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{\bf Question}

A rocket starts from rest in free space by emitting mass.  At what
fraction of the initial mass is the momentum of the rocket a
maximum?

\vspace{.25in}

{\bf Answer}

From theory $v = v_0 + u \ln \left( \frac{m_0}{m} \right)$

The rocket starts from rest so $v_0 = 0$

Rocket momentum $p = mv = mu \ln \left( \frac{m_0}{m} \right)$

Where is $p$ a maximum?

$\ds \frac{d p}{dm} = u \left[ \ln  \left( \frac{m_0}{m} \right) -
1 \right]$

Therefore  $p$ is max/min when $ m = e^{-1}m_0$

Check that $\ds \frac{d^2 p}{dm^2} < 0$ to confirm that $p$ is a
maximum when $m = e^{-1} m_0$



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