Throw a ball and it comes back. Throw it faster and it circles the world. Throw it faster still — past a precise speed set only by the planet’s mass and your distance from its centre — and it never comes back at all. That speed is the escape velocity, and it’s exactly √2 times orbital speed. Slide the world and find its edge.
Escape velocity is v = √(2GM/r): the speed at which an object’s kinetic energy exactly equals the gravitational potential well GM/r it must climb out of, so it just barely reaches infinity with nothing to spare. Orbital (circular) speed is √(GM/r), so escape is always exactly √2 × orbital — independent of the object’s own mass. A fail-loud self-check throws unless Earth’s escape speed comes out ~11.2 km/s AND the escape/orbital ratio equals √2 for every world.
Ignores atmospheric drag, the launch height above the surface, and the target’s own motion (real launches need less, thanks to Earth’s spin, or more, fighting air). The √(2GM/r) formula and the √2 ratio are exact.