A More General Linear Projectile Problem

TL;DR

This paper derives integral and closed-form solutions for projectile trajectories under complex conditions like wind and variable gravity, providing algorithms and code for accurate, parameterized trajectory computation.

Contribution

It offers the first comprehensive analytical solutions and algorithms for 3D projectile motion with linear drag, variable gravity, wind, and atmospheric effects.

Findings

Integral solutions accurate to O(ε) for position and velocity.

Closed-form solutions for flight time and extremities under constant wind.

Parameterizable, error-controlled algorithms with Matlab code.

Abstract

In a full 3D context, we study a projectile subject to linear drag, a non-uniform gravitational field, time-dependent wind, and parameterized atmospheric thinning. In this general context, we provide integral solutions, exact to $\mathcal{ O }( \varepsilon )$, for the position and velocity of the projectile, where $\varepsilon$ is a small perturbation parameter; in the special case of constant wind, we provide closed-form solutions, exact to $\mathcal{ O }( \varepsilon )$. Under the constant-wind assumption, we provide closed-form solutions of $\mathcal{ O }( 1 )$ for the time of tangency, times of flight, and extreme values of the radius achieved by the projectile. We provide physical interpretations throughout, including a physical interpretation of the branches $W_0$ and $W_{ -1 }$ of the Lambert W function in the context of flight time. We also provide parameterized, error-controlled algorithms to compute trajectories, complete with a full Matlab implementation that we make freely available. We compare the results of our implementation to a general-purpose, stiff ODE solver.