A rolling wheel is not a simple motion. While rolling is rotation plus translation, only the latter really matters for the paradox. Still, it is easy to be lured by the complexity. For example:
1. A point on the perimeter of a wheel travels a cycloid path. A more inner point travels a curtate cycloid path. In the paradox the center's path is a straight horizontal line.
2. A point's velocity in the direction the center moves is faster than the center's velocity during the top half of a rotation. A point's velocity in the direction the center moves is slower than the center's velocity during the bottom half of a rotation.
3. Pure rolling occurs when the circular object travels one circumference along the ground for every for every full rotation it makes. Slipping occurs when rotation is faster than pure rolling. A paradigm case is a car wheel stuck in snow. Skidding occurs when rotation is slower than pure rolling. A paradigm case is a car wheel that skids on ice after the driver brakes hard.
Slipping and skidding so described affect the two circles in the same way. For example, if the part of the tire in contact with the road slips or skids, then the metal rim the tire is mounted on is affected the same way. The rim can't slip when the tire doesn't. Yet the rim slips while the tire doesn't is one "solution" to the paradox given on Wikipedia. It makes no sense except as a far-fetched metaphor. In other words, the rim "slips" but it doesn't really slip. The rim "skids" would be less far-fetched.
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