Rolling object and Friction 
I am investigating the collision between two billiard balls. Ball 1 rolls down the ramp and collides with Ball 2 at rest on a flat surface. Because the collision between billiard balls is considered almost elastic, Ball 1 is momentarily stationary as seen in the graph. However, it's not ACTUALLY stationary: it spins (no horizontal displacement). Little bit later, Ball 1 starts rolling away on the flat surface as seen in the "secondary displacement" part of the graph. 
I am looking for some explanations for this phenomena. I suspect that friction and the interaction between rotational momentum and linear momentum, but I have not been able to connect all the dots together.
Thank you!
 A: Ball 1 rolls down the incline and gains speed. Since the first part of your graph is linear (showing no increase in speed), I suspect the incline to have ended and the surface to be horizontal when the measurement is started, am I right? 
During this horizontal rolling the ball has constant linear and rotational speed (first part of the graph). That means constant linear and rotational momentum. 
When the impact happens,


*

*linear momentum gets transferred from one ball to the other, because one ball completely blocks the linear path of the other.

*Rotational momentum could also have been transferred. That would similarly require the other ball to completely block it's rotational path. But it doesn't. 
Two cog wheels would be able to do that. They can interlock. But the billiard balls can't interlock. Ball 2 can't block the rotational motion of ball 1. There might be friction, but firstly I expect them to be smooth and secondly I expect the impact to be very short. Ideally no friction happens between the balls, meaning no rotational momentum transfer.
The graph
So, looking at your graph, what happens at the stationary middle part is that ball 1 hits ball 2 and is linearly slowed down to zero. But it still rotates unchanged. Rotating while being linearly stationary means that it slides over the surface. Wheel spin. There is kinetic friction, unless the surface is very smooth right there. This kinetic friction might cause some loss in kinetic energy, meaning a loss in rotational speed.
But clearly, before all the rotational speed is gone, the ball and surface gets grip again. And when they grip, the ball starts rolling, meaning linear motion starts (it "pushes itself off" from the surface).
The rotational kinetic energy was the only energy left inside the ball at this point. Some of this energy is now being used for linear kinetic energy. Both the linear and rotational motion will therefore be slower than before, since there is less energy present to be shared between them. 
