newtonian mechanics - What happens when the spacecraft velocity equals the velocity at it's exhaust?

So there I was resting me eyes thinking about rocket drives, and what-not. The thought struck me that, perhaps, even before Mr. Einstein interferes with the increasing velocity of the spacecraft Mr. Newton may have something to say.

Please poke me in the rib if my comprehension is wrong -

A rocket, basically a reaction mechanism, must push gases out of it's exhaust to impart velocity to the spacecraft. Assuming adequate fuel is available, what happens when the spacecraft velocity is equal to velocity at the exhaust? At this stage, does the rocket still accelerate the craft?

Answer

At this stage, does the rocket still accelerate the craft?

If by "velocity of the exhaust" we are talking about its velocity measured in the frame of the rocket, then Yes. Let $\mathbf u$ be the exhaust velocity as measured in the rocket frame, then in free space, the non-relativistic rocket equation is \begin{align} \frac{d\mathbf v}{dt} = \frac{\mathbf u}{M} \frac{dM}{dt} \end{align} where $M(t)$ is the mass of the rocket plus whatever fuel is on board at time $t$ and $\mathbf v$ is the rocket velocity in some inertial frame outside of the rocket. Let's say, for simplicity, that the exhaust velocity is constant, then this equation has solution \begin{align} \mathbf v(t) = \mathbf v(0) - \ln\frac{M(0)}{M(t)}\mathbf u \end{align} The rocket keeps going faster and faster until its fuel is exhausted. In particular, there is nothing preventing the rocket from going faster as measured in the inertial frame than its exhaust as measured in its own frame.