Groetsch, C. W. and S. A. Yost. 2014. Vertical Projection in a Resisting Medium: Revelations on Observations of Mersenne. The American Mathematical Monthly. 21(6): 499-505.
Article Abstract: This article, inspired by a 17th-century woodcut, validates empirical observations of Marin Mersenne (1588–1648) on timing of vertically-launched projectiles for a general mathematical model of resistance.
This is a beautifully and carefully written work with excellent exposition, flowing mathematical development, and rich historical detail. The issue addressed is one noted by Père Marin Mersenne in 1636, “. . . the time of ascent of projected body is always less than the time of descent, the difference increasing width the velocity of the projection.” Groetsch and Yost first render the case of symmetry in the Galilean model in which there is no resistance in a very readable (especially for students) development.
When it comes to the asymmetric case the authors show that “. . . the kinetic energy a launch exceeds that at impact owing to energy conversion caused by the medium’s resistance to the motion.” They offer a wonderfully clear analysis in which they show that the rate of change of the difference between descent and ascent times relative to initial velocity is positive for all initial velocities greater than zero.
The last paragraph of the article tells us about the intent and style of the two authors, “At first sight the experiment of Mersenne and Petit seems a bit wacky (and potentially dangerous). But in the early days of the development of dynamics, it treated the crucial relationship between time and motion. We suggest that the story of Mersenne’s experiment, in addition to introducing undergraduates to the rich history of dynamics, can be a useful vehicle for illustrating some principles of basic mathematical analysis.” And they do just that with flair!
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