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Question 9.127: During a lunar mission, it is necessary to increase the spee......

During a lunar mission, it is necessary to increase the speed of a spacecraft by 2.2 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 1000 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?

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We use Eq. 9-88 and simplify with v_f – v_i = Δv, and v_{rel} = u.

v_f-v_i=v_{rel}\ln \frac{M_i}{M_f}        (second rocket equation).  (9-88)

v_f-v_i=v_{\text{rel}}\ln \left(\frac{M_i}{M_f}\right) \Rightarrow \frac{M_f}{M_i}=e^{-\Delta v / u}

If Δv = 2.2 m/s and u = 1000 m/s, we obtain \frac{M_i-M_f}{M_i}=1-e^{-0.0022}\approx 0.0022.

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