Abstract
This article presents a detailed discussion of the shape of the trajectory traced by a projectile under the forces of gravity and air resistance. In particular, our results confirm the insight of the English scientist Thomas Harriot into the motion of a projectile before the development of Newtonian classical mechanics. Our approach is based on the fact that the flight path of a resisted projectile is implemented by a strictly concave function for which the derivative is also strictly concave.
Highlights
Notable among their many and varied common undertakings is Galileo’s and Harriot’s study of an airborne projectile
E curves shown in Figure 1 are based on the numerical solution of the differential equations resulting from Newton’s law for projectile motion under air resistance quadratic in speed, as will be detailed in the last section
The modern approach is certainly far beyond the theoretical and computational tools available to preclassical mechanics, in his more advanced models, Harriot was able to construct ballistic trajectories that are astonishingly similar to those shown in Figure 1, see the folios shown in Figures 8 and 9 of [2], Figure 58 of [1], and Figures 4 and 5 of [3]
Summary
Notable among their many and varied common undertakings is Galileo’s and Harriot’s study of an airborne projectile. In remarkable contrast to Galileo’s idealized version in which gravity is the only active force, Harriot followed the practitioners’ knowledge on projectile trajectories and tried to develop models that account for the effect of air resistance.
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