Whiffle Ball

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!

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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 time-reversal of the ascent phase, unlike the no-friction 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).

Su, Francis E., et al. “Whiffle Ball.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

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Michael Moody

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