Volume 16, Number 3, Fall 2008
OPTIMAL ANGLES FOR LAUNCHING
PROJECTILES: LAGRANGE VS. CAS
ROBERT KANTROWITZ AND MICHAEL M. NEUMANN
Abstract. This article centers around the motion of a
projectile that is launched from the top of a tower and lands
on linear or parabolic mountains. The main goal is to provide
explicit and manageable formulas for the angle of inclination
that maximizes the distance the projectile travels. A general
approach based on Lagrange multipliers is developed to obtain
useful information not only for motion without air resistance,
but also for the case when air resistance is proportional to the
speed of the projectile. In this case, the optimal angle is expressed
in terms of the Lambert W function. The explicit formulas
for the optimal angle are employed to answer a number
of natural questions. For instance, it turns out that the optimal
angle is a decreasing function of the height of the launch tower.
Also discussed is the extent to which computer algebra systems
are helpful in this context.
(Subscribers Only)
