How do communication satellites stay in one spot in the sky?

They're actually falling - just fast enough to keep missing the Earth.

The answer

Geostationary satellites orbit at exactly 35,786 kilometers above the equator, where their orbital period matches Earth's 24-hour rotation - making them appear stationary from the ground. This principle, first proposed by Arthur C. Clarke in 1945, is why satellite dishes can be fixed in one direction. Lower-orbit satellite constellations like Starlink (at 550 km) move across the sky rapidly but use thousands of satellites in coordinated orbits so that several are always overhead, handing off connections seamlessly.

Source: NASA

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