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 aims to show a derivation of the solution to the problem that is as simple as possible from a technical point of view, so that it can be understood 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 in order 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 to the differential equation of the trajectory, which allows its Taylor series expansion to be derived in an easy way. A numerical comparison of the polynomial approximants obtained by truncating such series with the solution recently proposed through a homotopy analysis is also presented.