You are piloting a spacecraft whose total mass is 1210 kg and attempting to dock
ID: 2009236 • Letter: Y
Question
You are piloting a spacecraft whose total mass is 1210 kg and attempting to dock with a space station in deep space. Assume for simplicity that the station is stationary, that your spacecraft is moving at 1.13 m/s toward the station, and that both are perfectly aligned for docking. Your spacecraft has a small retro-rocket at its front end to slow its approach, which can burn fuel at a rate of 1.47 kg/s and with an exhaust velocity of 117 m/s relative to the rocket. Assume that your spacecraft has only 33.7 kg of fuel left and sufficient distance for docking.a) What is the initial thrust exerted on your spacecraft by the retro-rocket? What is the thrust's direction (positive if in the direction of the velocity of the spacecraft, or negative if in the opposite direction to the velocity of the spacecraft.)
b) For safety in docking, NASA allows a maximum docking speed of 0.0210 m/s. Assuming you fire the retro-rocket from time t = 0 in one sustained burst, how much fuel (in kilograms) has to be burned to slow your spacecraft to this speed relative to the space station?
c) How long should you sustain the firing of the retro-rocket?
d) If the space station's mass is 501000 kg (close to the value for the ISS), what is the final velocity of the station after the docking of your spacecraft, which arrives with a speed of 0.0210 m/s?
Explanation / Answer
(a) Thrust is mass flow rate times exhaust velocity T = -1.67 kg/s*139m/s = -232.13 kg m/s^2 = -232.13N. Since this is slowing down the rocket, it is negative. (b) Impulse, or change in momentum = force times time. Rocket mass decreases slightly during burn. (1.09-0.021)1145 = 1224 (guessed it would burn 10 kg so put in 5kg lower average mass) 1224 Ns/232.13N = 5.27 s 5.27s*1.67kg/s = 8.8 kg fuel burned (c) solved as part of part (b), time is 5.27 seconds (d) conservation of momentum. Mass of rocket is 10 kg less. 1140kg(0.021m/s) = v(1140+501000)kg v =1140kg(0.021m/s)/(1140+501000)kg = 4.77 x 10^-5 m/s
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