Rocket Science A rocket has total mass Mi = 360 kg, including 330 kg of fuel and oxidizer. In interstellar space it starts from rest, turns on its engine at time t = 0, and puts out exhaust with relative speed ve = 1 500 m/s at the constant rate k = 2.50 kg/s. The fuel will last for an actual burn time of 330 kg/ (2.5 kg/s) = 132 s, but define a “projected depletion time” as Tp = Mi/k = 144 s. (This would be the burn time if the rocket could use its payload and fuel tanks as fuel, and even the walls of the combustion chamber.) (a) Show that during the burn the velocity of the rocket is given as a function of time by
v (t) = ─ve In [1─(t/Tp)]
(b) Make a graph of the velocity of the rocket as a function of time for times running from 0 to 132 s.
(c) Show that the acceleration of the rocket is a(t) = ve /(Tp ─ t)
(d) Graph the acceleration as a function of time.
(e) Show that the position of the rocket is x(t) = ve(Tp ─ t)ln [1 ─ (t/Tp)] + ve t
(f) Graph the position during the burn.