A ball is released from rest in the frame of a rotating cylindrical space colony, at a point halfway between the rotation axis and rim.

(a) Sketch the subsequent path of the ball as seen by an inertial observer who sees the colony rotating counterclockwise with angular velocity \(\omega\).

(b) If the colonist is directly "beneath" the ball when it is released (i.e., at the rim at a point along a line connecting the rotation axis and release point), how far must the colonist run to catch the ball, in terms of the colony radius \(R\) ? Sketch the path of the ball as seen by the colonist.