I'm trying to show that any tunnel through the Earth (not necessarily through the center) will have a free-fall time that is the same. I heard this was true somewhere.
acceleration of free-fall = GM / r^2 where r is changing
I believe this involves trig and calculus.
The Attempt at a Solution
I attempted a solution (attached-- please bear with me. I'm sorry if it's not very clear or if my process is weird).
I considered the acceleration of free-fall for a particle at an arbitrary position in a tunnel x that makes an arbitrary angle theta with the radius of the Earth. I eventually solve for the acceleration as a (complicated) function of x and theta.
Not sure what to do after that... If it involves integration, I feel like it's hopelessly complicated.
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