[Jared]

I'm not sure how friction factors into this. A really wide wheel will have much more turning scrub than a thin or donut shaped wheel.

Here's what I've got so far:
I broke the wheel velocity into two components, one in the direction from the wheel to the center of the bot, and the other perpendicular to this one. The inward pointing one is all lost to friction. The other one has a magnitude of 0.707 * 2 = 1.414 feet/second.

The distance from the center to the wheel is 2.828 feet. This means one rotation is 2*pi*r=17.768 feet. This means it's rotating at 1.414/17.768 = 0.079 rotations per second.

I think I'm missing something eight the friction.