Notes for Instructors: Bouncing Ball and Geometric Series Lab
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. But since we teach a combined math-physics class, we also used 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 “cognitive
dissonance” which probes our students’ understandings of infinity.
Primary Goal: to create cognitive dissonance before beginning the formal study of infinite series
that will undergird our later insistence on careful definitions and proofs.
Secondary Goals: to introduce geometric series, to review kinetics and energy from physics.
In-Class Mini-Laboratory: 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 pingping 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 about ten minutes
maximum 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.
Problems they will probably encounter are: the problem of determining height because of
parallax, exactly what point of the ball to use for the measurement, the error in their seeing how
high it bounces, and which way to hold the ruler (the zero point should be on the table!). Note we
let each group reason out why to use the bottom of the ball for measuring rather than the middle
or the top (since the meter stick standing on the table defines y=0 as the bottom), but unless one
intends to personally supervise each group, one should probably just tell them to use the bottom
of the ball. Typical starting heights should be about a meter down to about half a meter; starting
heights below half a meter tend to have 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 of the bounce height to the start height, which
is the elasticity coefficient R. 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 s = ½ 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 “Proofs without Words” referenced in the
module.
We slow down at the end, after deriving the symbolic formula for the total time (the final page of
this module before the acknowledgements). 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. For those students, we have achieved the cognitive dissonance that allows us to emphasize
later in the course how important careful definitions and proofs for convergence are.
Note: 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.
Note: 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, in the end, when your calculation of the total time is within ten
percent of what they measured with their stopwatches, they are impressed.
Note: If you are genuinely interested in what happens for very small bounces, this lab will not be
useful, because one cannot see the low heights accurately. In this case, one can measure the
restitution coefficient 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, the
restitution coefficient, which is the square root of the elasticity coefficient found in the above lab
exercise. Also see the science fair project linked to our web page with title “How a ball actually
bounces” for more information.