Which is the fastest way?
Move both balls along the slope and the cycloid into the top position. Engage the mechanism and watch the balls move. Then move both balls into any positions in the two cycloids, e.g. put one at the top and the other in the middle of the track. Let them go and watch how fast they reach the bottom.
How does it work
A cycloid is a curve drawn by a point on the circumference of a disc sliding without friction. The time that a ball takes to move between two points (the way it is moving in the experiment) is shortest along a cycloid and not along a straight line, which is the shortest path between two points. If the cycloids are identical, the sliding time is the same regardless of the starting point. The curve of fastest descent is also called a brachistochrone.
Brachistochrones (the curves of fastest descent) are used in planning space shuttle flights.