Yup.
Its not uncommon to experience space sickness for the first day or two in space. Some people never adapt (see Jake Garn).
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Astronauts appear to be floating in a spaceship because they are in a state of constant free fall around the Earth. This creates an environment of microgravity, where they experience weightlessness. Gravity continues to pull on them and the spaceship, but both are falling at the same rate, creating the appearance of floating.
Yes. Gravity is what keeps them in orbit. If it were not for gravity, those astronauts would continue out into space in a straight line.
Astronaut beds on the International Space Station are small sleeping compartments with walls to keep the astronaut from floating around while they sleep. They typically have sleeping bags that are attached to the wall to keep them in place. Astronauts sleep in a sleeping bag that is strapped to the wall inside the compartment to keep them from floating around.
Astronauts appear to be floating in space because they are in a state of continuous free fall around the Earth. This creates a feeling of weightlessness, where they float inside their spacecraft. The apparent lack of gravity is due to the spacecraft and everything inside it, including the astronauts, falling towards the Earth at the same rate.
Astronauts in space experience microgravity, where they are in a state of continuous free fall around the Earth. This creates the sensation of weightlessness. They do not completely overcome gravity, but rather experience an environment where the effects of gravity are significantly reduced due to the spacecraft's orbital motion.