Approaching the brachistochrone using inclined planes--striving for shortest or equal travelling times

摘要

The classical brachistochrone problem asks for the path on which a mobile point M just driven by its own gravity will travel in the shortest possible time between two given points A and B. The resulting curve, the cycloid, will also be the tautochrone curve, i.e. the travelling time of the mobile point will not depend on its starting position. We discuss three similar problems of increasing complexity that restrict the motion to inclined planes. Without using calculus we derive the respective optimal geometry and compare the theoretical values to measured travelling times. The observed discrepancies are quantitatively modelled by including angular motion and friction. We also investigate the correspondence between the original problem and our setups. The topic provides a conceptually simple yet non-trivial problem setting inviting for problem based learning and complex learning activities such as planing suitable experiments or modelling the relevant kinematics.