HOMEWORK I

Due:  Wednesday, January 22

The Brachistochrone*

In the land of Oz, four friends, Albertine, Boris, Candy, and Dorothy agree to engage in a marble contest.  The contestants place a marble at the origin and let it roll down a ramp that has the shape of a continuos curve that passes through (0,0) and (p, -2).  Each contestant designs the shape of the ramp for his/her marble to follow.  The winner is the person whose marble reaches the point (p, -2) in the least time.

Albertine has chosen the straight line:

x = ps, y = -2s,  0 < s < 1.
Boris decides to use an arc of a parabola:
x = ps, y = 2(s-1)2 - 2,  0 < s < 1.
Candy uses an exponential curve given by:
Dorothy selects the famous Brachistochrone curve:
x = ps - sin(ps),  y = cos(ps) - 1, 0 < s < 1.





(A)  Verify that each curve passes through the points (0,0) and (p, -2) by graphing each curve with an appropriate domain.

(B)  Using basic physics one can show that the time required for a marble to roll down a path, x = x(s), y = y(s), 0 < s < 1, is given by:
 
 

where g = 32 feet per second per second is the gravitational constant at the surface of the Earth.

Determine which of the four friends wins the match.

(C)   Write a few paragraphs describing the history of the Brachistochrone problem.  Include the roll of  Johann Bernoulli in your account of this problem.
Hint:  Do a web search.

* This problem is a modification of one in Discovering Calculus with Mathematica by Knoll, Shaw, Johnson, and Evans, Wiley (1995).


Johann Bernoulli (1667 - 1748)
 

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