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My understanding of gravity-assists (slingshot) is that they use some of a more-massive object's orbital energy to gain massive acceleration.

What's the upper limit on how much velocity an object coming out of one can have?

I am not interested in the max total amount of acceleration gained from a single assist. Just the highest velocity achievable.

I presume the limit here would come from the effects of frame-dragging, if trying to get an assist from a very massive object.

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    $\begingroup$ Your question doesn't obviously make sense... if you look at Mark Adler's answer to your linked question, you'll see he provides a handy formula. You can always get just a tiny bit faster than the velocity you arrived at, right? $\endgroup$ – Starfish Prime Nov 7 '19 at 12:46
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    $\begingroup$ So, a hair below lightspeed? $\endgroup$ – bobsburner Nov 7 '19 at 12:49
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    $\begingroup$ Perhaps, thought you'd have to be travelling at almost a hair below lightspeed to start with, so it doesn't seem like a very interesting or informative answer (and adds nothing that wasn't already said in the other question). $\endgroup$ – Starfish Prime Nov 7 '19 at 12:50
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    $\begingroup$ Realistically you will be limited by time or the availability of objects. Once you reach solar system escape velocity your ability to play cosmic billiards is extremely limited and even without that planets are months or years apart--try too many encounters and your probe will wear out before reaching it's destination. $\endgroup$ – Loren Pechtel Nov 7 '19 at 12:53
  • $\begingroup$ Not enough points to close-vote, but recommend doing so in favor of the linked question in @StarfishPrime 's comment $\endgroup$ – Carl Witthoft Nov 7 '19 at 14:42
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I don't think it's easy to put a limit on the maximum achievable speed (other than $c$), without constraining the problem. There is no obvious speed, above which gravity assists can't help, if you can plan the interactions appropriately.

There are things that will make this tricky though. For example the large relative speed that bodies will likely be encountered when moving fast, and the necessity to align exit velocity with the velocity of the assisting body, you are quickly going to run into the problem of needing near infinite accelerations. I.e. skimming event horizons of progressively smaller black-holes.

This may bump into some other hard physics limit (like a black hole that small existing for long enough to complete the interaction). However I don't think there is a nice/simple statement that governs this sort of thing. Further You're starting to blur the lines of what constitutes a gravity assist when you start considering these sorts of things.

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