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?.
The cycloid curve is "brachistochrone",
i.e. a curve of least time: given two points A, B in a vertical plane,
a heavy point will take the least time to travel from A to B if it
is displaced along an arc of a cycloid.
While the "dip" may not have this profile I think we can infer that a vertically curved path will be quicker than one that is horizontal.