## Throwing a Ball as Far as Possible, Revisited

- Sept. 16, 2016
- 2:30 p.m.
- LeConte 312

## Abstract

What initial trajectory angle maximizes the arc length of a ideal projectile (in a constant gravitational field with no resistance or propulsion)? We show the optimal angle depends neither on the initial speed, nor on the acceleration of gravity, and is the solution theta to a surprising transcendental equation:* sin(theta) = tanh(csc(theta))*, i.e., *theta = csc^{-1}(alpha)* where *alpha* is the unique positive fixed point of *coth(x)*. Numerically, *theta* ~ 0.9855 ~ 56.47^o. The derivation involves a fun application of differentiation under the integral sign.