Gravity Assist (Slingshot effect)?

[!Live_4Ever!]

Hi, I have to make a small presentation on how the phenomenon of gravity assist works. Basically, as we saw in Armageddon (I know, bad movie for scientific inspiration XD ) the two spaceships "slingshot" through the Moon, and attains higher speed, due to the Moon's own gravity.

I tried to use the stuff I know (Centripetal force, Momentum, Universal Law of gravitation, Newton's laws, Pot/Kin Energy, Kepler's laws...etc) to apply them to the phenomenon, but I have no idea how to come up with a concise way to explain how gravity assist works...

I would really appreciate any help in this topic...

Thank you in advance, =)

 

[LowlyPion]

Here is a problem that touches on the issue.
https://www.physicsforums.com/showthread.php?p=2085258

Here is a Wikipedia entry:
http://en.wikipedia.org/wiki/Gravity_assist