A Gravity Assist Steals the Planet's Speed
A spacecraft can speed up without burning a drop of fuel. The trick is to skim past a planet. Ride along with the planet and the spacecraft's speed does not change at all; it comes in and leaves at the same speed, only its direction bends. Switch to the frame fixed on the Sun and the story changes. The planet's own velocity adds to the spacecraft's, so passing behind the planet speeds it up and passing in front slows it down. The same event, seen from a different frame, shows a different speed.
See the same close pass in two frames. In the frame riding with the planet the spacecraft leaves at the very speed it entered; the two arrows are equal in length and only the direction has bent. Press the button for the Sun frame and the planet's velocity adds to the spacecraft's. The outgoing arrow grows noticeably longer, and the readout shows the speed has climbed. There is one physics here; which frame you measure in decides the speed.
How closely you skim decides everything. The impact parameter b is how far the approach misses the planet's center. Drag the slider to shrink b and come in closer, and the planet's gravity pulls harder, so the path bends more. The larger the bend, the larger the boost you gain in the Sun frame. Come in too close and you hit the atmosphere or surface, so a real mission picks a safe b and bends just as much as it needs.
Bending alone can either speed you up or slow you down. Pass behind the planet and it tows you along its direction of travel, adding to your speed; this is a trailing pass. Pass in front instead and you are pulled against its motion and lose speed. Switch between the two cases with the button and the sign of the speed change flips. Heading to the outer planets you take a trailing pass to accelerate; falling inward toward the Sun you take a leading pass to slow down.
Velocity is a vector, so it adds as a triangle. Build the triangle one step at a time. First there is the relative velocity in the planet frame. Add the planet's velocity to it and you get the incoming velocity in the Sun frame. The close pass turns this relative velocity, keeping its length and only rotating its direction. Finally add the planet's velocity again to get the outgoing velocity, which is longer because the direction has turned favorably. The extra kinetic energy came from the planet's orbit; the planet slows by a tiny amount so the spacecraft speeds up with no fuel.
If one assist is not enough, you chain several together. Each time the spacecraft skims past another planet, its speed in the Sun frame builds up a little more. Add close passes one at a time with the button and the cumulative speed climbs past the escape threshold marked by the dashed line. From there the path heads out to the outer solar system. Real probes borrow the gravity of several planets like this to reach far beyond what their onboard fuel alone could carry them.