Brachistochrone
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 φ _{m} 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. T
_{N} (**)
is the total travel time from A to B, and S_{N}
(***) the total arc length along the
cycloid from A to B.The final velocity at B is: _{}^{(3)}Furthermore the theoretical arc length is: 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 x _{m}=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: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:

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: 20.10.2023