EDIT: Thank you all for your kind yet informative responses!
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Another factor, besides all the great descriptions of inertia throughout this thread, is that there’s barely any friction in space. If you stick your hand out of the window of a car driving fast, the friction of the air forces your hand back. But there’s no air in space (well, hardly any air) so when astronauts do space walks and such, no matter is going to push them away. This is also how objects in space can move continuously at high speed with no need of propulsion.
The same way you’re unaffected sitting in a car going 75mph in a straight line. You only feel anything when the car accellerates in any direction, such as speeding up, slowing down, or turning. When the car is moving at a constant speed with no changes in direction, you feel at rest. It works the same way in an airplane going 400mph. It’s no different for the people on the Internation Space Station.
I guess you could compare it to riding a car.
Lets say you’re in a car going 50 mph, you don’t really feel like your going 50 mph, and because you’re in the car you are also going 50 mph. But when the car brakes suddenly or slows down a lot in a short amount of time, you lean forward because unlike a car that has the ability to slow down or go from 50 to 0 in seconds, our bodies can’t slow down that quickly.
Probably the only way astronauts in the ISS would feel how fast they’re travelling is if the ISS were to move faster or suddenly stop.
Because the astronauts are moving at exactly the same speed and direction (formally: velocity) as the station.
Both are free falling toward earth under the influence of gravity. Gravity affects both identically.
Both are also moving “sideways” very very very fast, which is why they never hit the earth.
As Hitchhikers Guide to the Galaxy put it: the trick is to throw yourself at the ground and miss.
Motion is relative.
The ISS is traveling that fast according to a point on earth.
To the ISS, everything on it is stationary. They’ve all caught up to each other and are travelling at around the same speed with very little gravity acting on them. Inertia and relativity at play.
The ISS then becomes the frame of reference for the astronauts rather than the earth.
Just like for you. The earth is spinning at a mad speed, orbiting the sun at a madder speed, and hurtling through space who knows how fast. The earth is your frame of reference and you are a point on the earth.
A car then whizzing past at 45mph isn’t actually going that speed to the earth. It’s going the speed of the earth plus the velocity of the rotation plus its own momentum.
To you, it is going 45 mph because you are “stationary” according to earth and they are moving at a speed that is 45 mph different than your relative 0mph.
You are currently moving at hundreds of thousands miles per hour relative to the center of our galaxy and you are unaffected by that. Same for the astronauts in the space station.
Everything is moving, and steady motion feels like it’s motionless, so we only feel changes in motion.
In other words it’s changes in motion that affect us, not merely motion itself. Usually we’d feel any sudden change in motion if the change is substantial enough.
Earth itself is moving over 100,000 miles per hour around the sun. Our entire solar system with sun and Earth are moving about 4 times faster than that (over 400,000 miles per hour) around our galaxy. We don’t feel those because that motion is might change only slowly, so its fairly steady.
Astronauts feel more during takeoff and rocketing into space because of the sudden changes in speed, than they later feel coasting at a more steady speed in orbit around Earth.
Edits: clarified and elaborated bits
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