Parabolic Sandwiches for Functions on a Compact Interval and an Application to Projectile Motion

Robert Kantrowitz,Michael M Neumann

doi:10.1155/2019/4868106

Abstract

About a century ago, the French artillery commandant Charbonnier envisioned an intriguing result on the trajectory of a projectile that is moving under the forces of gravity and air resistance. In 2000, Groetsch discovered a significant gap in Charbonnier’s work and provided a valid argument for a certain special case. The goal of the present article is to establish a rigorous new approach to the full result. For this, we develop a theory of those functions which can be sandwiched, in a natural way, by a pair of quadratic polynomials. It turns out that the convexity or concavity of the derivative plays a decisive role in this context.

Highlights

Introduction and Historical BackgroundEvery differentiable real-valued function f on an interval [a, b] induces a canonical pair of polynomials of degree at most two, namely, the polynomials p and q that coincide with f at the endpoints a and b, while their derivatives satisfy p󸀠(a) = f󸀠(a) and q󸀠(b) = f󸀠(b)
We focus on the particular case when f is sandwiched by these polynomials in the sense that either q ≤ f ≤ p or p ≤ f ≤ q holds on the entire interval [a, b]
We prove that q ≤ f ≤ p if f󸀠 is convex on [a, b] and that p ≤ f ≤ q if f󸀠 is concave on [a, b]

Summary

Introduction and Historical Background

From a broader point of view, while every calculus student knows what the conditions f󸀠 ≥ 0 or f󸀠󸀠 ≥ 0 on [a, b] mean for a given function f, here we offer an interpretation and visualization of the estimate f󸀠󸀠󸀠 ≥ 0 on [a, b] in terms of the parabolic sandwich condition q ≤ f ≤ p on [a, b] and its localized version. Charbonnier discovered that the trajectory is always sandwiched by the two parabolas associated with the data at the starting point and a chosen 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 complicated case of air resistance.

Preliminaries on Convex Functions

Main Sandwich Results

Application to Projectile Motion