A cylindrical space colony of radius \(R\) rotates with angular velocity \(\omega\) about its symmetry axis. A colonist standing on the rim throws a ball straight "up" (i.e., aimed at the rotation axis) with speed \(v=R \omega\) from the colonist's point of view.

(a) Sketch the subsequent path of the ball as seen by an inertial observer to whom the colony is rotating counterclockwise. First find the initial velocity of the ball in the inertial frame.

(b) Sketch the ball's path as seen in the colony frame.

(c) How far around the rim must the colonist run to catch the ball?