The task is to design a rocket which will accelerate a payload (mass = 100 kg) f

Question

The task is to design a rocket which will accelerate a payload (mass = 100 kg) from velocity 0 to velocity 6000 m/s, in interstellar space (gravitational field = 0). Two constraints are: (i) the relative speed of the exhaust gas is 1500 m/s; (ii) the mass of rocket fuel must be 10 x the mass of the rocket itself, for any rocket stage. Prove that the job cannot be done by a single stage rocket. So, use a two stage rocket. Let M10 and M20 be the initial masses of the two stages (including fuel). Calculate the minimum total initial mass (M10 +M20) required to raise the speed of the payload to 6000 m/s.

Explanation / Answer

(A)

It's given that mass of rocket (i.e, rocket engine and fuel container) = mass of fuel /10 for every stage.

We have the rocket motion equation

V = c ln(mo/m)

where 

V = Velocity final - Velocity initial = 6000m/s

c = Relative velocity of exhaust gas with respect to rocket = 1500 m/s

m = Final mass of rocket [In this case it's payload = 100 kg]

Mo = Initial mass of Rocket = 11 * Mro

[ since it's given that mass of rocket fuel = 10 * mass of rocket, Mro = massof rocket]

so the change in velocity that canbe obtained is

V = 1500 * ln(11Mro/100 + Mro) = 6000

11Mr0 = (100 + Mro)  * e^4

which gives Mr0 = 100*e^4/(11- e^4) =  -125.230428

but mass cannot be negative.

So using a one stage rocket is not possible.

(B)

using two stage rocket.

we assume that half the speed is achieved in the first stage and the other half speed is achieved in the second stage.

V1 = 3000 m/s

V2 = 3000 m/s

after first stage is completed the first stage rocket engine and fuel container are discarded and after second stage the second stage rocket engine and fuel container are sicarded.

First stage

M1i = Mass of rocket a the start of stage 1 = M10

M1f = Mass of rocket at the end of stage 1

V1 = C * ln(M1i/M1f)

we get M1i/M1f = e^(3000/1500) = 7.3890561

M10 = 7.3890561 M1f ---------------(1)

Mass of fuel used in first stage = M1f - M1i = 6.3890561 M1f.

Mass of rocket used in first stage = fuel mass in first stage /10 = 0.63890561 M1f

Mass of Rocket at the start of stage 2 = M20 = M1f - 0.63890561 M1f (mass of first stage rocket)

[ since the first stage rocket engine and first stage fuel container are discarded after stage 1]

this gives M20 =( 1-0.63890561) M1f

M20 = 0.36109439 M1f ---------------(2)

Second stage

M2i = Mass of rocket at the start of stage 2 = M20

M2f = Mass of rocket at the end of stage 2

V2 = 3000 = 1500 * ln(M2i/M2f)

M2i = 7.3890561 * M2f --------------------(3)

Mass of fuel used in second stage = 6.3890561 M2f

Mass of rocket used in second stage = fuel mass in second stage /10 = 0.63890561 M2f

Mass of payload = M = M2f - 0.63890561 M2f (mass of second stage rocket)

M = 0.36109439 M2f = 100 kg

M2f = 100/0.36109439 = 276.9359 kg

M10 = M2i = 7.3890561 *276.9359 = 2,046.2949 Kg

M1f = 2046.2949/0.36109439 = 5,666.9252 Kg

M10 = 5666.9252 *7.3890561 = 41,873.2282 Kg

Minimum mass of M10 + M20 = 4187302282 + 2046.2949 = 4.18730433 × 10^9 Kg

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