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Skateboard ramps

Skateboard ramps are an example of how the fastest path between two points isn't always the shortest in distance. So, what is the fastest path? You might think that it is circular, like a half pipe, but that isn't true. Assuming a frictionless ramp and gravitational acceleration, the fastest path is given by a cycloid. This problem is known as the Brachistochrone Problem.

Related: The shortest path between two points is a straight line. Showing this is more involved than you might expect, and it can be done with the calculus of variations.