Satisfying explanation of Aristotle's Wheel Paradox. The paradox:



We have a circle and there is another circle with smaller radius. They are co-centeric.

If circle make full turn without sliding, both smaller and bigger circle make full turn too. If we assume that the passed road is equal to the circumference of circles. We have got smaller circle's radius is equal to bigger one's.

Unsatisfying Solutions I found: 


"Do not assume that smaller circle's circumference is equal to passed road since the surface that contacts to the ground is bigger one. " // Okey but it does not explain the paradox, it explains just what is the wrong assumption (even does not explain why it is a wrong assumption.)
It is undeniable that every point on both smaller and bigger circle will contact exactly one and only one point on their path. Therefore we can think that this is a bijective maps and smaller circle is isomorphic to bigger one. (Okey but ....)


Question: What is the true answer? What is wrong with the definition of circumference of a circle and relationship with its taken path.
The velocity of any point $P$ on a wheel can be written as the sum of two velocities: the velocity $\vec V$ of the center $O$ and the velocity $\vec\omega\times\vec{OP}$ of rotation about the center, where $\vec\omega$ is angular velocity (perpendicular to the plane of the wheel).

A wheel turns without sliding with respect to a given path if the velocity of the contact point between wheel and path vanishes. Let then $C$ and $C'$ be the contact points of the two wheels. We have
$$
\vec v_C=\vec V+\vec\omega\times\vec{OC}
\quad\text{and}\quad 
\vec v_{C'}=\vec V+\vec\omega\times\vec{OC'}
$$
If $\vec v_C=0$ then $\vec V=-\vec\omega\times\vec{OC}$ and
$$
\vec v_{C'}=-\vec\omega\times\vec{OC}+\vec\omega\times\vec{OC'}
=\vec\omega\times(\vec{OC'}-\vec{OC})=\vec\omega\times(\vec{CC'}).
$$
This cannot vanish, unless $C=C'$. So the assumption that both circles turn without sliding is false.
Its really quite simple and doesnt even need math to show..
Take any given point on the outside  larger circle as it moves along..an do the same for the smaller one...neither one travels in a straight line from point A to point B....and they dont travel the same distance. Its seems like they do but they dont..Thats they key..
.
For proof..stick a pencil  in the outermost part of the larger circle and one in the smaller one..rotate the circle and notice the paths traced out..
They will not be straight lines but curved lines..and the curved line not the same length for both..Bigger the circle, the longer the path traced out from A to B.