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 problem (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 .
Using notations of mechanics, we can write
Thus we get the function
The only zero of is , where the sign changes from minus to plus as increases. It means that is the searched minimum point. The difference in altitude is thus equal to the http://planetmath.org/node/11642base, and the inclination must be .
|Title||speediest inclined plane|
|Date of creation||2013-03-22 19:19:11|
|Last modified on||2013-03-22 19:19:11|
|Last modified by||pahio (2872)|