Displacement of a point on a circle There is a mark on a wheel of radius $30$ cm. The mark is in contact with a horizontal plane. The wheel rotates to a distance of $10\pi \approx 31.4$ cm. 
$1.$ What is the angle that the old position of the mark makes with the new position? 
$2.$ What is the distance between the new position of the mark with that of the ground?
[point B can be anywhere on the circle]


The mark takes a trip on the dotted curve which is displayed on the animation below. The parametric equations of this curve, named as cycloid, are
$$\begin{align}
x_A&=R\phi-R\sin\phi \\
y_A&=R-R\cos \phi
\end{align} \qquad 0 \le \phi \le \pi/3$$
Where $R$ is the radius of the circle. 
Take a look at this animation. It is exactly made by your data included in the question. It may help to imagine better.
Can you derive the parametric equation?
Can you answer your questions based on the parametric equations?

