Why do wheels appear to revolve opposite to the direction they are rotating?

Question

When viewing cars that are driving along side of us, sometimes their wheels appear to be turning backwards even though they are traveling in the same direction as our car. Why do they look that way?

Answer 1

The issue appears to be rather complex, so I do not aim at providing an exhaustive answer.

At a toy model level it is reasonable to model the eye as a "camera". Specifically, let us assume that a human eye "samples" at a maximum frequency of $\nu$, so that we may make use of the Nyquist-Shannon sampling theorem. Basically, given an instantaneous angular velocity of $\omega$, if the wheel has $n$ spacings, then the "highest frequency" component is $n\omega\over{2\pi}$ (i.e., in a full rotation, there are $n$ wheel bars passing at a given angle). Therefore, writing $\omega = {v\over r}$ with $v$ being the car speed and $r$ the wheel radius (here I am assuming pure rolling of the wheel), when $$ v > {{\pi\nu r}\over{n} } $$ we may assume that some kind of aliasing took place, i.e., I guess you would be unable to reconstruct correctly the wheel motion.

So assuming that a typical wheel has 10 bars and a radius of about 0.3 meters and your eye samples at ~30 Hz (typical frame rate of most first person shooter videogames, so it may be used as an upper limit since there one has complete illusion of movement), a rule of thumb calculation yields about 30 meters/second as a reasonable threshold speed for aliasing phenomena.