THE TIME ELAPSED FOR AN OBJECT SLIDING DOWN AN ARBITRARY PATH

Abstract

This is a presentation of a student project that was to be a synthesis of a few of undergraduate analysis topics. The project's aim was to provide the student an overview and applications of some of the concepts from linear algebra, calculus and numerical analysis. At the same time, the project was part mathematical modeling with the following question in mind: On a prescribed path, how fast can an object slide from one point to a second and lower point? On a frictionless slide starting at the origin and to the point (pi,-2), it is a known fact that the cycloid is the fastest slide. We attempted to model the problem for paths that need not have an a priori mathematical formulation and may merely be a path sketched by hand and one that includes some form of normal friction. The task put forth calls for utilizing results from curve-fitting for finding a reasonable mathematical approximation to the path, basic physics and manipulation of vectors for setting up the system of differential equations, and numerical analysis for finding approximate solutions to the differential equations and the elapsed time. Computing technology primarily Mathematica had made possible and tenable the solving of the problem.