Speed of earth compared to rockets

as the Earth's rotational velocity is 465 m/s at the equator, a rocket launched tangentially from the Earth's equator to the east requires an initial velocity of about 10.735 km/s relative to Earth.

Our artists concept

A rocket continuously accelerated by its exhaust, need not reach ballistic escape velocity at any distance since it is supplied with additional kinetic energy by the expulsion of its reaction mass. It can achieve escape at any speed, given a suitable mode of propulsion and sufficient propellant to provide the accelerating force on the object to escape.

The maximum speed of any rocket so far is the Parker solar probe which reached a maximum velocity is 430,000 mph or ~192.2 km/s relative to the Earth.

In the case the Earth is the winner 🏆.