I am trying to find the initial velocity to slingshot a planet around the sun and through a gap.
The green ball is the planet, and the yellow ball is the sun. In this trial I need to get the planet to go around the sun and through the gap at 278Gm. I have tried different approaches, but nothing seems to be even remotely correct. Anything under 20k m/s will land you in the sun and anything over 50k will slingshot you out of the system.
I want to know what formula to use so that I can solve this type of problem.
Answer
If you are in a circular orbit what you need is a Hohmann transfer, from Wikipedia:
In orbital mechanics, the Hohmann transfer orbit /ˈhoʊ.mʌn/ is an elliptical orbit used to transfer between two circular orbits of different radii in the same plane.
It works like this assuming the planet is in a circular orbit.
Then the amount of delta v needed to go from the green orbit to the yellow orbit is. 
where units are
- $ v \,\!$ is the speed of an orbiting body
- $\mu = GM\,\!$ is the standard gravitational parameter of the primary body, assuming $ M+m$ is not significantly bigger than $M$ (which makes $v_M \ll v$)
- $r \,\!$ is the distance of the orbiting body from the primary focus
- $a \,\!$ is the semi-major axis of the body's orbit.
Using an online calculator I deduce that the delta v you need is 25.07 km/s
This is independent of the mass of the planet.
Ok, let's start over with a different approach, what is the velocity exactly. Lets just use our trusted elliptical orbits. 
Then using equations from this link you can calculate the speed at any point of a eclipse with,
$$ v^2 = \mu\left(\frac{2}{r} - \frac{1}{a}\right) $$
which leads to 44.31 km/s at perihelion.
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