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Brachistochrone

Brachistochrone. Joshua Blaskowski. Greek Words. Brachistos - The Shortest. Chronos -Time, delay. Brachistochrone. The problem is to find the curve that gives the shortest amount of time for a block of ice to slide from point A to point B. Johan Bernouli. Proposed the problem in June 1696

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Brachistochrone

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  1. Brachistochrone Joshua Blaskowski

  2. Greek Words • Brachistos - The Shortest. • Chronos -Time, delay.

  3. Brachistochrone • The problem is to find the curve that gives the shortest amount of time for a block of ice to slide from point A to point B

  4. Johan Bernouli • Proposed the problem in June 1696 • Newton, Jacob Bernoulli, Leibniz and L'Hôpital • Galilieo in 1638

  5. Johann Bernoulli • The problem he posed was the following:- “Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.”

  6. Some of the solutions • Newton: Had received the problem at 4 p.m. and worked on it until he finished it 4 a.m. the next morning only taking 12 hours to solve the problem • Jakob Bernoulli: in attempt to outdo his brother he created a harder version of the problem which then became later known as the calculus of variations

  7. Solution • Bernoulli showed that the solution to be one cycle of an upside down cycloid. • They found the slope of the curve by using calculus and optimal control. • Optimal Control: A mathematical optimization method for dealing with dynamical systems(I.e. a swinging pendulum, water flowing in a pipe, a ball rolling down a curb.)

  8. Upside down Cycloid

  9. One segment of a cycloid

  10. Work Cited • The Brachistochrone. Paul Kunkelhttp://whistleralley.com/brachistochrone/brachistochrone.htm • The Roller Coaster or Brachistochrone problem. Eric Hiobhttps://www.google.com/search?client=safari&rls=en&q=http://commons.bcit.ca/math/entertainment/coaster/index.html&ie=UTF-8&oe=UTF-8

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