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Incidentally the UMass Amherst PER group had a similar problem that
stumped many experts and students alike. It was a car rolling down
a slight incline vs a car rolling into a hole and then to the same
final location. It was surprising how many were surprised to see
that the longer path yielded a shorter time. They showed the result
with movies and simulations.
There is a very good chance that very many on this list know about
the brachistochrone problem since it has been around for literally
hundreds of years. Galileo showed by explicit demonstration
(remember, he loved to roll things down ramps) that the cycloid-
shaped path is the minimal time path. The problem was solved
analytically by J. Bernoulli over 300 years ago and is a classic
(probably *the* classic) example of the application of the
calculus of variations.
A point to keep in mind about this problem is that the over all shape
of the minimal time path depends on the ratio of the vertical height
of descent between the endpoints to the horizontal range between the
endpoints because this ratio determines the total parameteric
rotation angle of the cycloid. For instance, if the net height of
descent is greater than 2/[pi] times the horizonal range then the
minimal time path does *not* dip below the level of the final
endpoint and there is no dip or "hole".