An energy threshold with defined assumptions
For a small object outside a spherical central mass, the Newtonian threshold is vₑ=√(2GM/r). Here M is the central mass, G the gravitational constant and r the distance from its center; only a surface departure sets r equal to the body’s radius.
With drag, subsequent propulsion and collisions excluded, the initial kinetic energy just supplies the potential-energy increase to infinity. The remaining speed tends to zero at increasing distance. This is an initial energy threshold, not a constant speed maintained throughout flight.
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Mass and size both matter
Within the same ideal model, larger M or smaller r increases the threshold. At four times the original distance, the formula gives half the escape speed, although other bodies can matter in a real system.
NASA’s fact sheet lists roughly 11.2 kilometers per second for Earth and 2.4 for the Moon. Such values characterize the respective surface gravitational environments rather than marking a height where gravity abruptly disappears.
Orbiting and continuous propulsion are different cases
An orbiting body remains accelerated by gravity but has enough sideways motion to keep missing the surface. It can remain bound without reaching the escape threshold, so being in space does not prove permanent departure from Earth.
Continuing propulsion adds energy afterward and removes the requirement to supply this unpowered threshold all at one initial instant. Real launches also involve atmosphere, rotation, direction and losses; the surface value is not every mission’s engine velocity budget.