To shoot a rocket straight up and remain 'static' while the earth turns below would require constant thrust by the rocket to avoid being drawn back to earth via gravity. This is why satellites and space stations orbit earth - they only need to use enough fuel to get to a speed at the balancing point between leaving orbit (too fast) and being pulled in by earth's gravity (too slow). Eventually, all satellites will slow down enough to renter the atmosphere.
There are numerous companies and vast numbers of employees in the private world who make their living putting satellites into orbit. I'm curious how flat-earthers explain GPS and other communication satellites?