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Spacecraft acceleration using a planet's gravity

When a spacecraft travels to the outer solar system it picks up speed by using the gravity of planets along the way. In fact they sometimes would whizz by the earth a few times to pick up speed before continuing their journey. It does work, but I don't understand how it can work.

Suppose a spacecraft is heading to saturn. It approaches jupiter and gradually picks up speed as it nears the planet. However, as it pulls away from jupiter it must now fight the gravity of jupiter which would slow it down. Give the "conservation of energy" and that the distances approaching and leaving the planet are the same, how can the space craft gain energy?

In addition, these spacecraft normally change direction which would use enegy.
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Sean Stuber

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Thanks Nova - I'll dig out the old physics books so I can fully understand your explanation.
The short and oversimplified explanation follows.
The sling short method adds angular momentum to the spacecraft. The planet looses a corresponding amount of angular momentum. (But because it has so much the reduction of rotational speed is unnoticeable.) Because of a relationship between angular momentum and kinetic energy, the net result is a transfer of kinetic energy from the planet to the spacecraft.
Warning: If too many spacecraft use Jupiter for a slingshot, its day will be affected. (Exercise for the reader: If 25 slingshort events pet year involve Jupiter, how many centuries will have to elapse before the day of Jupiter decreases by 1 second?}
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NovaDenizen

Will it really affect the day length of the planet, or will it affect the planet's year length?  Unless the spaceship is large enough to cause detectable transient tidal forces on the planet, I don't think the rotation of the planet will be affected.
It doesn't affect the day length of the planet, it affects the year length of the planet.

The tidal forces won't be detectable on the spacecraft, much less the planet.
And the effect on year length of the planet is not even detectable,
Anyway, the undetectable effect of tides is more likely to slow down the spacecraft and speed up the planet