Conservation of Momentum in Rocket Propulsion

[VVS]

Hi,

I really need clarification for this problem.

Homework Statement

A rocket with initial mass M0 is accelerated by expulsion of a partial mass dm of its mass m with a constant velocity u. Write down the differential equation corresponding to the conservation law of momentum for the velocity v(m) of the rocket and evaluate it under the condition v(m = M0) = v0. What is the terminal velocity when the final mass of the rocket is Me?

The Attempt at a Solution

I have attached three different approaches (all using the conservation of momentum principle) of mine in this pdf file.
But unfortunately I get three different results.
Please point out which approach is wrong.

thanks a lot
VVS

 

[verty]

What are your thoughts about the three methods? I think you can apply some physics insight to decide which is correct.

 

[VVS]

I really can't tell. All three methods appear to be correct to me. I can't see what is wrong about any of the methods. But obviously because I get different answers something has to be incorrect, but I just can't tell what.