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My question is a 2 part question.

First if a vessel in space is going very fast and suddently stops (maybe it is not possible but that is not the point) will things/humans inside the vessel keep going forward or stop with the vessel?

Is it the same if the cabin is oxygened or not?

I know the question is pretty vague but please explain possibilities.

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This happens in moving vehicles on Earth, too. –  Peter Shor Dec 24 '12 at 3:04

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going very fast and suddently stops (maybe it is not possible but that is not the point)

Well, "stops" isn't actually well defined, but if it is subject to a thrust "backwards" any unsecured contents will all bang up against the "forward" bulkheads.

So, the answer is to your first part is yes.

Now, if there is an atmosphere present that will also slosh forward, but since it is a gas this will have no appreciable effect unless you have a helium baloon along--that will float toward the back of the vessel because buoyancy goes against the local effective gravity--or things get up to enough speed to encounter significant air resistance.

So the answer to your second part is mostly yes, things will still move forward with some specific exceptions.

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Yes the things/humans inside the vessel will keep going forward due to inertia. Since the things/humans have kinetic energy due to motion, they would keep moving.

Whether the cabin is oxygened is not important: If you are comparing between a vessel filled with oxygen (or some other gas) and a vacuum vessel, the difference would be that the things/humans inside the vessel would encounter air resistance in the vessel filled with oxygen (or some other gas). Due to the air resistance, the things/humans would slow down (the slow down is insignificant if you have a long time frame)/

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