Restricted Brachistochrone

The motion of a bead along a path restricted to straight lines (restricted brachistochrone), sliding without friction from rest and accelerated by gravity, is considered. For two shapes of path, the geometry of the route optimized to provide the least travel time from one point to another is obtained. The bead’s travel times, path lengths, and average velocities are compared between the two presented models, and with travel along a cycloid path, which (as the solution to the original brachistochrone problem) provides the lowest possible travel time. The calculations are made with and without the use of calculus, and therefore the problems presented are comprehensible for a large variety of students.