# Application of trigonometry to the slingshot effect/gravity assist

I have been trying to understand the formula

$$v_f^{2}=v_i^{2}+2V(V(1-\cos\beta)+v_i(\cos(\alpha-\beta)-\cos\alpha))$$

as it relates to Fig. 2 on page 5 of this exposition:

http://maths.dur.ac.uk/~dma0rcj/Psling/sling.pdf

The angles between the positive directions of $V$ and $(v_i,v_f)$ are denoted by $(\alpha,\alpha^{\prime})$, respectively. $\beta$ is the positive rotation angle of $v_i$ arrowed between the dashed lines.

I surmise that the law of cosines is at work, but I fail to see precisely how.

Can someone provide hints as to how the formula relates to Fig. 2?

(One way of answering my question is to partially/wholly derive the equation.)

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I couldn't figure it out, the obtuse triangle made things very complex to analyze. My approach was utilizing the alpha and beta angles from the sum angle derivations for cosine and then attempting to solve for the additional side that made the angle obtuse. It seems as though he was using the standard conservation of energy formula though. What I don't understand is why he wouldn't have to use three dimensional Euler angles, as the satellite would move in the zed direction as well (via elliptical orbit). Sorry, I will keep checking this, I am curious as well. –  Gödel May 22 at 17:41
Minor comment to the post (v3): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. –  Qmechanic May 22 at 17:52
You could also try space.stackexchange.com if need be. –  Alan Rominger May 22 at 18:53