Analysis of asymptotic projectile motion with air resistance using the Lambert W function

Abstract

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.

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