Gravity Assist Slingshot

Hyperbolic flyby with turning angle $\delta = 2 \arcsin\left(\frac{1}{1 + \frac{r_{\min} v_{\infty}^2}{GM}}\right)$. Solar-system $\Delta v$ emerges from vector addition with the planet velocity; presets reproduce Voyager 1 at Jupiter and Cassini at Venus to within 20%.

Figure 1. Gravity-assist slingshot: a spacecraft's path bent by a flyby of a moving planet, gaining or losing speed in the Sun frame while its speed relative to the planet is unchanged. Method: planar restricted three-body flyby with before/after velocity vectors.

WHAT TO TRY

Relative to the planet the spacecraft only turns, its speed never changes. The slingshot lives in the lower panel, in the Sun frame.
There, the same flyby adds the planet's velocity to the craft's: line them up and it comes out faster (a boost) for free, no fuel burned.
Pull the periapsis $r_{\min}$ in closer for a sharper turn and a bigger boost; flip the approach to lose speed instead (braking, used to settle into orbit).