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Chapter 7: Problem 22

During a lunar mission, it is necessary to make a midcourse correction of \(22.6 \mathrm{~m} / \mathrm{s}\) in the speed of the spacecraft, which is moving at \(388 \mathrm{~m} / \mathrm{s}\). The exhaust speed of the rocket engine is \(1230 \mathrm{~m} / \mathrm{s}\). What fraction of the initial mass of the spacecraft must be discarded as exhaust?

Short Answer

Expert verified
After computing, we find that the required fraction of the mass to be discarded is approximately 0.018 or 1.8 %.

Step by step solution

01

Write down the known values

From the given problem, we know the change in velocity (\(\Delta v = 22.6 \, \mathrm{m/s}\)), the velocity of the spacecraft (\(v = 388 \, \mathrm{m/s}\)), and the exhaust speed of the engine (\(v_e = 1230 \, \mathrm{m/s}\)).
02

Write down the rocket equation

The rocket equation is \(\Delta v = v_e \ln(\frac{m_i}{m_f})\). Here, we need to solve for \(m_f\), which is the final mass of the spacecraft after the midcourse correction.
03

Use the rocket equation to solve for \(m_f\)

First, arrange the rocket equation to represent \(m_f\), giving \(\frac{m_i}{m_f} = e^{\frac{\Delta v}{v_e}}\). Then, we find that \(m_f = \frac{m_i}{e^{\frac{\Delta v}{v_e}}}\).
04

Compute the ratio

We then should find the ratio \(m_r\) of the discarded mass to the initial mass which gives us \(m_r = \frac{m_i - m_f}{m_i} = \frac{m_i}{m_i} - \frac{m_f}{m_i} = 1 - \frac{m_f}{m_i} = 1 - e^{-\frac{\Delta v}{v_e}}\).
05

Substitute the known values into the formula to get the answer

Substituting the known values gives \(m_r = 1 - e^{-\frac{22.6}{1230}}\).

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Most popular questions from this chapter

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