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During a lunar mission, it is necessary to increase the speed of a spacecraft by 2.76 m/s when it is moving at 400 m/s relative to the Moon. The speed of the exhaust products from the rocket engine is 1100 m/s relative to the spacecraft. What fraction of the initial mass of the spacecraft must be burned and ejected to accomplish the speed increase?

Answer :

Explanation:

Formula to calculate initial mass of given spacecraft is as follows.

         [tex]v_{f} - v_{i} = v_{rel} \times ln(\frac{M_{i}}{M_{f}})[/tex]

The given data is as follows.

      [tex]v_{f} - v_{i}[/tex] = 2.76 m/s

        [tex]v_{rel}[/tex] = 1100 m/s

        [tex]\frac{M_{f}}{M_{i}} = e^\frac{-dv}{v_{rel}}[/tex]

So,    [tex]\frac{M_{i} - M_{f}}{M_{i}} = 1-e^-(\frac{2.76}{1100})[/tex]

                        = [tex]2.51 \times 10^{-3}[/tex]

Thus, we can conclude that [tex]2.51 \times 10^{-3}[/tex] fraction of the initial mass of the spacecraft must be burned and ejected to accomplish the speed increase.

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