Becoming Weightless
There are a couple of different ways to create weightlessness on Earth. Probably the easiest way would be to get on an elevator, ride to the top floor and cut the cable!
If you were to do that -- you would also have to jam the safety brakes, what would happen is that both you and the elevator car would start falling. Gravity is affecting both you and the car in exactly the same way, so you and the car would be falling at exactly the same speed. This would give you the feeling that you are "weightless." You would be floating in mid-air in the car because you and the car are falling at the same speed. This would work fine until the elevator car hit the bottom of the shaft. Then it would get a little messy when you crashed into the elevator floor at 100 mph or more.
By the way, this is exactly why the astronauts feel like they are weightless while they are orbiting Earth. They are in the spacecraft, and the spacecraft is falling toward Earth. The astronauts inside the spacecraft are falling too, so they feel weightless. The only difference is that the spacecraft does not crash into the ground. A spacecraft in orbit takes advantage of the fact that the Earth is a sphere. The spacecraft falls, but the spherical shape of the earth causes the spacecraft to miss the Earth. When in orbit, it's as though the spacecraft is falling forever.

Photo courtesy NASA
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Using an airplane to create weightlessness is somewhere in the middle between the elevator and the spacecraft. The airplane does not crash into the ground like an elevator, meaning that you can experience weightlessness over-and-over again. But, it cannot "fall forever" like a spacecraft.
Here's what actually happens. The airplane climbs to 32,000 feet. Then it falls and gives you about 25 seconds of weightlessness. Then it levels off and climbs back up to a higher altitude. That takes 40 seconds or so. Then it falls again for 25 seconds. On a typical flight, you get 20 or 25 minutes of weightlessness, but it is chopped up into little 25-second segments.
If you are careful and you do a lot of planning, you can get a lot done in these little 25-second segments. When NASA flies students on the Reduced Gravity Research Program flights, they are performing experiments especially designed to test out different things in weightlessness.
If you have ever seen the movie "Apollo 13," you have watched another success for these weightless flights. There are several scenes where the actors appear to be in weightlessness. The movie studio built a complete model of the inside of the Apollo 13 spacecraft. They put that model inside the Reduced Gravity Research Program airplane, and then they filmed the weightless scenes 25 seconds at a time.
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