How fast does a rocket have to travel to leave the atmosphere?

The orbit insertion velocity near the Earth's surface is practically the same as the Earth escape velocity of 11.2 kilometers per second, or 25,805 miles per hour.

Even at an altitude of 200 kilometers, a rocket is still inside the outer reaches of the Earth's atmosphere. To really leave the atmosphere it probably has to get to a distance of 27,000 miles near the 'geosynchronous' limit. There is no sharp boundary to Earth's atmosphere. It just decreases steadily in density until it eventually matches the density of the interplanetary medium.

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