Volume 19, Number 1, Spring 2011
OPTIMIZATION OF PROJECTILE MOTION
IN THREE DIMENSIONS
ROBERT KANTROWITZ AND MICHAEL M. NEUMANN
Abstract. This article addresses the motion of a projectile that is launched from the top of a tower and lands on a given surface in space. The goal is to determine explicit and manageable formulas for the direction of launch in space that allows the projectile to travel as far as possible. While the classical critical point method fails to produce a solution formula even in the special case of shooting to a slanted plane, here we develop a different general approach that leads to remarkably simple equations and solution formulas. Our method is in the spirit of Lagrange multipliers and tailored to motion in three dimensions. Essentially, the optimization problem is reduced to the solution of three equations of geometric type. One of these conditions just means that the Jacobian determinant for the position function of the projectile has to vanish at an optimal solution. It turns out that the theory works well even in the more general setting of launch from a moving vehicle. Applications are given to the case of motion without air resistance and also to the case when the retarding force is proportional to the velocity of the projectile. In particular, surprisingly simple explicit solution formulas are derived for the case of shooting to a slanted plane.