On the locus formed by the maximum heights of projectile motion with air resistance

TL;DR

This paper analyzes the geometric locus of maximum heights of projectiles with linear air resistance, expressing it using Lambert W function and studying its curvature, revealing insights into projectile behavior under drag.

Contribution

It introduces a novel analytical expression for the apex locus using Lambert W function and examines its curvature properties in projectile motion with air resistance.

Findings

Locus of apexes expressed with Lambert W function in polar coordinates.

Curvature analysis identifies angles of maximum curvature.

Synchronous curve remains a circle despite air resistance.

Abstract

We present an analysis on the geometrical place formed by the set of maxima of the trajectories of a projectile launched in a media with linear drag. Such a place, the locus of apexes, is written in term of the Lambert $W$ function in polar coordinates, confirming the special role played by this function in the problem. In order to characterize the locus, a study of its curvature is presented in two parameterizations, in terms of the launch angle and in the polar one. The angles of maximum curvature are compared with other important angles in the projectile problem. As an addendum, we find that the synchronous curve in this problem is a circle as in the drag-free case.