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Brachistochrone Select the run mode. You can add and compare an inclined plane, a root curve, or a free fall. Select the difference ∆H between start point A and end point B on the brachistochrone. Select the number of steps. The precision is of course depending on the step size. The button draws diagrams v(t), and v(x).

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: (1)
Then the radius R can be calculated by: (2) 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: (3)

Furthermore the theoretical arc length is: (4)
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 (##): Tautochrone

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: (#)

Select "store times" from the "Details" menu and repeat running different paths.
Connecting points of least times we get a curve similar to the brachistochrone:

 LinksLL Galileo and the Brachistochrone Problem The Brachistochrone The brachistochrone problem Brachistochrone curve (Wikipedia) Cycloid (Wikipedia) Zykloide (Wikipedia) (#) Time of Travel down Brachistochrone (ProofWiki) Courbe Brachistochrone Das Brachistochronenproblem Die Eigenschaften der Zykloide aus mathematischer, physikalischer und historischer Sicht (Diana Heuer) Tautochrone curve (Wikipedia) (##) Parabolic Segment (WolframMathWorld)

Updated: 2017, Dec 16