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LANS

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## Homework Statement

A rocket of initial mass [tex]m_{0}[/tex] accelerates from rest in vacuum in the absence of gravity. As it uses up fuel, its mass decreases but its speed increases. For what value of [tex]m[/tex] is its momentum [tex]p = mv[/tex] maximum?

## Homework Equations

Tsiolkovsky rocket equation:

[tex] v(m) = v_e ln \left( \frac{m_0}{m} \right) [/tex]

## The Attempt at a Solution

multiply both sides of rocket equation by m to get momentum in terms of current mass.

[tex] mv = v_e ln \left( \frac{m_0}{m} \right) *m [/tex]

[tex] p(m) = v_e ln \left( \frac{m_0}{m} \right) *m [/tex]

Find the maximum p(m) by differentiating and letting [tex]\frac{dp}{dt} = 0[/tex]

[tex] \frac{dp}{dt} = v_e \left( \frac{m_0}{m} \right) + v_e ln \left( \frac{m_0}{m} \right) [/tex]

[tex] 0 = v_e \left( \frac{m_0}{m} \right) + v_e ln \left( \frac{m_0}{m} \right) [/tex]

moving the ln term to the other side

[tex] -v_e ln \left( \frac{m_0}{m} \right) = v_e \left( \frac{m_0}{m} \right)[/tex]

[tex] v_e ln \left( \frac{m}{m_0} \right) = v_e \left( \frac{m_0}{m} \right)[/tex]

[tex]v_e[/tex] cancels out.

[tex] ln \left( \frac{m}{m_0} \right) = \left( \frac{m_0}{m} \right)[/tex]

Here's where I get stuck. I can't seem to define m in terms of m_0. Any suggestions? Am I approaching the problem wrong?

Any help is appreciated.

Thanks.

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