A spaceprobe of mass \(M\) is propelled by light fired continuously from a bank of lasers on the moon. A mirror covers the rear of the probe; light from the lasers strikes the mirrors and bounces directly back. In the rest-frame of the lasers, \(n_{\gamma}\) photons are fired per second, each with momentum \(p_{\gamma}=h u_{\gamma} / c\), where \(h\) is Planck's constant, \(c\) is the speed of light, and \(u\) is the photon's frequency.

(a) Show that in a short time interval \(\Delta t\) the change in the probe's momentum is \(2 n_{\gamma}^{\prime} p_{\gamma}^{\prime} \Delta t\), where \(n_{\gamma}^{\prime}\) is the number of photons striking the mirror per second, and \(p_{\gamma}^{\prime}\) is the momentum of each photon, both in the probe's frame of reference.

(b) The photons are Doppler-shifted in the probe's frame, so their frequency is only \(u^{\prime} \approx u(1-v / c)\), where \(v\) is the velocity of the probe. Show also that \(n_{\gamma}^{\prime}=n_{\gamma}(1-v / c)\), and then show that the ship's acceleration has the form \(a=\alpha(1-v / c)^{2}\) where \(\alpha\) is a constant. Express \(\alpha\) in terms of \(M, n_{\gamma}\), and \(p_{\gamma}\).

(c) Find an expression for the probe's velocity as a function of time. Briefly discuss the nature of this result as the probe travels faster and faster.