Video narration
How fast must you throw a ball so gravity never brings it back? Throw harder, and it rises higher before slowing. Now imagine the limiting throw: it reaches infinitely far away with zero speed left. That's the key turn. At launch, its kinetic energy must exactly pay for the gravitational potential energy needed to reach infinity. With potential energy zero there, write: one-half m v_e2 equals G M m over R. Cancel m: one-half v_e2 equals G M over R. Multiply by two and take the square root: v_e equals the square root of two G M over R. The object's mass disappears, so escape velocity depends on the body and distance R from its center, not the object. The launch speed is enough; no continuous thrust is needed. So how fast? At v_e, the ball reaches arbitrarily far away with zero speed.