / (1 − r ).

From the video (using a quadratic curve to fit data obtained using VideoPoint software), one can get somewhat rough measurements for the height of each bounce:

1. 48.2 cm
2. 38.0 cm
3. 30.0 cm
4. 23.7 cm
5. 19.0 cm

The ratios (we labeled this ratio R before) are all close to R = 0.80. This would mean that r = R^{1 / 2} is about 0.9. From t₁ = 2 (h₁ / 4.9)^{1 / 2} we can calculate that the first full bounce takes t₁ = 0.63 seconds.

The geometric series giving the total time is thus T = 0.63 / (1 − 0.9) = 6.3 seconds.

Replay the video, and listen again to the time from the first bounce till it dies out. Mentally estimate the number of seconds and compare your estimate to our calculation of the time for a ball bouncing infinitely often. An analysis of the audio file shows the time from the first bounce to the end to be about 6.42 seconds, consistent with our calculated time.