Very Simple Conceptual Question About Rolling

[Derezzed]

Hello everybody,
I am having a hard time understanding a very simple principle involving a rolling wheel. I know that the velocity at the bottom "contact point" of a rolling wheel is zero relative to a stationary observer.. yet I don't see how this is true.
So I made a quick sketch and here is my reasoning: As the wheel rolls, it translates horizontally over time. The contact point is no exception. Thus it must have a non-zero velocity, otherwise the wheel is stationary. Basically, the 'delta d'/dt will give the velocity of the contact point, which is non-zero. Why am I wrong? Any light anybody could shed on the matter would be much appreciated!
http://img130.imageshack.us/img130/4258/rolling.png

Uploaded with ImageShack.us

 

[zhermes]

You're right that the average velocity is not zero. But at the instant of contact with the surface, the point on the wheel has zero velocity (and non-zero acceleration).
An instant before contact, and an instant after contact---the point will have a velocity.

See: http://www.google.com/images?hl=en&...QB4qCsQOEt4GnBA&ved=0CD4QsAQ&biw=1045&bih=702

 

[rcgldr]

At any instantaneous moment in time, the relative speed of point of the surface of the wheel in contact the ground is zero, this is different than the 'contact point', which is the point where the wheel touches the ground independent of movment at the wheels surface. The 'contact point' (often called 'contact patch' in the case of tires) moves at the same speed as the wheel. The wheel surface speed relative to the center of the wheel is the same as the speed of the wheel wrt the ground. The surface speed at the bottom of the wheel wrt ground is zero, while the surface speed at the of the wheel wrt ground is 2 times the wheels speed wrt ground.

 

[tiny-tim]

Welcome to PF!

Hello Derezzed! Welcome to PF!

see http://en.wikipedia.org/wiki/Cycloid" , including the .gif

 

[Derezzed]

Thanks everybody for the quick and effective responses!