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I still think that the launch velocity should be dependent on the
launching point and direction, and it should be slightly different from
11.2 km/s.
To see this clearly, let us whirl earth-on-a-string so fast
that its centripetal acceleration is equal to 9.8 m/s^2. Then the force
from the ground on a rocket standing on the far side of the earth is
zero, and the rocket is on the verge of being thrown off. Its escape
velocity is zero. At the point where the string is attached, the escape
velocity will be much larger then 11.2 km/s.