Abrupt changes in direction and loss of energy 
In the picture that I drew above, I was thinking that there is definitely loss of energy when the ball rotates off of the inclined plane to the small section of horizontal plane. However, I do not know the mathematical equation to model the theoretical change in velocity. Is there a definite relation between the angle of incline and how much energy is lost? 
I was thinking that the velocity at the horizontal section is $v\cos\phi$, where $v$ is the speed of ball at the bottom of the ramp while still on the ramp, and $\phi$ is the angle of incline. However after doing experiments on it, this relation seems not to hold as my final values in the experiment is larger than even the value in theory.
If this relation is correct, how can the result of my experiment be explained?
 A: There are two types of coordinates. 


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*The angle of the ball with respect to the surface, and the conjugate momentum is the angular velocity in this case. (Rough metal surface)

*The ordinary linear translation along the tilted ramp, with momentum ordinary linear momentum. (Slippery ice surface)


The louder the ball when in motion, the more interference between angular motion and linear motion. We want silence.
In the transformation, imagine there was only angular acceleration, and then:


*

*The horizontal part is made of slippery slippery ice.

*The horizontal part is some very rough metal.


In both cases the ball will keep it's angular velocity, but in the ice case it will ideally rotate almost "in place" but might reach the edge.
In the metal case it will uniformly rotate to the edge as if it is still on the slope but gravity was switched off.
Think about it. Also on the icy slope case.
Another aspect to imagine is if the transition is very violent between coordinates. Think of the dribble sounds. Think of a spring..
Try to give a positive argument to why and how the ball WILL lose energy in the motion.
