Consider a Ferris wheel rotating in the vertical plane, with the position of a cabin modeled as a point moving in a circular path. The position in the xy-plane is given by $\(\vec{r}\)=(15.0\(\mathrm{\ m}\))\(\cos\) [(0.500\ \(\mathrm{rad}\)/\(\mathrm{s}\))t]\(\hat{i}\)+(15.0\(\mathrm{\ m}\))\(\sin\) [(0.5\(\mathrm{00\ rad}\)/\(\mathrm{s}\))t]\(\hat{j}\)$ , where r is in meters, and t is in seconds. Calculate the velocity and acceleration of the cabin as functions of time.

$\(\begin{array}{l}\) \(\vec{v}\)=-7.50 \(\mathrm{~m}\) / \(\mathrm{s}\) \(\sin\) [(0.500 \(\mathrm{rad}\) / \(\mathrm{s}\)) t] \(\hat{i}\)+7.50 \(\mathrm{~m}\) / \(\mathrm{s}\) \(\cos\) [(0.500 \(\mathrm{rad}\) / \(\mathrm{s}\)) t] \(\hat{j}\) \\ \(\vec{a}\)=-3.75 \(\mathrm{~m}\) / \(\mathrm{s}\)^{2} \(\cos\) [(0.500 \(\mathrm{rad}\) / \(\mathrm{s}\)) t] \(\hat{i}\)-3.75 \(\mathrm{~m}\) / \(\mathrm{s}\)^{2} \(\sin\) [(0.500 \(\mathrm{rad}\) / \(\mathrm{s}\)) t] \(\hat{j}\) \(\end{array}\)$

$\(\begin{array}{l}\[\vec{v}\)=7.50 \(\mathrm{~m}\) / \(\mathrm{s}\) \(\cos\) [(0.500 \(\mathrm{rad}\) / \(\mathrm{s}\)) t] \(\hat{i}\)-7.50 \(\mathrm{~m}\) / \(\mathrm{s}\) \(\sin\) [(0.500 \(\mathrm{rad}\) / \(\mathrm{s}\)) t] \(\hat{j}\)\\ \(\vec{a}\)=3.75 \(\mathrm{~m}\) / \(\mathrm{s}\)^{2} \(\cos\) [(0.500 \(\mathrm{rad}\) / \(\mathrm{s}\)) t] \(\hat{i}\)+3.75 \(\mathrm{~m}\) / \(\mathrm{s}\)^{2} \(\sin\) [(0.500 \(\mathrm{rad}\) / \(\mathrm{s}\)) t] \(\hat{j}\]\end{array}\)$

$\(\begin{array}{l}\[\vec{v}\)=-7.50\(\mathrm{\ m}\)/\(\mathrm{s}\]\cos\) [(0.500\(\mathrm{rad}\)/\(\mathrm{s}\))t]\(\hat{i}\)-7.50\(\mathrm{\ m}\)/\(\mathrm{s}\[\sin\) [(0.500\(\mathrm{rad}\)/\(\mathrm{s}\))t]\(\hat{j}\)\\\(\vec{a}\)=-3.75\(\mathrm{\ m}\)/\(\mathrm{s}\)^2\(\sin\) [(0.500\(\mathrm{rad}\)/\(\mathrm{s}\))t]\(\hat{i}\)+3.75\(\mathrm{\ m}\)/\(\mathrm{s}\)^2\(\cos\) [(0.500\(\mathrm{rad}\)/\(\mathrm{s}\))t]\(\hat{j}\]\end{array}\)$

$\(\begin{array}{l}\[\vec{v}\)=-7.50\(\mathrm{\ m}\)/\(\mathrm{s}\]\sin\) [(0.250\(\mathrm{rad}\)/\(\mathrm{s}\))t]\(\hat{i}\)+7.50\(\mathrm{\ m}\)/\(\mathrm{s}\[\cos\) [(0.250\(\mathrm{rad}\)/\(\mathrm{s}\))t]\(\hat{j}\)\\\(\vec{a}\)=-1.88\(\mathrm{\ m}\)/\(\mathrm{s}\)^2\(\cos\) [(0.250\(\mathrm{rad}\)/\(\mathrm{s}\))t]\(\hat{i}\)-1.88\(\mathrm{\ m}\)/\(\mathrm{s}\)^2\(\sin\) [(0.250\(\mathrm{rad}\)/\(\mathrm{s}\))t]\(\hat{j}\]\end{array}\)$