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Below are all of the formulas that are utilized to model the motion of our capsule as it falls through the Earth. Explanations for these formulas are included nearby.

g=\frac{G}{R_{Earth}}\times r

G = Universal Gravitational Constant ---- R(Earth)=6378 km ---- r=distance from capsule to center of Earth

This equation is derived in the video by rewriting the univeral law of gravity equation with density and volume taking the place of the mass of Earth, seeing as how the total mass that pulls you towards the center decreases as you fall, if you realize that all of the mass that is farther from the center than the capsule is in effect canceled because its net pull on the capsule is 0. If this is not intuitive, remember the Faraday Cage. Despite the charge that exists on the metal surface, the net voltage in the center of the sphere, no matter of where inside the sphere, is 0. A detailed explanation of this phenomenon is found in the source included -


g=\frac{4\pi G\times r\times\rho}{3}

g=Net acceleration due to gravity ---- G=Universal Gravitational Constant 

r=Distance between capsule and center of Earth ---- p=Average density of Earth beneath capsule/still considered to be pulling on capsule

This more accurate model for the acceleration of gravity is needed when we remove the uniform density of Earth. Otherwise, it is the same as the equation above. The actual equation changes between the interface of the Outer Core and the Mantle to a new linear approximation for the density, but the general equation was used throughout to account for the changing density of Earth. If you would like more information, feel free to explore the 1st included Spreadsheet which explores how this changing density of Earth changes the motion of the capsule.



p(air)=density of air ---- C(D)=Drag Coefficient (Shape dependent) ---- A=Cross Sectional Area

v=Velocity of capsule

This is a general formula for the force of air resistance that a moving body experiences. Many of these attributes of objects intuitively seem like they would affect air resistance - a denser medium or larger body obviously would have a greater resistance to motion in a fluid. The fact that the velocity term is squared is also important, as it explains why as things speed up, air resistance does not increase proportionally, as the air resistance when driving at 25 mph is not 1/4 of that produced when driving at 100mph. This equation is explained in the page below, and is showcased in the 2nd and 3rd included Spreadsheets.


P_{air}=P_0\times e^{-\frac{mgh}{kT}}\:----P_{\left(t\right)}=P_{\left(t-1\right)}\times e^{-\frac{mgh}{kT}}

This is the most complicated formula, so don't worry, its over now. This equation models the changing pressure/density of the air as we progress down the tube, as the air at the center has to support all the air above it, and therefore will be more compressed. Note the exponential relationship, which will prove extremely important when we start using it in our model. Now, I did not want to model the changing temperature as we move down, partially because I think we would have to artificially cool the tunnel walls so they wouldn't vaporize and cave in (not joking), and partially because it was really hard. So if we did allow the air to heat up, the density/pressure would increase slower than our model predicts. Anyway, the first equation is the general equation, but I noticed it had a flaw. When it predicts the pressure near the center, because g is so small, the pressure/density of air at the center would be predicted to be a near vacuum. Which is wrong. Thus, I tried to invent a work-around which is the second equation. It only makes sense in terms of an iteritive process, such as the 4th spreadsheet. In this way, when we go down, the pressures keep increasing, but they never can decrease, just level off. If someone finds a flaw with this model, please let me know, because I'm pretty sure this is wrong in some way, but I don't know how.

So again, if you want to see what the effects of this are, please explore either the source included below and/or the 4th Spreadsheet.

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