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.)