Some remarks on projectile motion with a linear resistance force

TL;DR

This paper analyzes projectile motion with a linear resistive force, deriving analytical expressions for trajectory maxima and radial distance oscillations, and examining energy and phase space effects, useful for advanced mechanics education.

Contribution

It provides an analytical solution for projectile trajectories with linear resistance without Lambert W function and explores the influence of resistance on various physical quantities.

Findings

Analytical expression for trajectory maxima

Radial distance oscillates beyond a critical angle

Energy and phase space are significantly affected by resistance

Abstract

In this article we revisit the projectile motion assuming a retarding force proportional to the velocity, $\vec{F_r} = -mk\vec{V}$. We obtain an analytical expression for the set of maxima of the trajectories, in Cartesian coordinates, without using the Lambert $W$ function. Also, we investigate the effect of the parameter $k$ on the radial distance of the projectile showing that the radial distance oscillates from a certain critical launch angle and find an approximate expression for it. In our analysis, we consider the impact of the parameter $k$ in the kinetic energy, the potential energy, the total energy and the rate of energy loss, and in the phase space. Our results can be included in an intermediate-level classical mechanics course.