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Imagine Apollo 13 had fuel to burn engines on the way back to earth but instead used earths gravity to accelerate and the slingshot around earth

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A slingshot maneuver doesn't just use the gravity of a planet. It uses that planet's motion. From the point of view of Jupiter, the Voyager probes left with exactly the same speed they came in with, only in a different direction. But, in the frame of reference of the sun, Jupiter is moving, and so the Voyagers picked up speed as they swung past behind it.

But Apollo 13 was already moving around the sun along with the earth. There was no way for earth's gravity to shoot it away from the earth. As they approached reentry, they had picked up speed to about the speed they had when they left low earth orbit, a little under earth's escape velocity of about 7 miles per second.

A space probe coming from Venus could use the earth for a slingshot, and they actually did that with the Galileo probe to Jupiter. That is, it was launched from earth, picked up speed from Venus, then swung by earth twice before going to Jupiter.

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    $\begingroup$ A little additional detail: technically, yes, you can do a slingshot around Earth, but the velocity you can gain is on the order of the orbital motion of the object you're doing a slingshot around. For the Earth-moon system, the barycenter is 4671 km from Earth's center, and it's Earth's motion around that that matters for the slingshot...a whopping 12.4 m/s. $\endgroup$ Oct 20, 2023 at 13:21
  • $\begingroup$ @ChristopherJamesHuff 12,4m/s! That is probably just enough for me to throw a ball on the roof of my house $\endgroup$ Oct 20, 2023 at 13:42
  • $\begingroup$ @TheRocketfan If you think that's slow, try calculating the mean speed of the Sun relative to the Earth-Sun barycentre. ;) $\endgroup$
    – PM 2Ring
    Oct 20, 2023 at 15:24
  • $\begingroup$ Yeah, if Earth-relative speed is the priority, probably best to burn every stage dry when departing low Earth orbit and avoid the Moon altogether. $\endgroup$
    – notovny
    Oct 20, 2023 at 18:31

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