This group/cooperative learning problem is designed to see if you can understand how adding friction dramatically alters the equations of motion along a slid.
 
 

QUESTION: A block of wood is launched with a velocity v₀ up a slide, with angle $\theta$ referenced to the horizontal. The slide is designed with kinetic ($\mu_k$) and static ($\mu_s$)coefficients of friction so that the block will, respectively, accelerate down the slide if in motion but remain motionless if initially at rest. Use this information to construct  space-time plots of the acceleration and velocity.
 
 

Qualitatively the frictional force acts in combination with the surface parallel component of gravity when moving up the slide and against gravity when its velocity is down the slide.

The constituent relationships are: Normal Force, $N = mg \cos \theta$ with $\mu_k N <mg \sin \theta < \mu_s N$
and
$ma = mg \sin \theta \pm f$ where $f=\mu_k N$. Putting it all together gives two different accelerations$a = g \sin \theta \pm \mu_k g \cos \theta$.

The really tricky part is to realize that the acceleration is always down the slide and that even at the turnaround point the block is in motion although there is an infinitesimal point at which the velocity of the block passes through zero. The final space-time plots are therefore: