Equal and opposite reactions in space.

Suppose that my spaceship crew mutinies and puts me in a pressurized suit and dumps me into space. They came to a dead stop first (if that means anything). Naturally, I start flailing my arms and legs (to use up my oxygen faster, I suppose). Where is the equal and opposite reaction to my movements?

When you flail your arm in one direction, your arm cannot continue going in that direction forever. As you start your flail, the rest of your body moves in the opposite direction, when your flail slows down to reverse direction, so does the rest of your body. The net effect is a bunch of flailing and not much else.
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First, coming to a dead stop does not mean anything except relative to some other object. Let’s just say that the ship is moving in a straight line at constant speed (so it constitutes an inertial frame).

The equal and opposite reaction to your movements is the motion of your body about your fixed center of mass (relative to the spaceship). If you swing your arm back forth, your body will rotate somewhat so that your center of mass stays in the same place. Try this in a swivel chair; same principle. You will be able to get the chair to spin back and forth, but it stays in the same spot on the floor (although in the chair there is some friction in the wheels so you might be able to get the chair to move a little bit).

The only way you can move your center of mass relative to the spaceship is to poke a hole in your suit and aim it so the air shoots out in the opposite of the direction you wish to go (see also “The Martian”). However, you would die from loss of pressure and oxygen.

Likewise, there is a stuck in space a few meters from safety version of Aron Ralston - you could cut part of your body off and fling it in the opposite direction. (After which, it occurs to you that if you’re going to cut through the suit, you could have taken the suit off and just flung the suit.)

If they were kind enough to let you have a blacksmith’s anvil, you could jettison that in the opposite direction you wish to move. The center of mass of the you/anvil system remains “stationary”, but we’re only concerned about you.

To illustrate what would happen in space in an earthbound situation for which your intuition may be better, watch the classic “cat jump fail” video:

The cat is expecting that the wall will provide a stable base to push back against. When it doesn't (or barely does) the cat essentially just stays almost in the same place, its powerful hind legs extending backwards and its body moving slightly forward, its center of gravity staying in roughly the same place. With nothing stable to push against in space, any attempt to displace your body by extending your limbs would have a similar result (ignoring the cat's subsequent fall due to gravity, of course).

Didn’t think of that one. A good reason to always bring a cannonball on EVAs.

You could also poke a tiny hole in your suit and let the air released under pressure push you back to the ship. Probably work as well as trying to throw an anvil, cannonball, your suit, or some part of your body, and you’d have some chance to adjust your trajectory if you’re off at the start. Kind of sad to watch your bowling arm head off into space as you glide past the space station just out of reach.

You deserve your fate if you didn’t think ahead to save a few loose fingers to lob to fine-tune your trajectory.

Even then, your center of mass isn’t moving. The ejected air moves in one direction, the rest of you move in the other direction. The center of mass of the whole system (the spacesuit and everything that was in it) stays in the same place.

Not only that, but the mass of the air isn’t much compared to the mass of your body.

From conservation of momentum, m[sub]1[/sub]v[sub]1[/sub] = m[sub]2[/sub]v[sub]2[/sub]

Your body won’t be moving much at all. You will want that bowling ball.

That is one of the oldies but goodies, but honestly I thought I was going to see either:

A cat and heavy object in the space station or in outer space in a little catsuit

Or, probably,

The same thing, minus the cat suit, over some appreciable but safe height. It would be the feather-and-cannon ball of Galileo with a twist, literally.

In Moontrap, Walter Koenig’s character uses a machine gun as a thruster. Equal and opposite reaction: the bullets (and propellant gases) go one way, he’s pushed the other way, while the combined center of mass of him and his bullets remains where it was.

True but it depends equally on the *velocity *of the air.

And if you’re only a few meters from safety, even the small mass of air might be enough.

And on how much time you have to wait.

Although you can’t change the position of your center of mass, you can change your orientation by flailing about in the right manner.

Cats use this technique to right themselves mid-fall. They do not depend on aerodynamics to land on their feet. See here for the vector analysis.