An astronaut of mass \(M\), initially at rest in some inertial frame in gravity-free empty space, holds \(n\) wrenches, each of mass \(M / 2 n\).

(a) Calculate her recoil velocity \(v_{1}\) if she throws all the wrenches at once in the same direction with speed \(u\) relative to her original inertial frame.

(b) Find her final velocity \(v_{2}\) if she first throws half of the wrenches with speed \(u\) relative to her original inertial frame, and then the other half with speed \(u\) relative to the frame she reached after the first throw. Compare \(v_{2}\) with \(v_{1}\) from part (a).

(c) Then find her total recoil velocity \(v_{n}\) if she throws all \(n\) wrenches, one at a time and in the same direction, and each with speed \(u\) relative to her instantaneous inertial frame just before she throws it.

(d) Find her total recoil velocity in the limit \(n ightarrow \infty\), and compare with the rocket equation.