A proposed interstellar ram-jet would sweep up deuterons in space, burn them in an onboard fusion reactor, and expel the reaction products out the tail of the ship. In a reference frame instantaneously at rest relative to the ship, deuterons, each of mass \(m\), approach the ship at relative velocity \(v\). They are burned, and the burn products, with essentially the same total mass, are ejected from the rear of the ship at velocity \(v+u\). The ship mass \(M\) stays constant, the cross-sectional area of the ship is \(A\), and the number of deuterons per unit volume is \(n\).

(a) Find \(d N / d t\), the number of deuterons swept up per unit time, in terms of \(n, A\), and \(v\).

(b) Find \(d P / d t\), the change in total momentum of the ship per unit time.

(c) Show that the velocity of the ship increases exponentially, with \(v=v_{0} e^{\alpha t}\) where \(v_{0}\) is the ship's initial velocity and where \(\alpha\) is a constant, which can be expressed in terms of given parameters. Assume that \(u\) is constant.