Primary
Goal: 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.
Secondary
Goals: to introduce geometric
series, to review kinetics and energy from physics.
Hands-On
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 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.
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 after a couple
minutes. 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
= ˝ 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.
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.
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.
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.
See the science fair project discussion
for more information.
copyright: Besson and Styer, Villanova University 12/22/2007