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Question

In a zero-gravity environment, can a rocket-propelled spaceship ever attain a speed greater than the relative speed with which the burnt fuel is exhausted?

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No since, in zero gravity condition, external force applied to the system is zero. So the total momentum of the system should remain unchanged. Now since the mass of the rocket is much greater compared to the mass of fuel that is exhausting, the relative speed of the brunt fuel must be greater than the speed of the rocket to keep the momentum unchanged.

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