$2\mathrm{.}5($Robot moving along a planar circular curve$).$ This exercise develops differential equations for the continuous dynamics of a robot that is moving in the two$-$dimensional plane. Consider a robot at a point with coordinates $\backslash (($x$,$ y$)\backslash )$ that is facing in direction $\backslash (($v$,$ w$)\backslash ).$ While the robot is moving along the dashed curve, this direction $\backslash (($v$,$ w$)\backslash )$ is simultaneously rotating with angular velocity $\backslash (\backslash$omega $\backslash ).$

Develop a differential equation system describing how the position and direction of the robot change over time. Build your way up to that differential equation by first considering just the rotation of $\backslash (($v$,$ w$)\backslash ),$ then considering the motion of $\backslash (($x$,$ y$)\backslash )$ in a fixed direction $\backslash (($v$,$ w$)\backslash ),$ and then putting both types of behavior together. Can you subsequently generalize the dynamics to also include an acceleration of the linear ground speed when the robot is speeding up$?$