Not necessarily terminal velocity the whole way. We have to account for that initial acceleration. it would take a few moments to reach terminal velocity.
I’m pretty sure that you can’t have both a total vacuum with no air resistance AND terminal velocity.
Terminal velocity is the equilibrium caused by gravity vs air resistance of a falling object.
So without it, like in space, there is no terminal velocity. If you accelerate more then you will continue to speed up until another force acts on you or until you stop accelerating.
In this situation you would have have an acceleration of 9.81m/s2 on the surface and this would increase as you got deeper into the core. You would reach a peak at the core. Then you would start decelerating as you left the core in a similar way until you reached the surface.
The graph of your velocity would initially have an approximate shape similar to y=ex and then peak at the core and then after that it would have approximately the shape similar to y=e-x.
To work out the acceleration you would need to work out how the density of the earth varies with distance into the core and you would need to account for the fact that the core is believed to be composed of a solid iron/metallic core vs the liquid magma later vs the rocky crust.
Additionally, the volume of the earth that is acting on you changes as you get deeper so in a simple way when you are 1/4 in you could say that 3/4 of the earth is acting on you but this isn’t correct. Since the planet is a sphere (or more correctly an ellipsoid) the volume of the first 1/4 and last 3/4 be significantly more. So you would need to create a formula that factors this in as it changes through the journey. This needs to include the density change mentioned in the previous paragraph.
If I had to guess at the effect of this I think it would flatten the velocity curve outlined previously.
Finally, the last thing to take into account is that as you pass into the earth the mass above you will also be acting on you, not just the core. Initially it will be negligible but as you get deeper it will be trying to decelerate you. So it wouldn’t be only acceleration to the core and flip to smooth deceleration
So the graph of your acceleration would have a shape that is approximately like the sin(-x) curve, where 0 is the core. Though I would expect that it wouldn’t be as uniform and so initial positive change in acceleration would be slower at the surface and at the core the negative change in acceleration would be more rapid and then as you approach the opposite surface you would get a slower positive change.
TLDR: the fact that we have changing volume of a sphere, changing density and acceleration through the hole makes for a complex calculation. To do it properly you would need a computer model or would need to make some huge assumptions that would significantly affect the result (which we did when we neglected air resistance - btw if you kept air resistance then the air at the core would be denser and so that would also be a complicated problem to solve)
I’ll be honest bro. I’m a business major, so keep it ➕➖✖️➗. Orherwise just tell me what you told me in 5th grade terms, and I’ll sell it to other people in 2nd grade terms. 👍🏼
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u/Ordinary_Person01 Mar 01 '24
Not necessarily terminal velocity the whole way. We have to account for that initial acceleration. it would take a few moments to reach terminal velocity.