Escape Velocity — An escape velocity is the minimum speed at which an object without propulsion can move away from a source of a gravitational field indefinitely if there is no friction.
This definition may need modification for the practical problem of two or more sources in some cases. In any case, the object is assumed to be a point with a mass that is negligible compared with that of the source of the field, usually an excellent approximation. It is commonly described as the speed needed to break free from a gravitational field, but one should keep in mind that gravitational fields are infinite in extent (in theory).
One somewhat counterintuitive feature of escape velocity is that it is independent of direction, so that “velocity” is a misnomer; it is a scalar quantity and would more accurately be called “escape speed”.
One complication is that virtually all astronomical objects rotate. The frame of reference must not rotate for that statement to be correct. Moreover, the gravitational slingshot effect sometimes involves transfer of energy to the projectile from the slingshot body that depends on the spatial relationship between projectile and body.
The body loses some angular momentum — possibly rotational as well as orbital — adding kinetic energy to the projectile. Therefore, the complete gravitational field of the slingshot body must be included in the overall field, which then can no longer be approximatively treated as symmetric.
Moreover, not only do escape velocities vary from place to place, they vary with time in such cases. Even moreso, they may sometimes depend on direction. The simplest way of deriving the formula for escape velocity is to use conservation of energy. (For reasons given above, computers may often have to be used to compute solar-system escape velocities to some desired precision.)
Defined a bit more formally “escape velocity” is the initial speed required to go from an initial point in a gravitational potential field to infinity with a residual velocity of zero, relative to the field.
In common usage, the initial point is a point on the surface of a planet or moon. It is a theoretical quantity, because it assumes that an object is fired into space like a bullet.
Instead propulsion is almost always used to get into “space”. It is usually in “space” that the idea gets a more concrete meaning. On the surface of Earth the escape velocity is about 11 kilometres per second. However, at 9000 km from the surface in “space,” it is slightly less than 7.1 km/s.
Continual acceleration from the surface to attain that speed at that height is possible. At no time would the “escape velocity” of 11 km/s be attained; yet at that height, even with zero propulsion now, the object can move away from Earth indefinitely.