How Do Different Orders of Taylor Expansion Affect Free-Fall Landing Times?

[HomogenousCow]

Hello fellow PF-goers
As part of my high school curriculum I am required to write an extended essay on an academic field of my choice.
I plan on doing a physics paper about the free-fall landing times of objects from high altitudes in a classical Newtonian potential.
My plan is to separate my paper into these parts

1.Analyse free-fall landing times of initially stationary objects, expand the Newtonian force in a power series and analyse the issue in zeroth, first, second and third order. (F=-g+ar+br^2+cr^3+..). I will be using mathematica to solve for the second and third order results.

2.Analyse free-fall landing times for objects with a non-zero initial angular velocity. I will switch to the lagrangian formulation for this and repeat the same power expansion method as before.

3.Tally the data, and compare to "exact" results given by mathematica. Compare the inaccuracies as a I increase the order of the taylor expansion.

Advice?

 

[Dishsoap]

You are learning Lagrangian mechanics and power series expansions in high school? Geez, I feel like a failure.

Your paper sounds excellent.

 

[WannabeNewton]

The only thing I would make sure to add is a physically meaningful expansion parameter in your perturbation theory calculation of the 2nd and 3rd order expressions for the force.

Regardless, your paper idea is very nice so have fun! :)

 

[HomogenousCow]

The only thing I would make sure to add is a physically meaningful expansion parameter in your perturbation theory calculation of the 2nd and 3rd order expressions for the force.

Regardless, your paper idea is very nice so have fun! :)

I'm thinking about expanding around G and working to third order.
I can expand around any parameter I want right? As long as I plug it back in at the end.