What Goes Up Must Come Down

By Brian Winkel


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Brauer, F. 1999. What Goes Up Must Come Down. American Mathematical Monthly. 108(5): 437-440.

Article Abstract: NONE

This paper is a wonderfully general analysis of the following, “It is natural to ask whether a particle propelled upwards takes longer to fall to earth from its maximum height than it takes to rise to this maximum height for frictional forces that are nonlinear functions of velocity. Since linear and quadratic retarding forces are at best approximations, we would like to answer the question for a general force function. The purpose of this Note is to establish that the falling time is greater than the rising time in general.”

The paper attempts to determine some qualitative behavior notions using an explicit solution but in a qualitative manner. The proof and analysis are very easy to follow and a typical student could be assigned to read this work and report back to the class, perhaps with a request to do some specific situations of resistance functions, f(v) in the differential equation for velocity of an object thrown up into the medium to return to the ground due to gravity: mv = − mgf(v),v(0) = v0.

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Researchers should cite this work as follows:

  • Brian Winkel (2017), "What Goes Up Must Come Down," https://simiode.org/resources/3294.

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