Among his accomplishments were the determination of the arc length of the logarithmic graph, one of the solutions to the brachistochrone problem, and the discovery of a turning point singularity on the involute of a plane curve near an inflection point.
12.
In 1697 Johann Bernoulli used this principle to derive the brachistochrone curve by considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration ( that of gravity " g " ).
13.
If curved tunels are allowed then the brachistochrone is the fastes path and will lead to shorter travel times for shorter distances between initial and final positions . this shows a few examples . talk ) 04 : 13, 27 May 2010 ( UTC)
14.
Can anyone explain the Brachistochrone problem in a more simple way than " The curve connecting two points displaced from each other laterally, along with which a body, acted on only upon by gravity, would fall in the shortest time . "?
15.
Historically, the idea of finding the shortest path a particle can follow subject to a force motivated the first applications of the calculus of variations to mechanical problems, such as the Brachistochrone problem solved by Jean Bernoulli in 1696, as well as Hamilton, and others.
16.
In this way it is related to the tautochrone curve, for which particles at different starting points always take equal time to reach the bottom, and the brachistochrone curve, the curve that minimizes the time it takes for a falling particle to travel from its start to its end.
17.
For example, imagine a spaceship that can accelerate its passengers at 1 g ( or 1.03 lightyears / year 2 ) halfway to their destination, and then decelerate them at 1 g for the remaining half so as to provide earth-like artificial gravity from point A to point B over the shortest possible time ( Brachistochrone curve ).
18.
The bent wooden rail in scene 13 and the discussion that the quickest distance between two points need not be a straight line ( though a straight line offers the " shortest " path, the fastest descent of a rolling ball in fact follows a curve ) alludes to Galileo's investigation of the brachistochrone ( in the context of the quickest descent from a point to a wall ), which he incorrectly believed to be given by a quarter circle.
