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 one-g 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.

θ= 138.2°
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(1-cos)
Which gives: a = 57.2925

The time for the player to complete the descent (derivation shown at the end) on the slide is

and the path equation

x = a(θ-sinθ), y = a(1-cosθ) is differentiated and substituted in the integral of the time of descent for the slide shown in Figure 2

t = 3.23 sec

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

t = 3.53 sec

and for an arc of a circle slide the time of descent is

t = 3.27 sec

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 Euler-LaGrange Method

The Euler-LaGrange Equation

The time equation is then written as:

Applying the Euler-LaGrange Equation

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 mv² = mgy
solving for velocity yields
v = [ 2gy]1/2

Figure 2.
The time of descent for the player is therefore

differentiating the parametric equations
x=a(θ-sinθ), y= a(1-cosθ)
dx=a(1-cosθ)dθ, dy= asinθdθ
thus dy/dx= sinθ/(1-cosθ) = y'

and

1 + y' = 1 + [sinθ/(1-cosθ )]² = 2/(1-cosθ)

by substitution in the time integral: (note that g = g = 32 fps²)

t = 3.23 sec

For the flat board:
the slope is

then

where m = 1 results in

t = 3.54 sec

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:

Minimum time path
Pendulum using brachistochrone would have equal periods regardless of amplitude

Regardless of position on path a particle would come to rest at the same time and position (isochronous)

Used in gear design to minimize slippage
Ascent trajectory of Space Shuttle conforms to this curve to minimize travel time and energy

Optimal aircraft trajectory - curve followed to climb to cruise altitude in minimum time. The aircraft dives downward as Mach I is approached to break through the sound barrier in minimum time and then climbs to altitude (the curve between Mach I and attained altitude is a brachistochrone)

Optimal path for low thrust rocket trajectories

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: McGraw-Hill 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