I feel this is somehow a stupid question, but I don't know the true answer. What happens to an astronaut who's floating in a spaceship in space when it begins to move? Will the astronaut not move until he smashes onto a wall in the spaceship? Or will he move with the spaceship due to gravity? Or does it depend on the speed/acceleration of the spaceship; if it's too fast, the astronaut will smash onto a wall and if it's enough slow, he'll move with the spaceship?
He will start to move with the spaceship due to the air pressure inside (the ship pushes the air, and the air pushes him) and the gravity of the ship pulling him, but unless it is accelerating really slowly he will smash into something, it is not so different from beeing in an accelerating car, infact its exactly the same if you consider only the horizontal direction.
The gravitational force of the spaceship on the astronaut is tiny, effectively zero. It would take a really huge object (e.g. a small moon) to have any appreciable effect on an astronaut or anything else.
If an astronaut is in a spaceship and the spaceship accelerates, for all practical purposes, the astronaut will not move along with the spaceship unless he's attached or connected to it in some way, until one of the walls hits him.
Dear huy, when a spaceship is flying to the Moon or when the International Space Station is orbiting the Earth, both the spaceship and the astronaut are moving by the same speed, so the relativity velocity is zero. Moreover, gravity determines the acceleration of all of them. The principle of equivalence implies that when the previous sentence holds, all effects will proceed exactly as in the absence of gravity and acceleration.
However, rockets have to be accelerated to get them to the speed. When they're accelerated, astronauts are pushed to their seats and their faces get deformed by the inertial force that is pushing them from one direction and the force from the seat or wall that is pushing them in the opposite directions. Astronauts should be able to withstand the acceleration of several $g$ - multiples of the Earth's gravitational field. They're trained on the Earth - in centrifugal gadgets, the vomit comet (even Stephen Hawking tried it), and otherwise.
If a spaceship accelerates by a big acceleration, it's a good idea to fasten your belt because indeed, the astronaut body floating inside the spaceship has absolutely no reason to accelerate at the same moment. It will continue to float by the same speed, so if the spaceship accelerates, the relative position of the astronaut and the spaceship will accelerate, too. The astronaut will hit the wall much like when he falls - by a constant acceleration - from a wall on Earth. The gravitational force of the spaceship or the astronauts are unmeasurably tiny; they make no effect. Even the gravity of Mr Everest is hard to measure or perceive by "ordinary tools".