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Ok. I'm not sure if this belongs in this section. This is a problem relying on physics 1 concepts, but I do believe it requires MultiV Calc (which I haven't taken) to solve.
1. The problem
How long would it take a particle of distance D away from the center of the Earth to fall to the surface of the Earth (from rest)? This is of course assuming that the only force acting upon the object is the Earth's gravitational force.
Fg = GmM/d^2 (M is mass of Earth and m is mass of object. G is universal gravitational constant and d is the distance from the center of the earth.
This was way beyond my current skill level, but I tried it anyway.
from the equation for Fg, I derived acceleration as a function of distance from the earth, so a(d) = GM/d^2.
but to make this in terms of x, starting from position "zero": a(x) = GM/(D-x)^2
Then came my attempt to incorporate the time:
a = dv/dt; v = dx/dt → dx/v = dt → a = v*dv/dx = GM/(D-x)^2 → ∫vdv = ∫GM/(D-x)^2dx
v^2/2 = GM/(D-x) + C , but v initial is zero so C = 0. → v = √(2GM/(D-x))
v = dx/dt = √(2GM)/√(D-x) → √(D-x)dx = √(2GM)dt → -2/3(D-x)^(3/2) + C = √(2GM)t
Since we know at t = 0, x = 0, C = 2/3D^(3/2)
so → -2/3(D-x)^(3/2) + 2/3(D)^(3/2) = √(2GM)t
t = 2/3/√(2GM)(D^(3/2)-(D-x)^(3/2))
Conclusion:
As I was typing this up, I started to see things and made some educated guesses and progressed along. But being a novice at calculus, I don't know if the process I took was acceptable.
I would truly appreciate any comments, corrections, or hints as to what direction I should be taking. (FYI: just to plug in some values, using the avg distance from the center of the moon to the center of Earth, it came out to take a little over 49 hrs for the moon to first touch the Earth surface to surface.)
1. The problem
How long would it take a particle of distance D away from the center of the Earth to fall to the surface of the Earth (from rest)? This is of course assuming that the only force acting upon the object is the Earth's gravitational force.
Homework Equations
Fg = GmM/d^2 (M is mass of Earth and m is mass of object. G is universal gravitational constant and d is the distance from the center of the earth.
The Attempt at a Solution
This was way beyond my current skill level, but I tried it anyway.
from the equation for Fg, I derived acceleration as a function of distance from the earth, so a(d) = GM/d^2.
but to make this in terms of x, starting from position "zero": a(x) = GM/(D-x)^2
Then came my attempt to incorporate the time:
a = dv/dt; v = dx/dt → dx/v = dt → a = v*dv/dx = GM/(D-x)^2 → ∫vdv = ∫GM/(D-x)^2dx
v^2/2 = GM/(D-x) + C , but v initial is zero so C = 0. → v = √(2GM/(D-x))
v = dx/dt = √(2GM)/√(D-x) → √(D-x)dx = √(2GM)dt → -2/3(D-x)^(3/2) + C = √(2GM)t
Since we know at t = 0, x = 0, C = 2/3D^(3/2)
so → -2/3(D-x)^(3/2) + 2/3(D)^(3/2) = √(2GM)t
t = 2/3/√(2GM)(D^(3/2)-(D-x)^(3/2))
Conclusion:
As I was typing this up, I started to see things and made some educated guesses and progressed along. But being a novice at calculus, I don't know if the process I took was acceptable.
I would truly appreciate any comments, corrections, or hints as to what direction I should be taking. (FYI: just to plug in some values, using the avg distance from the center of the moon to the center of Earth, it came out to take a little over 49 hrs for the moon to first touch the Earth surface to surface.)
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