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Trajectory of a body in a resistant medium: an elementary derivation

A didactical exposition of the classical problem of the trajectory determination of a body, subject to the gravity in a resistant medium, is proposed. Our revisitation is aimed at showing a derivation of the problem solution which should be as simple as possible from a technical point of view, in order to be grasped even by first-year undergraduates. A central role in our analysis is played by the so-called "chain rule" for derivatives, which is systematically used to remove the temporal variable from Newton's law to derive the differential equation of the Cartesian representation of the trajectory, with a considerable reduction of the overall mathematical complexity. In particular, for a resistant medium exerting a force quadratic with respect to the velocity our approach leads, in an elementary way, to the differential equation of the trajectory, which is subsequently solved by series expansion. A comparison of the polynomial approximants obtained by truncating such series with the solution recently proposed through a homotopy analysis is also presented.