I guess that's the point of my confusion then: why is u relative to the rocket's velocity before the fuel is ejected? It's been quite some time since I studied physics, so I very well could be wrong, but my intuition tells me it should be relative to the velocity after the fuel is ejected. My...
Hold on. The fuel speed is u relative to the rocket. Meaning in case A, the fuel is traveling at velocity v_A - u. In case B, the first bit of fuel is traveling at velocity v_B - u. But v_A \neq v_B.
The first page of these lecture notes explain my point...
I stumbled upon a 3-year old article from Wired that poses this question on rockets:
Suppose I have two rockets with a mass M and fuel mass m. Rocket A shoots all the fuel at once (again, like a nuclear propulsion engine) with a fuel speed of u and rocket B shoots two blobs of fuel—first a shot...