Bernoulli's challenge



Collaborative assignment #3

Parametric equations

A Mathematical Challenge:

It is known with certainty that there is scarcely anything which more greatly excites noble and ingenious spirits to labors which lead to the increase of knowledge than to propose difficult and at the same time useful problems through the solution of which, as by no other means, they may attain to fame and build for themselves eternal monuments among posterity…

Let who can seize quickly the prize which we have promised to the solver. Admittedly this prize is neither of gold nor silver, for these appeal only to base and venal souls. Rather, since virtue itself is its own most desirable reward and fame is a powerful incentive, we offer the prize, fitting for the man of noble blood, compounded of honor, praise, approbation [and a good calculus grade!]

---Johann Bernoulli

I, Johann Bernoulli, of illustrious fame for my many and remarkable mathematical acrobatics, challenge the intellectual community to match wits with me and attempt solutions to the challenges given below. Solutions should be submitted by Wednesday, December 6th, complete with clear and well-explained solutions

Challenge I. As everyone now knows well, the curve connecting two points A and B that represents the path of quickest decent is an inverted cycloid. In the middle of my ingenious derivation, you noticed that we used a conservation of energy argument to show that the velocity of the sliding object would at any time be proportional to the square root of [pic]. That is, we derived [pic] and substituted this into the differential equation [pic]. (We also saw that [pic].)

The first challenge is to determine what curve solves the brachistocrone problem if we instead assume that velocity is proportional to [pic]. Thus, find the curve of quickest decent from A to B assuming [pic]. Again, I assure you, the solution is not a straight line, but is one well known to geometers.

Challenge II. The cycloid is made by rolling a “generating circle” of radius R. In theory, how many copies of the original generating circle could one fit underneath one arch of the cycloid (assuming you could cut up the pieces to make them fit.) In this same spirit, how does the arc length of one arch of the cycloid compare to the circumference of the generating circle?

Challenge III: The trochoid is the name given to the curve generated by rolling a circle of radius R along the x-axis, but with the pen located at a shorter radius r. The parametric equations for this curve are very similar to that of the cycloid but not exactly the same.)

We all know that the standard cycloid has a “cusp” at the end of each arch where the “speed” equals zero and the derivative does not exist. What happens on the trochoid with r ................
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