Suppose that an asteroid) that moves at a velocity $v$ passes near to a planet: now I know that the gravity force of the planet will curve the trajectory of the asteroid so I tried to make a draw of the asteroid while passing near the planet using vectors but as result the planet ALWAYS falls into the planet that is obviously wrong because the asteroid in real life can also pass the asteroid. Can someone please help me and maybe show some math to calculate the orbits?
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The situation can be approximated by a restricted two body problem, namely that the mass of the Earth is much larger and therefore its velocity will basically not change.
The result from this approximation would be a Kepler orbit with an eccentricity larger than one (also known as a hyperbolic orbit). Therefore its trajectory could look like the blue curve in this image:
