Designing the Fastest Ramp for a Ball: Solving a Challenging Physics Problem

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jinkm
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I bumped into this physicis problem about 2 weeks ago and it is much harder than you might think:

A ball is dropped down a frictionless ramp an eliptical ramp. It must travel down the ramp and then back up the ramp to its original height 10 meters away. What is the equation that defines the fastest ramp for the ball?

Assume the only force acting on the ball is gravity - 9.8 m/s.
 
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