How would one calculate the path for the fastest velocity for a rolling object while accounting for frictions? Because ideally in the theoretical world, the path would not matter as long as there was no friction. 2-dimensional motion (x,y).
How would the first curve (extreme drop) compare with the curve in red and the last drop?
Does the first curve's drop have a faster end velocity than the red curve while accounting for friction?
In either instance, only gravity can do work on the rolling object because friction is acting on the bottom of the circle which has a zero velocity at that point (thus it is not moving and that point has no displacement) and the normal force is perpendicular to the displacement of the rolling circle meaning the dot product is 0. That means that even with accounting for friction, the final velocity is the same since gravity is conservative.
Note that for static friction:
$$\int F dx = \int m \frac{dv}{dt} * dx = \int v dv=0 $$ if $v(t) = 0$