Does a ball take longer to come down than go up, or
does it take the same amount of time either way?
Of course, in the absence of air friction, a ball
takes the same amount of time either way. But what
if air friction is taken into account? Doesn't
friction oppose the motion in both directions?
Shouldn't both up and down take the same amount of time?
In fact, no.
With air friction, the trip down takes longer!
Presentation Suggestions:
Ask students to think about what a whiffle ball
might do.
The Math Behind the Fact:
While this sounds like a physics problem,
it can be analyzed using a mathematical model
involving ordinary differential equations.
Any commercial ODE solver can numerically
solve such a model.
The analysis reveals that, in fact, a whiffle
ball takes longer to come down than to go up
to its maximum height. One way to intuitively
see this is to analyze extreme cases, such as a
whiffle ball. The whiffle ball can be thrown up
at any velocity, but when it falls, it will reach
terminal velocity and no more.
The key is to notice that, with friction,
the descent phase is not the timereversal
of the ascent phase, unlike the nofriction case.
The ball is generally moving faster on its way up
than at the corresponding point on its way down,
because in the descent phase, the frictional force
is opposing the force of gravity, rather than
pulling in the same direction (as they do when the ball
is rising).
How to Cite this Page:
Su, Francis E., et al. "Whiffle Ball."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.

