Only the centrifugal and Coriolis pseudoforces act upon a particular projectile moving within a rotating cylindrical space colony of radius \(R\).

(a) Find the differential equations of motion of the projectile in the rotating frame, using Cartesian coordinates centered on the rotation axis.

(b) Decouple and solve the differential equations, to find expressions for \(x(t)\) and \(y(t)\) in terms of four arbitrary constants of integration.

(c) Evaluate the constants in terms of the initial conditions \(x_{0}, y_{0}, v_{\mathrm{x}_{0}}, v_{\mathrm{y}_{0}}\).

(d) A colonist on the rim at \(\left(x_{0}, y_{0}\right)=(R, 0)\) throws a ball toward the rotation axis with velocity \(\left(v_{\mathrm{x}_{0}}, v_{\mathrm{y}_{0}}\right)=\left(-v_{0}, 0\right)\). Find a general expression for the time at which the ball returns to the rim, and show that in the limit as \(v_{0}\) becomes small, the time agrees with what you would expect in a uniform gravitational field \(g=R \Omega^{2}\).