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John CarneyFlag for United States of America

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How does a object's velocity get transferred to the horizontal in a satellite launch?

I may not have this right but my understanding is that an object's velocity must be tangential, not perpendicular, to the earth's surface in order to achieve orbit. If that's correct then how is its direction shifted 90 degrees when it reaches its desired distance from the earth?

I'm assuming that this is relatively easy to do with a man-made powered object. Someone in Houston presses a button and before you know it you and I have satellite tv and radio. But what factors could fortuitously come together to cause the presumably quadrillions of orbiting objects in the universe to achieve perfectly "horizontal" trajectories around their host objects and thus achieve stable orbit?

Thanks,
John
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Great answers, thank you all! This is like a mini course on orbital mechanics. :- )