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Paths of projectiles with different launch speeds

The path of a projectile or a satellite moving in the proximity of a body (e.g. the Earth) depends on the body's speed relative to the orbital speed $$v_{\text{o}}$$ and the escape speed $$v_{\text{e}}$$.

Note that the direction of velocity of the body is variable. The following equations only give the magnitude of velocity (i.e. speed).

$$$v_{\text{o}}=\sqrt{GM/r_{\text{o}}} \quad \quad v_{\text{e}}=\sqrt{2GM/r_{\ell}}$$$

Possible trajectories are:

$$v\lt v_{\text{o}}$$: the object will fall to the primary (e.g. the Earth).

$$v=v_{\text{o}}$$: the object will stay on a circular orbit around the primary.

$$v_{\text{o}}\lt v\lt v_{\text{e}}$$: the object will stay on an elliptical orbit.

$$v\ge v_{\text{e}}$$: the object will leave the gravitational field of the primary.

$$G=$$universal gravitational constant, $$M=$$mass of the primary (e.g the Earth), $$m=$$mass of the satellite (too small to affect primary), $$r_{\text{o}}=$$orbital radius of a satellite, $$r_{\ell}=$$distance between centre of the Earth and launch position (launch radius).

Speed is lower than orbital speed: A (black), Speed is equal to orbital speed: B (red), Speed is greater than orbital speed but smaller than escape speed: C and D (blue), and speed is equal to or greater than escap speed: E (green).