Should the Soapbox Derby be run on Brachistochrone?

I just learned about Brachistochrones. Just thought I’d share!

No, for the same reason that you shouldn’t run hovercraft races on a track with a parabolic cross-section.

Intersting, but I’m not entirely clear on one thing. There are multiple cycloid curves that will pass through two given points, depending on the size of the circle and the angle along which it was rolled. At one point in the video it showed a curve that dipped below its end point, so that an object rolling down the curve would actually go slightly uphill at the end. Is there some formula for determining which cycloid generates the quickest roll, or do the differences cancel out and one cycloid is as quick as any other?

I wondered about this very same thing. I wish Adam would have added that one to the experiment.