Two identical balls roll across the top of a table on parallel paths. One of the balls has to roll down into and up out of a dip in the table. The other ball rolls on the flat all the time. Which ball gets to the far side of the table first, and why?.

I'm not going to attempt any proofs but rather offer some observations.

http://mathworld.wolfram.com/BrachistochroneProblem.html

"Find the shape of the curve down which a bead sliding from rest and accelerated  by gravity will slip (without friction) from one point to another in the least time."  
In our circumstance the bead/ball is "at rest" in a vertical sense until it enters the curve.

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

As I look at the animation here I note that the red spot keeps pace with the black one.   
Let the graphic be inverted and the two spots be independent of each other.   If the curve/dip is a cycloid (and we dismiss any "jumps" due to entry-exit of the dip) then both both balls would arrive at the same time.

Should the dip be any other configuration then it would appear that the ball traversing it cannot keep pace with that on the horizontal. 

Posted by brianjn on 2009-11-15 23:50:02