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

TL;DR

This paper provides a simplified, educational derivation of a body's trajectory in a resistant medium under gravity, emphasizing the use of the chain rule and series expansion for clarity and accessibility.

Contribution

It introduces an elementary derivation method for the trajectory problem, making it accessible to first-year undergraduates and comparing series solutions with homotopy analysis.

Findings

Derivation reduces mathematical complexity using the chain rule.

Series expansion effectively approximates the trajectory.

Comparison shows the series method aligns with homotopy analysis results.

Abstract

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.