Analytic Approximations of Projectile Motion with Quadratic Air Resistance

Abstract

We study projectile motion with air resistance quadratic in speed. We consider three regimes of approximation: low-angle trajectory where the horizontal velocity, u, is assumed to be much larger than the vertical velocity w; high-angle trajectory where w u ; and split-angle trajectory where w  u . Closed form solutions for the range in the first regime are obtained in terms of the Lambert W function. The approximation is simple and accurate for low angle ballistics problems when compared to measured data. In addition, we find a surprising behavior that the range in this approximation is symmetric about  / 4 , although the trajectories are asymmetric. We also give simple and practical formulas for accurate evaluations of the Lambert W function.