Optimization of Projectile Motion Under Air Resistance Quadratic in Speed

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• 作者：Robert Kantrowitz ; Michael M. Neumann
• 关键词：Projectile ballistics ; Quadratic drag ; Optimal launch angle ; Enveloping function ; Lambert W function ; Tartaglia’s inverse problem
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：14
• 期：1
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1660-5454
• 卷排序：14

文摘

This article centers around the problem of maximizing the horizontal range of a projectile that is launched from atop a tower and is subject only to gravity and air resistance quadratic in speed. Here the surface to which the projectile is launched is represented by a convex impact function, while the projectile motion is described by a classical approximation model for flight curves that is widely considered acceptable for quadratic drag and launch angles up to moderate size. In this setting, the optimal range is given by the point where the impact function intersects the enveloping function induced by the family of flight paths. In the special case of a linear impact function, manageable explicit formulas for the range function, the maximal range, and the corresponding optimal launch angle are provided in terms of the Lambert W function. The article concludes with a solution to Tartaglia’s inverse problem in this context.