Hard time grasping the concept of the Geodesic

AI Thread Summary
Understanding the trajectory of an asteroid entering Earth's gravitational field involves applying Newton's law of gravitation. The challenge arises due to the variation of gravitational acceleration (g) with distance, as it is expressed by the formula g = GM/(2r^2). This complicates calculations, especially when attempting to model the motion as projectile motion at significant distances. Advanced calculus and tensor analysis can aid in accurately determining the path. Seeking guidance on these calculations can help clarify the concept of geodesics in this context.
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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.
 
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You find the trajectory using Newton's law of gravitation.
 


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?
 
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