# Gravitational Slingshot

If a spacecraft slingshots around a planet P (with escape velocity V) at an angle $$\theta$$, I understand that the resulting velocity is $${ v }_{ 2 }=({ v }_{ 1 }+2u)\sqrt { 1-\frac { 4u{ v }_{ 1 }(1-cos(\theta )) }{ ({ v }_{ 1 }+2u)^{ 2 } } } .$$

However, this equation does not involve the mass of the assisting planet, nor the distance/altitude from which the spacecraft must slingshot from. After reading the answer to To what extent could a single Triton flyby slow down a direct Hohmann transfer to Neptune for NOI?, I have a few questions:

• How is the formula for the bent angle and eccentricity derived?
• What exactly is the geometry behind the maneuver? More specifically, where does the hyperbola come from?
• How is the formula for the turning angle, $$\delta =2\sin ^{ -1 }{ \left( \frac { 1 }{ 1+\frac { { r }_{ p }{ v }_{ \infty }^{ 2 } }{ \mu } } \right) }$$ derived?

Please note that I'm a high school student with a knowledge of Calculus I & II, and that visual diagrams would be greatly helpful.

• If you know the escape velocity you should be able to get the mass of the planet - see wikipedia for the formula, $v_e = \sqrt{2GM/r}$ . After that, if you can map your various "velocity" parameters from one equation to another for us, that might help. Also keep in mind that the minimum altitude at least has to be exoatmospheric! Jan 16 '20 at 20:28
• What is $u$ in the equation? Jan 17 '20 at 8:37
• I see some similar stuff at Wikipedia: Hyperbolic trajectory but would appreciate a link to the source of your first equation. Jan 18 '20 at 19:52