Throwing a Ball as Far as Possible, Revisited

Abstract

What initial trajectory angle maximizes the arc length of an ideal projectile? We show the optimal angle, which depends neither on the initial speed nor on the acceleration of gravity, is the solution x to a surprising transcendental equation: csc(x) = coth(csc(x)), i.e., x = arccsc(y) where y is the unique positive fixed point of coth. Numerically, $x \approx 0.9855 \approx 56.47^\circ$. The derivation involves a nice application of differentiation under the integral sign.