Oh my, where to begin...

The only maximum speed for rockets, *any* type of rocket given enough fuel, is the speed of light. Rockets are *not* limited by the speed of their exhaust. Do some googling if you don't believe me. In fact, the Apollo 11 mission reached speeds in excess of 11 km/s with an exhaust velocity of less than 5 km/s. As they were powering away from us, their exhaust was also moving away from us. What "currently accepted law of physics" did they break? None whatsoever.

If you are on a boat that's moving forward at 10 m/s, and you jump off the back with 5 m/s (so you are still traveling in the same direction of the boat at 5 m/s), won't that push the boat forward? Same thing with rockets.

The only thing that matters for a rocket in vacuum is the speed of the exhaust relative to the rocket. There are no absolute speeds in space, so if the rocket did have a maximum speed, what would it be relative to? Relative to earth? They're in space, earth is just one of trillions of rocks flying in all directions at vastly different speeds. Why would the rocket have a maximum speed relative to one rock in particular? (Apart from the speed of light, but I won't get into relativity here).

So no, rockets in vacuum produce the same thrust at any speed. This means that, if they are going faster, the engines are producing more kinetic energy per second.

Now I imagine you are probably thinking you caught me on an inconsistency here. Didn't I just say that there were no absolute speeds in space? How can they become more efficient at higher speeds then? Speed relative to what?

Indeed. All speeds are relative, but kinetic energy is *also* relative. A car coming towards you has lots of kinetic energy (and can therefore cause a lot of damage if it hits you) but if you are traveling in the same direction at the same speed, it has zero kinetic energy from your point of view.

When rockets are flying close to planets, we only care about their kinetic energy relative to that planet because we want to move to an orbit that corresponds to a certain amount of energy, or escape from the planet at a certain speed relative to that planet.

So how do we get as much kinetic energy as possible relative to that planet? By producing thrust (which is constant like I said) at the point where the relative speed of the rocket is the highest (which coincidentally happens to be at the deepest point in the gravity well).

I don't think you read the entire Wikipedia article I linked to, about the Oberth effect. Scroll down to "explanation in terms of work". The derivative of the kinetic energy is thrust (F) times speed (v). Same thrust, higher speed, more energy gain. It really is that simple.

You can also explain it as "leaving the fuel deep in the gravity well where it has less potential energy, so you are throwing away less energy that way" and in a way that's true, but it's not very practical for actual calculations. (If you think it's practical, I won't argue with you).