Absence of gravity is known as weightlessness. It is like floating, the feeling you get when a roller coaster suddenly goes down. Astronauts on the International Space Station are in free fall all the time.
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Without the earth's gravity to help the sensory systems in our ears and brain, astronauts in space can also feel dizzy. In fact, the sensation feels like constantly falling… like when you ride in an elevator or on a roller coaster.
If you're falling at the same rate as everything around you (that is, with no air resistance), you feel like you're floating. If you wanted to reach a point where Earth's gravity no longer has a hold on you, you'd have to fly out about 21 million kilometers, or 13 million miles.
The injuries come from hitting heads on the ceiling or being thrown around in the aisle like a rag doll. If you're seated with your belt on, it's like a roller coaster ride and nothing worse. Remember that I said it's rare, not unheard of, for turbulence to bring airplanes down.