In order for a rock thrown up from Earth to not fall back down, it would need to achieve what is called escape velocity. Escape velocity is the minimum speed an object must attain to escape the gravitational pull of a celestial body, such as the Earth.
The escape velocity from Earth's surface is approximately 11.2 kilometers per second (km/s), or about 40,270 kilometers per hour (km/h), or 25,020 miles per hour (mph). This means that for the rock to escape Earth's gravitational field entirely and not fall back down, it would need to be thrown with a speed equal to or greater than the escape velocity of 11.2 km/s.
It's important to note that this value assumes an idealized scenario where there is no air resistance. In reality, air resistance would affect the motion of the rock, causing it to lose speed and eventually fall back to the ground. The actual speed required to overcome air resistance and escape Earth's gravitational pull would depend on various factors, such as the shape and size of the rock, its aerodynamic properties, and the density of the atmosphere.
However, if we neglect air resistance, the escape velocity of 11.2 km/s would be the minimum speed necessary for the rock to leave Earth and not fall back down under the influence of gravity.