IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Throwing a Ball as Far as Possible, Revisited

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


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.

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