The Journal of Online Mathematics and Its Applications

Volume 7. May, 2007. Article ID 1550

Bouncing Balls and Geometric Series

Robert Styer and Morgan Besson

Villanova University


This teaching module explores the time and distance of a bouncing ball and leads to a study of the geometric series.




  1. The Question
  2. Typical Student Answers
  3. Infinitely Bouncing Balls
  4. Bounce Height and Time I
  5. Bounce Height and Time II
  6. Bounce Height and Time III
  7. Geometric Series I
  8. Geometric Series II
  9. The Answer!

Supplementary Material

Source Files


If a ball bounces an infinite number of times, it must take an infinite amount of time to finish bouncing! This obvious "fact" motivates our module on infinite series.


The ideas in this module were developed for a combined math-physics course. Before beginning infinite series towards the end of the second semester of calculus, we want to create "cognitive dissonance" in our students, who otherwise think they intuitively understand processes involving infinity. The bouncing ball geometric series is a nice example related to Zeno's paradoxes that forces students to think about how infinitely many discrete steps can sum to a finite answer. In the combined math-physics class, we also use this module to review kinetics and energy.

We actually use this example in conjunction with the "tower of bricks" harmonic series example, and the "towers of exponents" with √2 versus √3. These three examples create the "cognitive dissonance" which probes our students' understandings of infinity.

The module is a web version of our in-class hands-on activity. Please view the module first then use the laboratory notes to implement the hands-on lab in your own classroom.