A space traveler pushes off from his coasting spaceship with relative speed \(v_{0}\); he and his spacesuit together have mass \(M\), and he is carrying a wrench of mass \(m\). Twenty minutes later he decides to return, but his thruster doesn't work. In another forty minutes his oxygen supply will run out, so he immediately throws the wrench away from the ship direction at speed \(v_{w}\) relative to himself prior to the throw.

(a) What then is his speed relative to the ship?

(b) In terms of given parameters, what is the minimum value of \(v_{w}\) required so he will return in time?