New
Millennium Theme Park
This month's
Brain Game was inspired by a treatment of the challenge by Tom Logsdon
in his fascinating book Orbital Mechanics Theory and Applications
listed in the bibliography (who was assisted in the solution by Janis
Indrikis). He also provided the derivation of the minimum time equations.
New Millennium
Theme Park, on the fringes of the Santa Monica Mountains, is conducting
a competition for the design and installation of the fastest slide.
The slides are coated with zeromu a frictionless composite material.
Which one of the following proposed designs was selected? (Specification:
both vertical fall and horizontal displacement are 100 feet)
a)
Convex Solutions  a concave down shaped slide 

b)
Cyclex Systems  a cycloid shaped slide 

c)
Linear Mechanics  a flat shaped slide 

d)
Great Circle Travel Systems  an arc of a circle shaped slide 

The answer is
b. The minimum time path between two points in a constant oneg
gravitational field (at different elevations) is a cycloid also known
as the brachistochrone (shortest time from the Greek) when concave up.
The cyloid is the locus of a point on a circle (or rim of a wheel) as
the circle (or wheel) rolls along a smooth surface and the point on
the wheel returns back to its origin (one revolution through 2)
as portrayed in Figure 1. The solution of this path, as the optimal
minimum time of travel in 1696, was made by Isaac Newton in less than
a day in response to a challenge by the Bernoulli brothers (who had
allowed up to six months for its solution). Also claimed with solving
this challenge were Gottfried Leibnitz, Guillaume L'Hopital and the
Bernoulli brothers themselves. This occurred 31 years after the discovery
of calculus in 1665. The solution was not simple and required the development
of the calculus of variations.
Figure
1. Cycloid the locus of a point on the circle through one revolution
of rotation.
Inverting the
cycloid of Figure 1 to concave up yields the brachistochrone.
Solving the
two parametric equations for the value of (a) the radius and ()
the rotation angle through a horizontal and vertical distance of 100
feet on the cycloid is obtained by simple iteration of simultaneous
equations:
x = 100 = a(sin)
y = 100 = a(1cos)
Which gives: a = 57.2925
= 138.2°
The time for
the player to complete the descent (derivation shown at the end) on
the slide is
and the path
equation
x = a(sin),
y = a(1cos)
is differentiated and substituted in the integral of the time of descent
for the slide shown in Figure 2
For a flat slide
slope = dy/dx= m from
y= mx
substitute m for y' and mx for y in the time equation when m = 1 (since
slope is 45°) the time of descent is
and for an arc
of a circle slide the time of descent is
The derivation
of the minimum time path
Shown is : The
ordinary derivative
as contrasted
by Variation
To minimize
I in finding the optimum path the calculus of variations is employed
This function
will achieve a minimum value when its first variation is zero and its
second variation becomes positive. This parallels determining the minimum
slope when the first derivative is zero and the second derivative is
positive in differential calculus.
In minimizing
I, the second order differential equation is solved using the EulerLaGrange
Method
The
EulerLaGrange Equation

The time equation
is then written as:
Applying the
EulerLaGrange Equation
Implying
an arbitrary
constant
squaring and
rearranging terms
then
The physics
for the sliding board deals with the exchange of potential energy for
kinetic energy at any distance y
1/2 mv
= mgy
solving for velocity yields
v = [ 2gy]
Figure
2.
The time of
descent for the player is therefore
differentiating
the parametric equations
x=a(sin),
y= a(1cos)
dx=a(1cos)d,
dy= asind
thus dy/dx= sin/(1cos)
= y'
and
1 + y'
= 1 + [sin/(1cos )]
= 2/(1cos)
by substitution
in the time integral: (note that g
= g = 32 fps)
For the flat
board:
the slope is
then
where m = 1
results in
Performance
comparisons of the flat board, cycloid and circular arc slides are depicted
in Figure 3. If the horizontal displacement were increased to 160 feet,
the advantage of the cycloid would be more pronounced: flat board 4.64
sec, cycloid 3.91 sec and circular arc 4.54 sec.
Figure
3. Performance comparison
Commentary
on the cycloid/brachistochrone:
Bibliography
Hartog J.P.
Mechanics. New York: Dover Publications Inc., 1961
Logsdon, Tom. Orbital Mechanics Theory and Applications. New
York: John Wiley and Sons, Inc., 1998 Misner,
Charles W., Kip S. Thorne and John A. Wheeler. Gravitation. New
York: W. H. Freeman and Company, 1970
Solkolnikoff, I. S. and R. M. Redheffer. Mathematics of Physics and
Modern Engineering. New York: McGrawHill Book Company, Inc., 1958
Tenenbaum, Morris and Harry Pollard. Ordinary Differential Equations.
New York: Dover Publications, Inc., 1985
Click
Here To View A dynamic demonstration
of the brachistochrone
Or
A dynamic demonstration of the
brachistochrone can be viewed at:
http://galileo.imss.firenze.it/museo/a/brachi.avi