Hard time grasping the concept of the Geodesic

[Philosophaie]

Have a hard time grasping the concept of the Geodesic. If you are given a velocity and position of say a small astroid entering the gravitational field of the Earth. How do you find the trajectory or path it takes as it falls to Earth? Not afraid of tensors and advanced calculus.

 

[dx]

You find the trajectory using Newton's law of gravitation.

 

[Philosophaie]

I am trying to do this by sort of projectile motion but at greater distances from the Earth g varies by a factor of:

g = GM/(2r^2) making that calculation difficult. Any suggestions?