Originally Posted by
zaqwert6
I understand the spirit of the question but yeah, if you travelled 10k , so did your shoes.
Its like saying you drove your car 10k but your hat went farther cause you were inside waving it all around.
Or how about the chain on the bike? Or the Spokes then? Or the pedals?
Dang, I think the value of all my used parts is going down the drain with this thread.
The wikipedia article on cycloids shows you how to do it:
http://en.wikipedia.org/wiki/Cycloid
I think the only difference is that the bike is travelling more quickly along the x-direction than a true cycloid (your feet are spinning just the same, but you're moving forward more quickly). So you have to modify the equation in the x-direction:
x = r(t-sin(t)) + (v-r)t,
where v is the average velocity in the x-direction.
y stays the same:
y = r(1-cos(t))
The arc length is now more difficult to compute, because the dx/dt derivative has an extra (v-r) term. You can plug the integral into Mathematica and it can probably do it for you (I could probably do it by hand by collecting like terms but I'm lazy as hell):
It's the integral from t = [0,2*pi] of sqrt((r*sin(t))^2 + (r*(1-cos(t))+(v-r))^2).
That integral gives you the distance your foot travels when your bike moves a distance v*2pi.