Hills' mechanism for making Hypervelocity Stars
Consider two unit masses with velocity $V$. Give them a small push apart so they have velocities $V+v$ and $V-v$ with $v\ll V$ . Their kinetic energies are then $$\frac{(V+v)^2}{2} \approx \frac{(V)^2}{2} +vV$$ and $$\frac{(V-v)^2}{2} \approx \frac{(V)^2}{2} -vV$$
We have transferred energy $\approx vV$ from one mass to the other with the expenditure of a much smaller  energy $v^2$.

Now suppose the masses are initially moving with some tiny velocity and then travel into a potential well where they have velocity $V$. If we push while they are in the well then one will have enough kinetic energy to escape the well plus $vV$, while the other will be trapped in the well.

This is the same principle as the Oberth effect, where the ejected reaction mass plays the role of one of the masses.