Level 1: Fired Sideways at Escape Speed: Does It Still Escape?
From orbit, how much faster to escape?
A satellite orbits just above Earth's surface. By what percentage must its speed be increased so that it escapes from Earth?
In a low orbit, gravity supplies the centripetal force, which gives the orbital speed √(GM/R). The escape speed from the same height is √(2GM/R).
The satellite needs about 41% more speed, not twice as much. In energy terms it already has half the kinetic energy it needs, so doubling its kinetic energy is enough, and that takes only √2 times the speed.
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