Bop 'til you drop
by M. Gallant
Question:
A rubber ball is dropped from rest at a height of 10 feet. On each successive bounce,
the ball reaches half the previous height attained. How far has the ball traveled
when it finally comes
to rest? How long will this take?
In this puzzle, we wish to know how far and how long an idealistic bouncing ball
would travel, given that it attains half of its previous height on each bounce,
starting from rest at a height of 10 feet. It is easy to calculate the
total distance as thirty feet, recognizing that the well known infinite series
(1/2 + 1/4 + 1/8 + 1/16 +..... ) adds up to a FINITE sum of exactly 1.0.
Surprisingly, most people immediately and incorrectly guess that the
time involved would be infinite. The time of each bounce shortens quickly, and
using the simple expression d = 1/2*g*t^2 for the distance d traveled from rest
during the time t under gravity g=32 feet/sec/sec, a very similar infinite series
to that mentioned above leads to a FINITE time of 4.61 sec. required for the ball
to come to rest. In fact, most "reasonable" real balls bounce LONGER than this,
as common experience shows (our model ball is pretty "lossy" on bouncing ;
take a tennis ball and watch it carefully as it bounces, noting the height
attained on successive bounces). The concept of an infinite series adding up
to a finite number is one of the fundamental ideas that lead to the invention
of the Calculus.
Java applet simulation of the
bouncing ball.
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