escape velocity
Throw a ball upward and it falls back. Throw it harder and it goes higher before returning. Is there a speed so great that it never comes back at all? Yes — that threshold is the escape velocity, the smallest launch speed that lets an object coast away from a body's gravity forever, slowing the whole way but never quite stopping.
Escape velocity depends only on the mass M and radius R of the body you are leaving, not on the mass of the thing escaping — a pebble and a rocket need the same speed. The formula is v_esc = sqrt(2 G M / R). For Earth it is about 11.2 km/s (around 40,000 km/h); for the Moon, with far less mass, only about 2.4 km/s; for the Sun's surface, about 618 km/s. Notice it is just sqrt(2) ≈ 1.41 times the circular orbital speed at the same radius: a little extra speed turns 'falling around forever' into 'leaving for good'.
This single idea threads through astrophysics. It sets how big a rocket must be, why small low-gravity worlds like the Moon and Mercury cannot hold onto an atmosphere (their gas molecules move faster than escape speed and leak away), and it foreshadows black holes — push the escape velocity up to the speed of light and not even light can leave, which is the Newtonian whisper of an event horizon.
Jupiter's escape velocity is about 60 km/s, more than five times Earth's, which is why Jupiter has clung to a thick hydrogen and helium atmosphere since its birth while tiny low-gravity worlds long ago lost theirs.
Escape velocity decides which worlds keep an atmosphere and which leak it into space.
'Velocity' here is loose: escape speed is the energy threshold for a coasting object with no further push, so a rocket under continuous thrust can leave far more slowly. Escape speed also ignores air drag and assumes you are heading away, not into the ground.