The Journal of Online Mathematics and Its Applications, Volume 7 (2007)

Bouncing Balls and Geometric Series, Robert Styer and Morgan Besson

We now can find the total time. Our laboratory experience measuring the times with a stopwatch indicated that it is best to start timing from the instant the ball hits the table, because the noise of impact helps immensely in starting timing. So by "total time" we will mean the total time for all bounces, starting with he instant the ball first hits the table.

For convenience, we set `r` = `R`^{1 / 2}. In case you enjoy big vocabulary words, recall that `R` is called the elasticity coefficient; the new parameter `r` is the coefficient of restitution.

We have the relation of each bounce height and time to the first bounce height and time; now we will sum all the times.