Why is linear speed at the top of the wheel 2r * omega?

[BifSlamkovich]

I thought that in the formula v=wr, r refers to the distance from the center, i.e., the radius. However, if the wheel is rotating on the ground, the books says that the linear speed at the top of the tire is 2rw, as opposed to rw. Why is this?

 

[ZapperZ]

When a wheel spins, the center of the wheel is stationary.

When the wheel rolls, the center of the wheel is now moving with respect to a stationary ground. So now you have to add the two velocity vectors together.

Zz.

 

[Doc Al]

You can also view the rolling wheel as instantaneously rotating about the contact point at the bottom of the wheel. So the top of the wheel is a distance of 2R about that axis, thus v = ω(2R) = 2ωR.