speediest inclined plane

We set the problem, how great must be the difference in altitude of the top and the bottom of an inclined plane in that a little ball would frictionlessly roll the whole length of the plane as soon as possible (cf. the brachistochrone problemMathworldPlanetmath (http://planetmath.org/CalculusOfVariations)).  It is assumed that the http://planetmath.org/node/9475projection of the length on a horizontal plane has a given value b.


Using notations of mechanics, we can write


Thus we get the function

t2=2gx2+b2x=:f(x)  (x>0),

the absolute minimum point of which is to be found.  This function is differentiableMathworldPlanetmathPlanetmath, and its derivativePlanetmathPlanetmath is


The only zero of f(x) is  x=b,  where the sign changes from minus to plus as x increases.  It means that  x=b  is the searched minimum point.  The difference in altitude is thus equal to the http://planetmath.org/node/11642base, and the inclination α must be 45.

Title speediest inclined plane
Canonical name SpeediestInclinedPlane
Date of creation 2013-03-22 19:19:11
Last modified on 2013-03-22 19:19:11
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Example
Classification msc 26A09
Classification msc 26A06
Related topic ExtremumMathworldPlanetmath
Related topic CalculusOfVariations
Related topic BrachistochroneCurve