A point-like body that starts at the first point A with zero speed and is constrained to move along the curve to the second point B (B lower than A, constant gravity g, no friction).
The curve that is covered in the least time is a brachistochrone curve. It is an upside down cycloid passing vertically through A and B.
The cycloid through the origin A, with a horizontal base given by the line y = 0 (x-axis), generated by a circle of radius R rolling over the "positive" side of the base (y ≥ 0), consists of the points (x, y), with
where φ is a parameter, corresponding to the angle through which the rolling circle has rotated.
Fitting the maximum angle φm and the radius R and to reach point B(100m, H) WolframAlpha was used solving the equation:
Then the radius R can be calculated by:
For steps the following calculations are performed (i=0 .. i<N):
(*) applies because of the energy law. TN (**) is the total travel time from A to B, and SN (***) the total arc length along the cycloid from A to B.
The final velocity at B is:
Furthermore the theoretical arc length is:
and the theoretical total travel time (#):(5)
The precision of the results for v, S, and T by the formulas (3), (4), and (5) - compared with (*), (**), and (***) - is of course depending on the number of steps:
From (1), (2), and (5) we get:
T(φ) has a minimum of T=5.659 s at φ = π, using xm=100 m, g=9.81 m/s^2:
The minimum of travel time occurs at H=200m/pi = 63,66 m:
Example of results along the brachistochone (∆H=30m, n=100,00 steps):
t = 5.94177 s is the numerical traval time, T = 5.9417 s the theoretical result by formula (5),
s = 118.7115 m is the numerical arc length, S = 118.7115 m the theoretical result by formula (4).
You can add a root curve and compare it with the brachistochrone:
The arc length of the root curve
can be computed by the formula by (##):
A tautochrone or isochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point B is independent of its starting point A.
The curve is also a cycloid.
The coordinates of point B are x=R·π and y=2R
The time of descent is:
and the Brachistochrone Problem
The brachistochrone problem
Brachistochrone curve (Wikipedia)
(#) Time of Travel down Brachistochrone (ProofWiki)
Die Eigenschaften der Zykloide aus mathematischer, physikalischer und historischer Sicht (Diana Heuer)
Tautochrone curve (Wikipedia)
(##) Parabolic Segment (WolframMathWorld)