Escape Velocity – Calculating for Satellites in Newtonian Mechanics

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I know that the equation for it is $$v^2 = \frac{2GM}{r},$$ and with that, the rocket should be launched at that speed. But could it go much slower spending much more fuel to escape from gravity right?

Wouldn't it be easier to calculate it with energy? 

Best Answer

You've made the common mistake of thinking that the velocity needed to launch a satellite is the (initial) velocity needed to raise it to its orbital radius.

If you raise a satellite to e.g. 300km then let go the satellite will immediately fall straight back to Earth. You need to do two things:

  1. raise the satellite to 300km

  2. increase its tangential velocity to $\sqrt{GM/r}$

The rocket used to launch a satellite doesn't travel straight up. It travels in a curve that looks something like:

Satellite launch

(Image from nasaspaceflight.com)

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