Does Oberth Effect work in free space or it works only in gravitational field (gravitational well)?

From Wikipedia article about Oberth Effect it is not clear if it works in free space (no gravity, no air, etc).

For example, in very far space between 2 galaxies, gravity is almost zero. There are 2 identical rockets. Every rocket has the same amount of fuel, the same mass, the same type of engine, etc. It means if both rockets have the same initial condition, for example, zero speed, then after switching on the engine for the same time, they will get the same speed.

1st rocket has initial speed (relatively to one of the galaxy center) $$v_{10}=0$$ zero, 2nd rocket has initial speed $$v_{20}=10\frac{km}{s}$$. Then both rockets switch on engines for small amount of time, for example, 5-10 seconds. 1st rocket with zero initial speed after the engine is switched off (after this 5-10 seconds) gets the speed $$v_1=1\frac{km}{s}$$ and what will be the speed of the 2nd rocket $$v_2=10+1=11\frac{km}{s}$$ or it will be something else because of Oberth Effect? If it is not 11 km/s, then how to calculate it?

There is a similar question, but it is not answered.

• I suggest you reconsider the As to the almost duplicate Q. The v of your rockets will be dictated by the rocket equation. However if they start equal the moving rocket will reach in principle a speed slightly less than 11 (the engines use the same amount of energy during the firing). But I do not see space for invoking or even thinking of Oberth effect here. The effect requires a bound conservative system of two terms one of which is a quadratic one. – Alchimista Jan 3 at 16:30
• @Alchimista "almost duplicate Q" which question exactly do you mean? About quadratic function, energy is always quadratic by speed $\frac{m*v^2}{2}$. This is why I was thinking it might work in free space if speed will be something like $v+dv$. Maybe I need to solve Momentum Energy conservation equations and will get something... – Zlelik Jan 8 at 12:06
• Sorry! I forgot to link that Q that I have found here on SE Physics. Dig for that. For the rest: you need a quadratic term but bound as in E = K + U for the benefit to appear (or to think about). You must see it for an orbiter! Else your energy is better spent at low speed as usual but is even another point. Look for Oberth here you will find the Q/A I referred about. – Alchimista Jan 8 at 13:59
• physics.stackexchange.com/questions/62958/…. Be cautious of some comments as for to my understanding Oberth effect does not make sense with no well. It is when is more convenient to manoeuvre having a fix delta V budget. Without a gravitational term is like you on your car. A deltaV when you are already fast of course results in much bigger energy. But this is finally the fuel! This last point should clarify the situation. – Alchimista Jan 8 at 14:29