  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.
