Rollercoaster Imagine that two cities, A and B, are 500 miles apart and are joined by a gigantic, powerless, frictionless rollercoaster. Assume that the passenger car starts at A from rest, is subject to uniform gravity but is not subject to air resistance, and finishes at B at rest. Same as part 1, except assume that the railway is further constrained to descend no deeper than 50 miles. For both cases, be accurate. Solution by Patrice Le Conte and others 1.
Travel time is .
This might seem a short time to travel 500 miles, but observe that the peak velocity reaches . The travel times for associated paths in this family are all greater than 11.967, as was to be shown. It is not hard to deduce that two half cycloids, connected by a horizontal line, provide the optimal path for the car to follow. Travel time is Reference E. Weisstein, Brachistochrone Problem , University of Virginia.