The brachistochrone problem is a common problem in sophomore/junior level "Classical Mehcanics" course. The idea of this problem is to find the path between two points for an object that will take the least time if the object is sliding under the influence of gravity.

In this demonstration, the point at (0,0) is fixed, but you can position the other point (default location of (1.00 m, -0.637m) ) by dragging the black circle around the graph. The graph shows the straight line path between the points and the "cycloid path" that will minimize the time of the sliding object. Other parameters are also calculated, including the average speed of an object on either path. The "Animate" button will start an object on each path. The animation time is slowed down by a factor of 10 so that you can observe the motion, that is, if the actual time is 0.5662 s, the animation will take 5.662 s to complete.

Animate