The astronaut has the same speed v as the space station
According to Newton’s second law, we have:
Since the force we are considering is a centripetal force, we can write Newton’s 2nd Law as follows:
Substituting the force F for the gravitational force in the above equation, and further simplifying the smaller masses, we have:
Looking at the equation above, to calculate the velocity, we realize that the velocity depends on the mass of the body that is orbiting, that is, it only depends on the mass of the central body ( M ) and the radius of the orbit ( R ). Therefore, we can say that any body that rotates in the orbit of radius R will have the same speed v, since its mass is much smaller than the mass M of the central body.
This is what gives the sensation of weightlessness to an astronaut who finds himself inside a spaceship. It doesn’t matter whether the astronaut is inside or outside the spacecraft: his speed will be the same. That way, he has the sensation of floating. We call this effect weightlessness .