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Could a people do all sort of gymnastics movement in vacuum space? I asked this because I am worry about that the astronaut leave the space shuttle during emergency could not go back to earth by himself if there are no fuel on the astronaut, could he swim in space to get back on earth even if there are no water?

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No, you can't move in space (aside from following your orbit), unless you have something to push against. See Newton's Laws of Motion. (Also, if said astronaut did manage to re-enter the atmosphere, he or she would burn up on re-entry. The impact wouldn't be survivable either.)

Astronauts are well aware that many sorts of emergencies will not be survivable.

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  • $\begingroup$ Er...while you can not influence either you linear or your angular momentum you can (in principle) influence your orientation (or the phase associated with a non-zero rotational motion). $\endgroup$ May 3, 2012 at 22:45
  • $\begingroup$ @dmckee: Really? Is that true in practice, or is it only some GR result that would be undetectable on human scales? $\endgroup$
    – Colin K
    May 4, 2012 at 1:03
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    $\begingroup$ @ColinK: It's true in practice and is how springboard divers power half-twists (by contortion of their arms). Longer twists musts take advantage of setting up unstable rotations around $I_2$. Think of it this way. You can't change your $L$, but you can rotate one part of your body relative another part until you run out of range of motion, so the final result is that you continue with the same $\omega$ but a different phase. Not sure how practical it is going to be in a vacuum suit, however. $\endgroup$ May 4, 2012 at 1:40
  • $\begingroup$ @dmckee - Sure, but if he's worried about astronauts in orbit around the Earth, changing orientation isn't going to do the trick. $\endgroup$
    – Rex Kerr
    May 4, 2012 at 15:53
  • $\begingroup$ great video of playing with angular momentum in space. looks fun. youtube.com/watch?v=dmnmuTv4pGE $\endgroup$ Jul 19, 2014 at 7:58

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