Rotation of a wheel in the positive direction and clockwise?

Question

The Question

I’m riding my bike at a constant speed of 20 miles per hour on flat land when i ride over a dot of fresh yellow paint, thereby getting a yellow spot on my front wheel, which is 2 feet in diameter. I continue riding my bike in a straight line at 20 miles per hour. When t seconds passed since running over the dot, how many feet is the yellow spot on my tire from the yellow dot i ran over?

My Understanding

I faced some difficulties in understanding an exercise about parameterization (pre-calculus level). I understood the solution except this point; the spot starts at the bottom of the circle, corresponding to an angle of $\frac{3\pi}{2}$. We let the wheel move in the positive direction, so that the wheel rotates clockwise

Is it possible that we can move in the positive direction with a clockwise rotation?

Answer 1

If the bike is rolling ("moving") on the ground toward $+\infty$ (the positive $x$-direction, towards the right), then the wheel spins clockwise (negative rotation).

Answer 2

The positive direction is given to be the direction you are traveling. The clockwise vs counterclockwise direction of rotation depends on the side of the bike to the observer. If you are on the right side of the bike – it moving to your right – the wheel will appear to rotate clockwise.

The wheel moves horizontally according to a cycloid curve: $$\quad x=r(1-\cos \theta)\quad$$ and $\theta$ changes at a rate of $$\dfrac{d}{\pi}=\dfrac{20\times5280\space ft/sec}{3600}\bigg/2\pi\space ft=\dfrac{44}{3\pi\space}\dfrac{radians}{sec}.\quad$$ If we start at $\space \theta=\dfrac{3\pi}{2}\space $ and continue in the clockwise direction, we get $$d=x=r(1-\cos \theta)\quad\text{where}\quad \theta=\dfrac{3\pi}{2} -\dfrac{44t}{3\pi}$$