Home

Site Map

Search

physics,astronomy,software,applet,java,shareware,sun,moon,earth,java,applet
GeoAstro
Applets
astronomy
Astronomy
sundial,dial,astronomy,software,applet,java,shareware,sun,moon,earth,java,applet
Chaos Game
sundial,dial,astronomy,software,applet,java,sun,moon,earth,java,applet
Java
sundial,dial,astronomy,software,applet,java,shareware,sun,moon,earth,java,applet
Miscel-

laneous
sundial,dial,astronomy,software,applet,java,sun,moon,earth,java,applet
Physics Quiz

Brachistochrone





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

    brachistochrone cycloid
                applet   brachistochrone cycloid applet

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

table phi radius

For steps step size  the following calculations are performed (i=0 .. i<N):

calculations

(*) 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:
final
                              velocity(3)

Furthermore the theoretical arc length is:
arc length
                            cycloid (4)
and the theoretical total travel time (#):
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:

error table


relative error precision deviation

From (1), (2), and (5) we get:

minimum travel time

T(φ) has a minimum of T=5.659 s at φ = π, using xm=100 m, g=9.81 m/s^2:

minimum of travel
                        time

The minimum of travel time occurs at H=200m/pi = 63,66 m:

travel_time

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).

data brachistochrone


You can add a root curve and compare it with the brachistochrone:

root curve

The arc length of the root curve
square root
                            function

can be computed by the formula by (##):

arc length root
                            curve parabola


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.
tautochrone isochrone
                        curve
The coordinates of point B are x=Rπ and y=2R

The time of descent is:
tautochrone
                          isochrone time  (#)






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