It's been a while since I took physics so let's see how well I do:

The Oberth Effect is the observation that the total kinetic energy imparted by a rocket is greater when the rocket is moving at higher speeds. In simpler terms, a rocket engine is more efficient at harnessing the energy in its fuel when it is moving faster.

At first this seems counter-intuitive—force is force, right? The rocket should provide the same change in velocity no matter how fast it's going. However, we can prove it conceptually by centering the frame of reference on the rocket. If the rocket fires at the apoapsis of its orbit—the point furthest away from the body it's orbiting around and the point where the rocket is moving the slowest—you get a certain change in velocity based on the thrust. Now imagine the rocket moves towards periapsis, the closest and nearest point. As it moves towards the object, the rocket picks up speed due to gravity, following Newton's Second Law of Motion: force = mass X acceleration (in this case, gravity). If the rocket fires its engine at periapsis, not only does it get the thrust from burning the propellant itself, but it also reduces its mass so when it starts climbing back to apoapsis, the force of gravity is smaller than it was before. This means that its apoapsis will rise because it will take longer for gravity to counteract the rocket's velocity. But not only that, because the rocket was carrying that propellant on the fall down to periapsis, it got a little extra velocity in the fall. So the rocket is gaining energy from the propellant just by falling into a gravity well because the potential energy of the propellant is being converted into kinetic energy as it falls. By burning the propellant at periapsis instead of apoapsis, not only did the rocket gain the velocity in the chemical bonds of the fuel but it also stole a little bit of its kinetic energy too.

To prove it mathematically is a different exercise. We start with the mathematical definition of work (which in this case is a stand-in for kinetic energy): E (work or kinetic energy) = F (force) * S (distance traveled). From this it's fairly intuitive that if you increase the distance over which you apply the force, you also increase the total amount of energy imparted, despite keeping the force constant.

With a little calculus you can get a more literal explanation of the phenomenon. By differentiating the equation for work, we can calculate the instantaneous rates of change in each of the variables:

 dE  = F *  dS      (Since we're holding F constant, it is a constant factor and so is not differentiated)
 dt         dt    
In words, that equation says that rate of change in energy = force * rate of change in distance (velocity). Again, you can see that the energy imparted depends on the velocity, even if force remains constant.

This effect has obvious applications to space travel as it allows a spacecraft to get different amounts of delta-v from the same fuel depending on the rocket's speed and location. If a spacecraft's journey is planned to always take advantage of this effect, you can actually design it with less fuel than you'd nominally need to travel that far/fast, freeing up weight for instruments and other devices. The greatest gain in velocity due to the Oberth effect occurs on close approaches to gravitational bodies in a powered slingshot maneuver, a variation of a gravity assist. As a spacecraft approaches the planet/moon/star, the gravitational potential energy stored in the mass of the fuel begins to be converted into kinetic energy. If the spacecraft fires its rocket at closest approach, it has gained kinetic energy not only from the chemical energy of the fuel but also from the potential energy that was released as it fell toward the body. Since the closest approach is also when the spacecraft is moving the fastest, the Oberth effect adds additional energy to the spacecraft (by taking it away from the fuel). Finally, since burning fuel also reduces the mass of the rocket, the resulting velocity is greater due to the conservation of momentum.

ScienceQuest 2012

Sources: For some diagrams way over my head. And because I've completely forgotten everything about calculus by now.

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