The path that a satellite follows around the larger body is called its orbit. In order to get into an orbit, a satellite must move very fast. If you shot it even faster, it would travel perhaps ten miles before it hit the ground. It would be falling all right, but when it got back to you, it would still be at almost the same height. By the way, when any object is falling, it is weightless.
Objects only weigh something when there is a resistance to their falling.
There are many kinds of orbits for our manmade satellites. The Shuttle travels at about 17,500 miles an hour and goes around the earth every 88 minutes or so. It is not quite high enough to stay in orbit more than a few weeks, because there is still enough air up there to create drag. This drag is like a brake that gradually slows down the Shuttle each orbit. As it slows down, it slowly spirals toward the earth.