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Kantrowitz ’82 Revives Halley’s Optimization Problems


Robert Kantrowitz
Robert Kantrowitz

An article titled “A Halley revival: another look at two of his classical gunnery rules” by Professor of Mathematics Robert Kantrowitz ’82 and Michael M. Neumann of Mississippi State University, appears in the December volume of the journal The Mathematical Scientist.

The article is devoted to a revival of two optimization problems from the realm of ballistics that were considered by 17th-century British astronomer and mathematician Edmond Halley. The first problem is to determine the angles of launch that maximize the horizontal range of a projectile that lands on a slanted surface. The second is to find angles that minimize the projectile’s kinetic energy at launch while ensuring a strike of an intended target.

For each of the two problems, the authors revisit the ‘flat Earth model’ in which gravity is the only operative force. The framework is then enlarged by accounting for drag that is linear or quadratic in speed.

The unifying thread is an unassuming trigonometric function that sheds light on the duality to Halley’s problems and helps reveal analytic and geometric parallels between the optimal flight curves in the non-resistive environment and that in which air resistance is quadratic in speed.

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