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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.

 

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