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I've been trying to solve this problem for a few days but have been having a bit of trouble finding anything better than a partial, approximate solution, which is this (this isn't for homework):

### parametric graphing

www.desmos.com

For reference:

pg 116

http://classicalmechanics.lib.rochester ... f/vpcm.pdf

If there was no initial velocity, the quickest path would be a "cycloid," but when there is initial velocity the path is a partial cycloid, with the upper portion that gives the needed initial velocity "chopped off the top" so to speak. Despite knowing this I'm having a lot of trouble finding a formula that gives the exact solution and travel time. The above solution was found through trial and error and should be within about a percent or so of the exact answer.

A descent of 0.5102m gives a velocity of 1m/s on the Earth's surface...