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Down the hill (Posted on 2019-07-05) Difficulty: 4 of 5
There is a mouse at the top of a frictionless hill with equation y=-|x|n. The mouse is given an infinitesimal nudge and starts to slide down. What is the maximum value of n such that it never leaves the surface of the hill?

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution | Comment 1 of 6
Assume the mouse is sliding down the surface as described.  The only forces acting on the mouse are gravity, and the normal force from the surface.  If the mouse is about to leave the surface, that means the normal force is approaching zero, and it becomes zero if the mouse leaves the surface.  If the normal force is zero, then the motion of the mouse reverts to projectile motion, under gravity as the only force, where the vertical motion (in this case the "Y")is a squared function of "X".  Given that the initial velocity of the mouse in "X" is extremely small (essentially approaches zero), it's only the normal force on the surface that can supply a component of force to change the "X" velocity, so this fits.  If n=2, the mouse can slide down the resultant curve with zero normal force yet just follow the surface

Max "n" = 2.  Or I suppose n= I, lim I-->2

Edited on July 7, 2019, 7:40 am
  Posted by Kenny M on 2019-07-05 20:10:46

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