# The Locus of the apices of projectile trajectories under constant drag

## TL;DR

This paper derives an analytical expression for the apex locus of projectiles under constant drag, revealing that the optimal launch angle is smaller than in free-flight conditions, with implications for mechanics education.

## Contribution

It provides the first analytical solution for the apex locus in projectile motion with constant drag, enhancing understanding of dissipative systems.

## Key findings

- Apex locus is analytically characterized under constant drag.
- Optimal launch angle is smaller than in free-flight.
- Numerical results for range and flight time are provided.

## Abstract

We present an analytical solution for the projectile coplanar motion under constant drag parametrised by the velocity angle. We found the locus formed by the apices of the projectile trajectories. The range and time of flight are obtained numerically and we find that the optimal launching angle is smaller than in the free drag case. This is a good example of problems with constant dissipation of energy that includes curvature, and it is proper for intermediate courses of mechanics.