We calculate the range of a projectile experiencing air resistance in the asymptotic region of large velocities by introducing the Lambert W function. From the exact solution for the range in terms of the Lambert W function, we derive an approximation for the maximum range in the limit of large velocities. Analysis of the result confirms an independent numerical result observed in an introductory physics class that the angle at which the maximum range occurs, θmax, goes rapidly to zero for increasing initial firing speeds v0≫1. We show that θmax∼(ln v0)/v0.
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 ; and split-angle trajectory where . 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 , although the trajectories are asymmetric. We also give simple and practical formulas for accurate evaluations of the Lambert W function
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