### New Millennium Theme Park

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

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(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

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

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

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