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Alice in the tunnel.

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#21 Raider5678



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Posted 17 May 2017 - 09:16 PM

So in order to find the correct amount of time, you'd have to integrate g' twice for dt?

By figuring out the height, you can figure out the constant acceleration.

Except then the depth of the earth comes into play.........

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"My good sir,

I feel the need to inform you that your argument has reached rock bottom and has proceeded to dig.

 Good day."

#22 studiot



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Posted 17 May 2017 - 09:35 PM

No integration is necessary.


Perhaps this definition will help as a hint



When particle moves in a straight line in such a way that its acceleration is always directed towards a fixed point of the line and is proportional to its distance from that point, the particle is said to move with simple harmonic motion

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#23 Function



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Posted 18 May 2017 - 04:20 AM

Will it act like a pendulum?

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So I guess we are who we are for a lot of reasons. And maybe we'll never know most of them.

But even if we don't have the power to choose where we come from, we can still choose where we go from there.

We can still do things. And we can try to feel okay about them.


– Steven Chobsky in The Perks of Being a Wallflower

#24 studiot



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Posted 18 May 2017 - 12:24 PM

Will it act like a pendulum?


Almost let's see




Let P be any point on the straight tunnel AB and O be the right bisector of AB, through Earth centre C.


Then the geometrical relationships are as shown in the first sketch.


The acceleration due to gravity at P is g' and acts towards C as shown in the second sketch.


This therefore has a component acting along APOB, given by the triangle as also shown in the second sketch.


Substitution shows that this acceleration component is proportional to the distance of P along AB from O and by definition must be directed towards O.


This is the definition of simple harmonic motion.


So the ball executes SHM between A and B with a period


2\pi \sqrt {\frac{a}{g}}


Travel time from A to B is half this or


\pi \sqrt {\frac{{4000*5280}}{{32}}}  = 100\sqrt {66} {\rm{seconds}}


or about 42.5 minutes.


Note that this is like the pendulum, independent of the Amplitude, (ie the distance along y) but depends upon the movement arm.


Perhaps you can also see why the use of a, the distance to the Earth's centre is better than the depth, d.


That is the High School derivation, if you really want to do a double integration then the equation of motion is


\frac{{{d^2}y}}{{d{t^2}}} + \frac{g}{a}y = 0

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#25 Zolobrine



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Posted 22 May 2017 - 02:03 PM

I don't know about the answer but that tunnel will defenatelly solve the global CO2 problem xD since it's heavier then air it should theoretically collect in the tunnel. However due to this we might even experience a huge pressure drop xD Anyway Ill stop overthinking cuz itll lead to bad things
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