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

Question

The task is the design a rocket which will accelerate a payload (mass=100kg) from velocity 0 to velocity 6000 m/s, in interstellar space (gravational field= 0). The two constraints are: (i) the relative speed of the exhaust gas is 1500 m/s; (ii) the mass of the rocket fuel must be 10x the mass of the rocket itself, for any rocket stage.

A) Prove that the job cannot be done by a single stage rocket

B) So, use a two stage rocket. Let M_10 and M_20 be the initial masses of the two stages (including fuel). Calculate the minimum total initial mass (M_10 + M_20) required to raise the speed of the payload to 6000 m/s.

(Explain your calculations)

Explanation / Answer

consider (rocket fuel + payload) as a system. Since there is no external force acting on the system, the net moment of the system will be conserved.
momentum of rocket fuel must be equal to momentum of payload
=> 10*100*1500 = 100*6000
=> 15* 10^5 = 6* 10^5   ==>the task can't be done by a single stage rocket.

now considering the total mass of the two stages as (M1 + M2)
=> momentum of rocket fuel must be equal to momentum of payload
=> (M1 + M2)*1500 = 100* 6000
=> (M1 + M2) = 400 kg

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