Rocket Question - 2 solutions?

[unscientific]

Homework Statement

A rocket of initial total mass m_i leaves a space station in deep space, far from all sources of gravitation. The rocket ejects burn fuel at a constant speed u << c relative to the rocket.

(a) Derive an expression for the rocket relative to the space station when the total mass of the rocket plus remaining fuel is m_f.


(b) Some time after the rocket has left the space station, a small quantity of the ejected fuel remains stationary in the frame of the space station. What limit does this observation place upon the fraction of the original total mass that was burnt as fuel?

(c) The rocket ejects fuel at a rate proportional to its remaining total mass m, so that (dm/dt) = -αm. Derive an expression for the distance x between the space station and the stationary burnt fuel. Obtain a value for x given that u = 1000 ms^-1, and that m is reduced to half its starting value after 10 minutes.

The Attempt at a Solution

Part (a)

Part (b)

Part (c)

Method 1

Method 2

 

[mfb]

In the second line in method 2, there is "m" missing in the denominator of the right side.
And the units of the result do not match as a consequence.