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The weightlessness that is experienced in freefall or in orbit around the earth is actually microgravity. These situations create a sense of floating, but gravity is still acting on the person or object, so the term microgravity is used. If you have ever ridden a roller coaster that took you quickly up and down a hill, or that stopped at the very top of a quick rise, you've experienced this phenomenon.
When you fall at the same speed as your surroundings, you experience microgravity, a feeling that you are floating in space. Astronauts orbiting the earth experience long periods of near-weightlessness as they orbit. The astronauts and their space shuttle or space station are both falling at the same speed, so the microgravity makes it seem that they are floating inside it.
A vehicle, like the space shuttle, that is orbiting the earth is actually constantly falling toward it. The space shuttle is moving at a very high speed in just the right direction, so even thought it's falling toward the earth, it keeps "missing." By moving at such a high speed, it essentially falls around the earth over and over. Gravity is still pulling it down, but because the shuttle and the astronauts inside fall at the same speed, the astronauts experience weightlessness, or microgravity.
NASA has a plane known as the "vomit comet" which is used to create this experience without going into orbit. By flying the plane up and down in parabolic arcs, the same shape as the repeated bumps in a roller coaster, the passengers experience short periods of microgravity. The scenes in the movie Apollo 13 that showed the actors experiencing near-weightlessness were filmed in this plane.
You can also experience this phenomenon without riding in a roller coaster or on the "vomit comet". Whenever you jump, you will be in a very short period of microgravity at the peak. It is easier to notice when you jump especially high, using a pogo stick or trampoline. Even though you feel weightless, gravity is still pulling you back toward earth. When you land, remember that it wasn't really weightlessness, but microgravity.