Rocket Science

9.52: Rocket Science

A rocket has a total mass Mi = 360 kg, including 330 kgof fuel and oxidizer. In interstellar space, it starts from rest atthe position x=0, turns on itsengine at time t=0, and puts outexhaust with relative speed ve = 1500 m/s at theconstant rate k= 2.50 kg/s. The fuel will last for an actual burntime of 330kg/ (2.5 kg/s) = 132 s, but define a “projecteddepletion time” as Tp = Mi/k = 360 kg / (2.5kg/s) = 144 s (which would be the burn time if therocket could useits payload and fuel tanks, and even the walls of the combustionchamber as fuel). (a) Show that during the burn of the velocity ofthe rocket as afunction of time is given by v(t) = -veln( 1 – 1/Tp). (b). Make a graph of the velocity of therocket as a function of time for times running from0 to 132 s. (c)Show that the acceleration of the rocket is a(t) = ve/Tp – t. (d). Graph the acceleration as a function oftime. (e). Show that theposition ofrocket is x(t) =ve (Tp – t) ln( 1 – t/Tp) +vet. (f) Graph the position during the burn as afunction oftime