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Inertia? What Inertia?
I spacewalked out of my ship to visually inspect some hull damage and check my exterior collision model. Ship was moving at the time. Instead of floating alongside me, it zoomed away as if I'd just finished hopping freight and jumped off the railcar.

This is highly unexpected, unrealistic and, when in a vehicle with collision damage, potentially deadly. Say you have a ship the shape of a Star Destroyer. You open a hatch on top of the nose and float out, but, instead of gently coasting along in the serene ballet "2001" has taught us to desire, the superstructure slams into you from behind like a stagecoach and six panicking horses.

It's also the source of the classic newbie (and not-so-newbie) complaint "I accidentally pressed E trying to steer my rocket and it dumped me".

When a ship is not in an atmosphere, can objects transiting from a hull void into space just inherit the ship's velocity (add it to their own velocity at the point of exit)? That way it's all intuitive and a bit safer. Doesn't need to apply to transporters, the most common method of travel, but should apply to launch tubes.
If this is technically possible, yes please.
Why should it be limited to outside the atmosphere? Just apply it everywhere.
I guess you could do it everywhere, if you can be bothered to factor atmospheric density etc. I was just keeping it simple to start with.

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