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Consider a rocket (initial mass mo) accelerating from rest
Chapter 3, Problem 3.10(choose chapter or problem)
Consider a rocket (initial mass \(m_o\)) accelerating from rest in free space. At first, as it speeds up, its momentum p increases, but as its mass m decreases p eventually begins to decrease. For what value of m is p maximum?
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QUESTION:
Consider a rocket (initial mass \(m_o\)) accelerating from rest in free space. At first, as it speeds up, its momentum p increases, but as its mass m decreases p eventually begins to decrease. For what value of m is p maximum?
ANSWER:Step 1 of 2
The momentum of inertia is given by the equation as shown below.
\(\begin{array}{l}
\mathrm{P}=\mathrm{mv}_{\mathrm{a}}+\mathrm{mv} \ln \frac{\mathrm{m}_{0}}{\mathrm{~m}} \\
\mathrm{mv}=\mathrm{mv}_{0}+\mathrm{mv} \ln \frac{\mathrm{m}_{0}}{\mathrm{~m}}
\end{array}\)
Then the value is given as,
\(\begin{aligned}
\frac{\mathrm{dP}}{\mathrm{dm}} & =\frac{\mathrm{d}}{\mathrm{dm}}\left[\mathrm{mv}_{\mathrm{a}}+\mathrm{mv} \ln \frac{\mathrm{m}_{0}}{\mathrm{~m}}\right] \\
& =\mathrm{v}_{0}+\mathrm{m} \frac{\mathrm{d}}{\mathrm{dm}}\left[\mathrm{v}_{\mathrm{ex}} \ln \frac{\mathrm{m}_{0}}{\mathrm{~m}}\right]+\left(\mathrm{v}_{\mathrm{ex}} \ln \frac{\mathrm{m}_{0}}{\mathrm{~m}}\right) \frac{\mathrm{dm}}{\mathrm{dm}} \\
& =\mathrm{v}_{0}+\mathrm{mv}_{\mathrm{ex}}\left[\frac{\frac{-\mathrm{m}_{0}}{\mathrm{~m}^{2}}}{\frac{\mathrm{m}_{0}}{\mathrm{~m}}}\right]+\left(\mathrm{v}_{\mathrm{ex}} \ln \frac{\mathrm{m}_{0}}{\mathrm{~m}}\right) \\
& =\mathrm{v}_{0}-\mathrm{v}_{\mathrm{ex}}+\mathrm{v}_{\mathrm{ex}} \ln \frac{\mathrm{m}_{0}}{\mathrm{~m}}
\end{aligned}\)
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