How does Lunar Orbit Injection (LOI) work? Just what the title states.
I read that LOI was used by the Apollo program (and possibly others) to achieve escape velocity. How does it work? Is it merely a matter of centrifugal/centripetal force?
 A: The trans-lunar injection doesn't achieve escape velocity. Instead it changes the orbit of the spaceship to an elliptical orbit that intersects the Moon's orbit.
The Wikipedia article on Hohmann transfers explains this very nicely. The spaceship starts out in a circular orbit around the Earth. The Moon is also in a circular orbit, except of course the radius of the Moon's orbit is a lot greater. If you suddenly accelerate the spaceship by firing the motors then the spaceship is moving too fast to stay in the original circular orbit. In fact you've put it into a highly elliptical orbit with the apogee near the Earth and the perigee near the Moon's orbit. You time this orbit so that as the spaceship reaches perigee it hits (not literally!) the Moon.
At perigee the spaceship is moving too slowly to stay in the Moon's orbit, so you fire the engines to accelerate it again. It may seem odd that you're accelerating instead of braking, but because your spaceship has lost lots of speed as it drew away from the Earth it would pass the Moon and fall back towards the Earth. Firing the motor gives it the extra speed to keep up with the Moon.
When you want to return to earth you fire the motors to decelerate. This puts the spaceship back into an elliptical orbit with the apogee near the Earth. As you reach the Earth one last burn puts the spaceship in an orbit that takes it into the atmosphere, and atmospheric friction slows it down enough to parachute into the sea.
A: The typical Apollo TLI burn results in an Earth orbit with an apogee approximately twice the lunar mean distance. It is targeted close enough to the Moon that at some point the Moon's gravity has more influence on the Apollo's path and pulls the spacecraft toward it [the Moon]. The velocity of Apollo once it reaches the back side of the Moon is too fast for lunar orbit so the Apollo must use it's big SPS engine to slow down enough to be captured by the Moon. As a safety measure against slowing down too much and crashing into the Moon, the initial Moon orbit was planned to be elliptical (further from the Moon on the Earth facing side) then a shorter burn circularized (early missions) or reshaped for a close perilune (later missions with heavier LM's) to eliminate the PDI burn to save LM fuel for the descent.
