gravity assist slingshot explained
A spacecraft can speed up without burning any fuel. The trick is to skim close past a planet. Seen from a frame moving with the planet, the craft's speed does not change at all; it leaves exactly as fast as it arrived, only turned. Switch to a frame fixed on the Sun and the story changes. The planet's own velocity adds to the craft's, so passing behind the planet speeds the craft up and passing in front slows it down. One and the same event yields different speeds depending on the frame you measure it in.
Here is the same flyby seen side by side in two frames. In the planet frame the craft swings around the gold planet and leaves as fast as it came, so the green vin and vout differ only in direction. Choose the Sun frame with the button and a gold arrow vp appears from the origin; it is the planet's velocity, and it adds equally to both. Now the blue arrow vout is clearly longer than vin, and the readout shows a higher speed on the way out than on the way in. The physics is the same; the frame you measure in decides the speed.
How closely the craft skims the planet decides everything. The impact parameter b measures how far off center the craft would pass; in the figure it is the gap between the planet and the dashed line. Shrink b with the slider and the craft swoops closer, gravity grips it harder, and the blue path bends through a larger angle δ. The bigger the bend, the bigger the boost in the Sun frame. Come too close, though, and the craft hits the atmosphere or the surface, so real missions pick a safe b that bends the path only as much as needed.
Depending on which way it bends, a flyby can speed a craft up or slow it down. The planet in the figure moves to the right; pass behind it and the planet tows the craft along its direction of motion, adding speed. Pass in front and the craft is pulled backward and loses speed. Switch between the two cases with the buttons and, of the two bars below, vout goes from longer than vin to shorter as the sign of the speed change flips. That is why missions to the outer planets use trailing passes to speed up, and missions heading inward toward the Sun use leading passes to slow down.
Velocities are vectors, so they add as a triangle. Press Next and the triangle goes up one side at a time. It starts with the relative velocity in the planet frame (the green arrow); adding the planet's velocity vp gives the incoming Sun-frame velocity vin. The flyby rotates the relative velocity while keeping its length, and adding vp to the rotated vector gives the outgoing velocity vout. Because the direction has swung toward the planet's motion, vout is longer. The extra kinetic energy is borrowed from the planet's orbital motion: the planet slows by a tiny amount, and the craft speeds up without fuel.
If one assist is not enough, chain several. Each press of Add flyby sends the craft past the next planet; its speed relative to the Sun builds up and the bar below grows. On the second pass the cumulative speed crosses the dashed escape threshold, and from then on the orbit heads for the outer solar system. Real probes do exactly this, borrowing the gravity of several planets to reach places far beyond what the fuel they can carry would allow.