The best way to give an accelerating boost to a spacecraft, without using any fuel, is to get a "gravity assist" from a large planet or moon. It's not immediately obvious how this could work: yes, gravity can accelerate falling objects, but it also slows down rising objects. A "flyby" of a moon would do both: accelerate on the approach and then decelerate on the departure. In fact, conservation of energy would seem to imply that you couldn't gain any net velocity at all!

So how can gravity assists ever work? The trick is that the flyby is past a body (moon or planet) that is orbiting something else. True, you don't change your velocity relative to the flyby object; but you do change it with respect to the sun (and, in this case, Jupiter).

For example, the moon Io is going around Jupiter at an orbital velocity of 8.5 km/sec. Say you are in a spaceship at a standstill with respect to Jupiter, with Io coming right towards you. (Assume this all happens so fast that Jupiter's gravity isn't interfering). The spaceship, although it's not moving, is still moving at 8.5km/sec with respect to Io, and if you swing around a quarter of the moon when it arrives (a flyby), you will now still be moving at 8.5km/sec (after speeding up and slowing down), but now you can be moving away from Io in a different direction -- say directly away from Jupiter as well. Now you're moving at 8.5km/sec with respect to Jupiter, when at first you weren't moving at all -- that's a big acceleration! The spaceship didn't gain any energy with respect to Io, but it did with respect to Jupiter, because Io was moving around Jupiter. Details abound, but that's the physical basis for gravity assists.

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