Throwing a Ball as Far as Possible, Revisited

TL;DR

This paper investigates the optimal launch angle for a projectile to maximize its arc length, revealing a surprising transcendental equation solution independent of initial speed and gravity.

Contribution

It derives a novel transcendental equation for the optimal angle, demonstrating a surprising independence from initial speed and gravity.

Findings

Optimal angle approximately 56.47 degrees

Derived a transcendental equation involving coth and csc functions

Applied differentiation under the integral sign in the derivation

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.