Rockets never actually need to reach escape velocity to leave Earth
Escape velocity from Earth's surface is about 11.2 kilometres per second, but that figure only applies to an unpowered object thrown upward once. A rocket firing its engines, or a climber on a hypothetical space elevator, could leave at walking pace. What never changes is the minimum energy the trip demands.
Escape velocity is the lowest speed at which a coasting object, pushed by nothing else and ignoring other bodies' gravity, will never fall back. Strictly it is a speed, since direction does not matter. It follows from conservation of energy: gravity's pull creates a well of finite depth, and once an object's kinetic energy matches its negative potential energy, it can reach arbitrarily far away while slowing endlessly toward zero. The formula is the square root of twice the gravitational constant times the body's mass, divided by distance from its centre.
Distance matters. The needed speed falls with altitude, so at a 200 km low Earth orbit it is about 11.0 km/s. Since a craft there already travels at roughly 7.8 km/s, the extra push is modest, which is why missions often park in orbit first. Launching straight out at 11.2 km/s through thick air would be impractical anyway: the heating and drag at those hypersonic speeds would destroy most objects. Escape speed is always about 1.41 times the circular orbital speed at the same height, earning the names first and second cosmic velocities.
Earth's spin helps too. At the equator the surface moves at 465 m/s, so a rocket fired eastward needs about 10.735 km/s relative to the ground, while one fired westward needs roughly 11.665 km/s. That bonus shrinks with latitude, which is one reason launch sites like Cape Canaveral, at 28 degrees north, and the French Guiana Space Centre, at about 5 degrees north, sit as close to the equator as practical.
Orbit shapes reveal whether something is bound. Circular and elliptical paths always stay below escape speed; a parabolic path sits exactly at it; a hyperbolic path exceeds it, leaving with a leftover speed at infinity. Spacecraft on elliptical orbits get the cheapest escape by burning at closest approach, the Oberth effect, and gravity assists from planets can lend extra energy. The same formula emerges from general relativity using the Schwarzschild radial coordinate.
Source: Escape velocity
