Send a spacecraft close past a planet. The pull of the planet bends its path into a hyperbola and it passes through.
Look from the planet. It comes in from far away and leaves for far away. Start and finish are both infinitely distant, so the energy of position is at each.
The energy does not change, so the speed coming in equals the speed going out. Not one kilometre per second has been gained.
Only the direction has changed. On this view the spacecraft has been given nothing at all.
Move to the Sun. The planet is itself moving, so the velocity in the frame of the Sun is the velocity seen from the planet plus the velocity of the planet.
The quantity added is the same on the way in and the way out. What differs is the direction it is added to, because the direction turned in the last scene.
Add the same vector to two vectors pointing differently and the results have different lengths. Pass behind the planet and it grows; pass in front and it shrinks.
The speed has changed with no fuel spent. Nothing happened in the frame of the planet, and yet in the frame of the Sun the craft is faster. Both views are correct, and there is no contradiction.
Put the same thing in more familiar terms. Throw a ball at a truck coming toward you.
In the frame of the truck the ball simply bounces back at the speed it arrived. Nothing has changed.
From the ground it is different. The ball comes back faster by twice the speed of the truck. A truck at m per second returns the ball m per second faster.
No energy has appeared from nowhere. What the ball gained, the truck lost. The truck is so much heavier that the loss cannot be measured.
The largest use of this was Voyager , launched in .
Jupiter, Saturn, Uranus, Neptune: four planets passed in turn. Each pass gave it speed, and that is how it reached the outer solar system. Going directly was beyond the rockets of the day.
The four line up so that one flight can take them all once in years. The nineteen seventies were such a time, and the next will not come until the twenty second century.
After Neptune, Voyager left the solar system. It is still receding on the speed those four planets gave it.
Energy does not appear from nowhere here either. What the spacecraft gained, the planet lost.
How much it lost follows from the ratio of the masses. Jupiter is more than orders of magnitude heavier than a spacecraft, so its orbit is retarded by about one part in . There is no way to measure it.
It works in reverse as well. Pass in front of the planet and the craft is slowed. Probes heading toward the Sun shed speed by repeated passes of Venus and Mercury.
The next article returns to spending fuel. The same burn does not do the same work: where it is made changes what it buys.