[math-fun] projectile with quadratic air drag solution

I found this on the web posted by "pelli" on "reddit". Brilliant! Fantastic! Here's a forward solution (found by reverse-engineering the answer): Consider a projectile moving in gravity with quadratic air resistance. The governing equations are u' = -a * u * sqrt( u^2 + v^2 ) v' = -a * v * sqrt( u^2 + v^2 ) - g where a is the coefficient of air resistance defined by |F| = ma|v|2 . Cross-multiply and rearrange to find a * sqrt( u^2 + v^2 ) * (u*v'-v*u') = g*u' Substitute v = s*u and separate variables: a * sqrt( 1 + s^2 ) * s' = g*u'/u^3 Integrate both sides to get the answer: g/u^2 + a*(v * sqrt( u^2 + v^2 )/u^2 + arcsinh|v/u|) = const