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Let's say I stay at point $x=0$ in vacuum. When I move my arm forward such that it will have a positive $x$ position (say $x=5$) will the rest of my body move backward such that it will have a negative $x$ position (like $x=-1$)?

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hmm, lets say one can lift oneself up by pulling the straps of one's shoes? –  Nikos M. Aug 9 at 22:31

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Yes.

There are only internal forces of your body. Without external forces, the center of mass of your body cannot change position. As your center of mass did not move, the main body should move in the opposite direction.

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+1, i almost got this wrong initially, since no (external) forces act on the person, momentum is conserved (similarly to a canon-ball system), when the ball is exploded the canon pushes back, of course since the arm is only a percentage of the remaining body's mass, the body offset will not be -1, but between 0 and -1 (assuming arm's offset is 1) –  Nikos M. Aug 9 at 23:43
    
So if my arm would be much more massive than the rest of the body when trying to move my arm my body would move backward instead of my arm forward? (it seems strange) –  hygrok Aug 10 at 0:40
    
@hygrok Your arm will move slightly forward, but your body a much larger amount backward. –  Bernhard Aug 10 at 7:51

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