gravity assist help?

[power5000]

Okay I don't understand how to calculate even a proximate about the advantage I would gain from a gravity assist and I've seen the picture on wikipedia about 2U+v but I don't get why it's 2U and what if you came in at a perpendicular angle?

[Pixel of Life]

You need vectors in order to calculate how big of a gravity assist you would get. I found these 2 pictures in my physics book and I think they explain the math behind it quite well (text photoshopped to English as the book is in Finnish):

A gravity assist:

Not a gravity assist:

Sorry for the different image sizes. I think the entry velocity vector thing in the first picture is wrong as it should only have the black vector (amirite? Someone confirm, please) because the instant you enter Jupiter's SOI, it has not yet changed your velocity at all. But you get the point. Vectors. Simply put, the gravity assist is the sum of your exit velocity vector and the planet's orbital velocity vector.