# The Journal of Online Mathematics and Its Applications

Volume 7. May, 2007. Article ID 1550

# Bouncing Balls and Geometric Series

### Villanova University

#### Abstract

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

#### Keywords

• geometric series
• bouncing ball
• conservation of energy
• conservation of momentum

## Introduction

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.

#### Background

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.