What is the fastest path between two points?

What is the fastest way to get from point A to point B? Well, you would think that moving in a straight, downward slope, correct?

Not quite! Newton was given the same challenge called the "Brachistochrone problem" and he and a few other fellow mathematicians/scientists came up with the cycloid.Here are some links to read about the problem and the solution.

• http://mathworld.wolfram.com/BrachistochroneProblem.html
• http://en.wikipedia.org/wiki/Brachistochrone_curve

And about the cycloid itself:

• http://mathworld.wolfram.com/Cycloid.html

And here are some videos demonstrating the same...

• In action - normal speed http://www.youtube.com/watch?v=nzk3ba-Oe00&sns=em
• In action - slow speed http://www.youtube.com/watch?v=Y05nIAGiIqw&sns=em

Here's an interesting observation: Wherever you start from on the cycloid slope, you will reach the bottom of the curve at the same time. Two demonstrations showing this:

• http://www.youtube.com/watch?v=Ib1TdgeYL4o&sns=em
• http://www.youtube.com/watch?v=FAYWccuLVvY&sns=em

Can anyone explain why?