TITLE: Question about square-wheeled cars
QUESTION [9 upvotes]: It's kind of an infamous problem in differential equations to find the correct road surface so that a car with square wheels (and an axle located in the center) keeps its axle level as it drives along.  I hope I won't offend anybody by saying that one smooth piece of the solution (for a wheel with sides of length 2) is $y = -\cosh(x)$ 
If you actually take this solution and describe the position of the axle at any given point, unless I have calculated incorrectly you find that the axle is always positioned directly over the point where the wheel makes contact with the road.  I've been unable to come up with a physical justification of this phenomenon and it seems fairly non-obvious to me.
Is there a straightforward reason why this must be true?  Is it specific to this wheel shape?

REPLY [3 votes]: The physical justification is quite simple. The point of contact between the wheel and the road is instantaneously stationary. Since the wheel moves rigidly, the velocity of any point on the wheel is as though the wheel were rotating about the point of contact. In particular, the velocity of the axle is perpendicular to the line joining it to the point of contact. Since we require that the axle stays level, it velocity must be purely horizontal, so it has to be directly above the contact point.