In a gravity assist maneuver, a spacecraft is allowed to fall toward a planet, gaining speed just as a rock gains speed when you drop it. Before getting too close, the spacecraft executes a controlled sideways burn of its engines which slightly changes the angle of approach. This allows the spacecraft to slingshot around the planet without falling into an orbit, and retain some of the extra velocity it acquired on the way in. The exact force, timing, and direction of the engine thrust must be carefully matched to the spacecraft's velocity vector and position, otherwise the spacecraft would end up moving in the wrong direction. A mistake of even 1/10 of one degree could mean missing the next flyby completely. 
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How does physics help us to understand how to send spacecraft to other planets? This chapter is about motion and the relationship between motion—position, velocity, and acceleration—and force. We will start with one-dimensional motion, but the techniques we develop carry over directly to three-dimensional motion. The techniques of this chapter are the very same techniques that Cassini-Huygen’s engineers and scientists used to calculate and control the spacecraft over its incredible journey.
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Cassini's mission planners calculated that a direct flight from Earth to Saturn would burn too much fuel, and also pass up opportunities to explore other planets. The mission planners devised a clever but complex trajectory that included two flybys of Venus, a flyby of Earth and a fourth flyby of Jupiter. Each flyby served two purposes. First, each was an opportunity for the spacecraft to make new scientific observations. Second, each close-approach allowed a gravitational “slingshot” maneuver providing extra velocity. Only after the four “gravity assists” was Cassini-Huygens able to reach Saturn.
