Office: F.W. Olin Science Hall, Room 209
Phone: (614) 587-6209
MAPLE BASED ILLUSTRATION OF
THE BRACHISTOCHRONE PROBLEM
This program allows a user ot illustrate the classical brachistochrone problem, proposed by Jacob Bernoulli in 1696. The problem is to find the curve between two given points which provides the least time of descent for a particle falling along a wire in the shape of that curve under the influence of gravity. A common initial response to this problem is that the curve should be the straight line between the two points. This straight line does indeed give the shortest distance betwwen the two points; however, this is not the curve which provides the least time. A shorter time may be acheived by picking a curve which is steep at the beginning, therby taking advantage of gravity. This program allows an instructor to illustrate this fact.
The program begins by prompting the user to enter a sequence of points. The program then calculates the piecewise linear path between the points and calculates the time of descent along that path. The program ends by animating the descent and displaying the accumulated time as the particle travels along the path.
An instructor may illustrate the brachistochrone problem by first choosing the sequence of points to be along the straight line between two points. Once the program has displayed the descent along the straight line path, the instructor then picks points which provide for a piecewise linear path between the two endpoints which is steeper at the beginning. The program will display the descent along this new path. If the instructor has exercised sufficient care in picking the points, the time along the steeper path will be demonstrably shorter.
Future enhancements will allow the user to pick parabolic, circular, and cycloidal (which, in fact, is the path with the least time of descent) paths of descent. Further enhancements will guide the user through the development of brachistochrones for force fields which differ from gravitational force fields.
[Math: Symbolic Computing]
[Math Faculty Home Page]
[Send e-mail]
Supported by grants from the Fund for the Improvement of PostSecondary
Education
(FIPSE grant P116B30079) and the W.M. Keck Foundation
Last Update: 6/11/96