An article co-authored by Professor of Mathematics Robert Kantrowitz ’82 and Michael M. Neumann of Mississippi State University was recently published in Volume 2019 of the International Journal of Mathematics and Mathematical Sciences.
“Parabolic Sandwiches for Functions on a Compact Interval and an Application to Projectile Motion” revisits a century-old result by the French artillery commandant Charbonnier about the trajectory of a projectile that is moving under the forces of gravity and air resistance.
Charbonnier discovered that the trajectory is always sandwiched by two parabolas associated with the data at the starting point and endpoint of the trajectory. These parabolas allow a natural interpretation in terms of projectile motion without air resistance in the classical sense of Galileo and serve to define a certain safety region for the more complicated case of air resistance.
Kantrowitz said this article fills a gap in Charbonnier’s original proof, and the analysis employed along the way lends a geometric interpretation to the third derivative of a function, extending the geometric implications of the first and second derivatives that students routinely encounter in calculus.