Why does a rocket have to go up so fast?

Answer

A rocket must not only get “above”, but above all must also get into orbit around the earth. A minimum horizontal speed of about 8 km/s is required for this. This is about 28,000 km/h. In an orbit around the earth, there is a balance between gravity, which wants to pull the rocket towards the earth, and centrifugal force, which wants to push the earth outwards. You always have gravity in the vicinity of the earth, but you only have centrifugal force when it rotates with a certain speed. Only with a horizontal speed of 8 km/s is the centrifugal force strong enough to maintain a low Earth orbit.

A rocket must therefore not only get out of the atmosphere, but above all must also build up horizontal speed. That’s why a rocket starts to fly more and more slanted as you see it on TV images. She has to build up that enormous speed of 8 km/s.

In summary : The speed is not so much needed to get “above” as in your reasoning, but to reach that minimum speed once you are above.

Nice to mention: to leave an orbit around the earth, for example to travel to the moon, the orbital speed must be increased even further to 11.2 km/s. The space shuttle, for example, does not have that capacity. So you can’t fly to the moon with a shuttle. At most you can reach an orbit of about 500 altitude. An Apollo also did not have that capacity. Therefore, the 3rd stage of the giant Saturn V rocket was not disconnected when the Apollo entered parking orbit around the Earth. They had to light it a second time to fly from the earth’s orbit towards the moon. Only after they were well on their way to the moon was the LEM pulled from the top of the 3rd stage and the 3rd stage itself ejected. So they sent him to the moon to crash there.