The Journal of Online Mathematics and Its Applications, Volume 7 (2007)
Bouncing Balls and Geometric Series, Robert Styer and Morgan Besson
Before beginning the formal study of infinite series, we create "cognitive dissonance" that will undergird our later insistence on careful definitions and proofs rather than intuition.
We also want to introduce geometric series, to review kinetics and energy from physics.
The introductory movie on this web module is not nearly as effective as a hands-on in-class mini-lab! Here is an outline of what we do in class.
We begin with the introductory question: if a ball bounces half its height each time, and if it is a "mathematical ball" that would continue bouncing half as high infinitely many times, how much time would it take to bounce an infinite number of times? After some prodding, students will answer, almost always saying "forever" or "infinitely long."
We now divide the students into groups of three or four. Each team is given a meter stick, a ping-ping ball or super ball, a stopwatch, and a piece of paper to record their data.
The goal of the mini-lab is to have each team find the elasticity coefficient R of its ball (sometimes this is referred to as the restitution coefficient but generally the restitution coefficient is the ratio of the velocities rather than the heights). Each team will drop the ball from some height and record how high it bounces. Each team creates a table with three columns: the first column records the starting heights, the second column the corresponding heights of the resulting bounce, and the third will be the ratio of the second over the first. We spend almost ten minutes on this; students need time to learn how to drop the ball near the ruler and how to determine the height of the resulting bounce.
The students will encounter several problems:
Ideally each group should figure out for itself why to use the bottom of the ball for measuring rather than the middle or the top. (The reason is that the meter stick standing on the table defines y = 0 as the bottom). But unless the teacher intends to personally supervise each group, it's probably best to just tell the students to use the bottom of the ball after a couple minutes. Typical starting heights should be from about a meter down to half a meter; starting heights below half a meter tend to result in large relative errors in the start height versus the bounce height. In five minutes, students can get three to ten measurements, the first column of each row on their paper listing the starting height, ranging roughly from 100 cm down to 50 cm, and the second column recording the height of the resulting bounce.
Finally, have the students drop the ball from one meter and record the total time it takes to stop bouncing.
Now the students calculate a third column, the ratio R of the bounce height to the starting height. Typically, the ratios are 65-80% for ping-pong balls or super balls. Each team should discuss their data and determine a consensus elasticity coefficient. We do not use a formal average partly because there are subtle numerical analysis issues involved, but mostly because we only need a rough estimate to make our point.
Most science and engineering students have had high school physics and college algebra classes, so they are familiar with kinetics formulas like h = (1 / 2) g t2 and the formula for summing a geometric series. This web module spends a fair amount of time discussing these points, but in class, we go through the symbolic calculations very quickly, giving a handout with some pictorial proofs of the sum of a geometric series taken from the book Proofs without Words (reference).
We slow down at the end, after deriving the symbolic formula for the total time. We have the students plug in their value of R to calculate the total bouncing time according to the geometric series formula, then compare the calculated total time to their measured total time. Generally, the answers are about ten percent off, which students seem to find satisfactory.
Finally, we ask again how much time it would take if the ball bounced infinitely many times. At this point, some of the students still answer "forever" and will go away from class scratching their heads. Many of the rest still think this though they will not admit it. For most students, we have achieved the cognitive dissonance that allows us to stress the dangers of intuition and the importance of careful definitions and proofs for convergence.
Please pretest the balls to make sure they bounce fairly straight. Small ridges on super balls or tiny cracks in ping pong balls are very annoying. Also, we originally got the idea from a calculus book that had the ball dropped from ten feet; obviously, the calculus textbook author had a basketball in mind. So we tried it in class with a basketball. But neither a basketball nor a tennis ball has a constant elasticity coefficient (it increases as the height decreases), so in the end the calculated total time was off by a factor of more than two. We discovered that super balls and ping-pong balls give a total time of bouncing that closely matches the answer given by the infinite series.
A hands-on lab has great advantage over a mere video. This lab enhances the students' understanding of the elasticity coefficient which is critical in the infinite series argument. This lab has the added benefit that students generate real data and see the role of error. Because they see how much error is involved, the students are impressed in the end, when the calculation of the total time is within ten percent of the measured time.
If you are genuinely interested in what happens for very small bounces, this lab will not be useful, because the low heights can not be measured accurately by sight. However, the restitution coefficient can be measured by recording the sounds of the bounces. Using any sound editing program, one can see the sound spiking for the first five to fifteen bounces, and thus find the time between bounces. The ratio of successive times is of course the same as the ratio of the velocities. This is the restitution coefficient, the square root of the elasticity coefficient. See the discussion of how a ball actual bounces for more information.