What is the maximum momentum of a rocket with given exhaust speed and mass?

[aznwei]

Homework Statement

A rocket that starts at rest with mass M ejects exhaust at a given speed u. what is the mass of the rocket(including unused fuel) when its momentum is maximum?

Homework Equations

p=mv

The Attempt at a Solution

well basically i used conservation of momentum to get the velocity of the rocket with respect to mass. after doing all the integrating work i got v=uln(M/m) assuming that m is the mass of the rocket at a later time. plugging the velocity into the p=mv formula i differentiate with respect to mass where the equation becomes dp/dm=u[ln(M/m)-1/m^2]. now i set dp/dm to zero to try to find the mass needed for a max momentum but unfortunately the dp/dm equation isn't easy to solve due to the "ln" function. Am i doing something wrong here?

 

[kuruman]

... the equation becomes dp/dm=u[ln(M/m)-1/m^2].

I don't believe you took the derivative correctly because dp/dm must have dimensions of speed. Your expression does not. Remember the chain rule.