I have learnt that when a body is projected with a velocity greater than escape velocity it escapes along a hyperbolic path, if it is equal to escape velocity a parabolic path. If the velocity is betweeb escape and critical velocity than it orbits in a elliptical path. I want to know why is it the case and is there a way to prove it say mathematically? If yes can someone tell me what do to do prove it?
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To the person who gave the link of derivations thank you! The derivation with cartesian coordinates helped a lot – Astrobuuz Dec 27 '23 at 13:55