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At what altitude does weightlessness occur?
Short answer
Weightlessness isn't tied to an altitude but to a cause: free fall. In a parabolic flight it appears just a few kilometres up, as soon as the plane only follows gravity. The edge of space is set at the Kármán line, 100 km up, though Earth's pull there is still almost full strength.

A matter of motion, not altitude
A body is weightless as soon as no force acts on it except gravity: no floor, no seat, no thrust. This happens in an elevator with a severed cable, in a parabolic flight just a few kilometres up, and also in orbit. By contrast, a rocket climbing with its engines firing still feels strong acceleration even at great altitude. The difference is an everyday one: when an elevator starts moving we feel momentarily heavier, and when it brakes, lighter. Weightlessness is the limiting case of that effect.
Where space begins
The most common boundary is the Kármán line, at 100 kilometres altitude, which the FAI aeronautical federation considers the threshold of space. It is a convention, not a physical leap: the atmosphere thins out gradually with height without ending at any specific point. The United States traditionally uses a somewhat lower threshold to award astronaut wings, so two definitions coexist. Gravity does not end at either mark; it decreases with the square of the distance from Earth's centre, and a hundred kilometres is small compared with Earth's radius.
That's why ISS crews float
At just over 400 kilometres altitude, Earth pulls on the space station with almost the same force as on an object on the ground. The station is permanently falling, but it moves sideways so fast that its fall carries it around the Earth instead of into it. The length of an orbit, however, does depend on altitude: just above the atmosphere, residual air slows things down so much that an orbit decays within days or weeks. A suborbital flight produces the same weightlessness for a few minutes, because the capsule is also in free fall after engine cutoff, only on a trajectory that brings it back to Earth.
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