Ken's POTW

A Bumpy Plane
Consider a perfectly flat sheet [of aluminum, for example], in which a rounded dip has been formed about two inches wide, running the width of the otherwise flat sheet. The dip is smoothly connected, with no sharp corners, such that a marble could roll into and out of it from the rest of the sheet. The sheet is cut lengthwise across this dip, and one of the halves is turned over so that it has a hump in the sheet. Of course, this is also smoothly connected.

The two sheets are placed side-by-side at the same angle to form two inclined planes, one with a dip and one with a hump. Two identical marbles are released from the top of the planes and roll to the bottom without slipping or jumping. Which marble reaches the bottom first, or is it simultaneous? Can you show why? Please feel free to assume any dimensions needed to provide a simple proof.

Now consider a sheet with a dip followed by an identical hump. Again the cut is made and one half turned over. Now one piece has a dip followed by a hump, and the other has a hump followed by a dip. When the two marbles run their race this time, which will reach the bottom first?

Source: Several places. I originally saw it in a UCD Physics 8A course. (I reworded this slightly, due to the answers received.)

Solution
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