Exact expressions for the range and the optimal angle of a projectile with linear drag

Abstract

The analytical expressions for the range and the optimal angle of a projectile with linear drag are obtained for the first time. The analytical results are expressed in terms of the Lambert W function, and from those a number of mathematical and numerical results obtained previously with the implicit relation for the range are obtained more easily. The explicit solution for the range of the linear problem is then used as the zeroth-order term for the series solution of the range of the projectile with resistive force proportional to the nth power of the projectile's speed, in powers of δ = (n – 1)/2. PACS Nos.: 02.30.Gp, 45.20.Dd, 45.30.+s, 45.50.Dd