We present an essentially geometric approach to the brachistochrone problem. We apply basic knowledge of trigonometry, accessible to the students in a one or two semester basic physics program, in such a way to allow the problem to be introduced at the beginning of an undergraduate course of Physics or Engineering. We show that the resulting acceleration on a particle describing a brachistochrone has gravitational acceleration of magnitude gand points towards the center of the circle that generates the cycloid. We present a geometrical method to deduce the differential equation for the cycloid, without using variational calculus. We show that the descending time along this trajectory is always the same, independently of the starting point.
brachistochrone; cycloid; geometry
