A Minimization Problem Based on Straight Lines

It is well known that the brachistochrone curve is the curved path of fastest descent under uniform gravity. In order to find such a curve, we need to rely on the calculus of variations and the Euler–Lagrange equation. In this contribution, we propose an alternative and simpler approach to the problem, based on straight lines. In our scheme, we divide the descending path into a series of segments and compute, analytically, the time it takes for a particle to cover each of them. Looking for the y-coordinate set that minimizes the total time, for a given x-point set, we are able to find a curve that resembles the brachistochrone. We show how accurate our approximation is, for a different number of x points and choices of the x-point set. This innovative way to solve this ancient problem not only allows the students to consolidate relevant kinematical and...