Question

A box of physics books slides down a ramp inclined at an angle of 40 degrees to the horizontal. The total length of the ramp is 10m. The coefficient of kinetic friction between the box and the ramp is .2 for the top 8m and zero for the bottom 2m. Upon reaching the bottom of the ramp, a spring bounces the box back upward.

(a) Find the distance that the box travels back up the ramp on the return bounce.

(b) Your answer shouldnt depend on the spring constant of the spring at the bottom. Why not?

Answer 1

The description of the problem is not very clear but I will asume that the situation is as illustated on figure below:

Let us Apply the work Energy Theorem:

from where

from where

from where

from where

from where

from where

from where

, and this is the speed at which it reaches the frictionless 2-meter road on the ramp.

Now since the rest of the road is frictionless, then once the book compresses the spring, it will pick up the same kinetic energy when it bounces back into the ramp as when it reached the spring at the bottom, becuase the spring exerts a conservative force on the book and therefore it will give its kinectic energy back when it bounces back.That is why the distance that it travels afterwards doesn`t depend on the spring constant as any spring will give the same kinectic energy back to the book since this arrival kinetic energy depends only on the friction and the potential energy of the book when it was realeased from the top at rest. Having clarified that, we can say that the book will travel a distance (x +2) meters when getting back up on the ramp given by the work energy theorem equation below:

from where

from where

from where

from where

from where

from where

but the distance that the box travels back up the ramp on the return bounce is (x+2) meters since we only made the calculation of the distance that it tavels on the friction road of the ramp, and therefore we need to add up the 2 meters of the frictionless path, so the final answer is 4.918m + 2m = 6.91 meters.

So 6.91 meters aprox is the distance that the box travels back up the ramp on the return bounce.