Finding Maximum Momentum of a Rocket in Vacuum

[LANS]

Homework Statement

A rocket of initial mass m_{0} 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 m is its momentum p = mv maximum?

Homework Equations

Tsiolkovsky rocket equation:

v(m) = v_e ln \left( \frac{m_0}{m} \right)

The Attempt at a Solution

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

mv = v_e ln \left( \frac{m_0}{m} \right) *m
p(m) = v_e ln \left( \frac{m_0}{m} \right) *m

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

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

0 = v_e \left( \frac{m_0}{m} \right) + v_e ln \left( \frac{m_0}{m} \right)

moving the ln term to the other side

-v_e ln \left( \frac{m_0}{m} \right) = v_e \left( \frac{m_0}{m} \right)
v_e ln \left( \frac{m}{m_0} \right) = v_e \left( \frac{m_0}{m} \right)

v_e cancels out.

ln \left( \frac{m}{m_0} \right) = \left( \frac{m_0}{m} \right)

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.

 

[diazona]

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

Looks like you made a mistake in taking the derivative - the right side of this isn't correct. (Also note that what you're calculating here is not dp/dt, since the variable you're taking the derivative with respect to is not t.)