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单词 SpeediestInclinedPlane
释义

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.

αbxx2+b2..

Using notations of mechanics, we can write

F=ma=mgsinα=mgxx2+b2,
x2+b2=s=12t2a=t22gxx2+b2.

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

f(x)=2gx2-b2x2.

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.

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更新时间:2025/5/4 21:07:05