A More General Linear Projectile Problem

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.