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To Everywhere in 42 Minutes

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Re: To Everywhere in 42 Minutes

#21

Post by ph0rque »

patrickg-zill wrote: post #19

There was work done on similar things to this in the 1970s, however it all went classified. The term is "subterrene" - a tunnel boring machine that keeps the drill tip at high temperature, melting the rock and allowing a smooth glassy tunnel to be made.

Fascinating... wikipedia mentions using nuclear power to achieve the 1300-1700C temperature needed for the rock melting. (BTW, the smooth glassy tunnel is a byproduct of that). I wonder if you could achieve the same thing using plasma arcs (http://en.wikipedia.org/wiki/Plasma_Converter), then somehow use the pressure and heat of the earth, once you're deep enough?

Re: To Everywhere in 42 Minutes

#22

Post by reconbot »

ars wrote: post #12
anigbrowl wrote: post #5

It sounds like great fun until your rapid transit pod plunges into a bath of superheated magma. Vomitous fellow passengers would not be my primary anxiety about this mode of travel.

Superheated magma is not really much of a problem. If you don't go too deep (a few thousand KM), the temperature is under 1000 degrees. We have plenty of materials than can handle that. For insulation use vacuum, or aerogel (which melts at 1,473 K) and is a phenomenal insulator. Add a large cold reservoir (liquid nitrogen) and you don't need a conventional A/C - it only has to last 42 minutes, and weight is not a pro…

The coolant would only need to last 42 minutes if you thought it could never get stuck.

Re: To Everywhere in 42 Minutes

#23

Post by bocajuniors »

dantheman wrote: post #17
gtufano wrote: post #13

Earlier quoted context omitted.

I find interesting that you assume that disparity within nations is caused by differences in intelligence. While this is true (i.e.: differences in intelligence make a difference, sorry for the joke) there are many, many, other causes for differences: geographics (easy), social (not all the countries are permeable to social status changes) and so on... I feel this "all is caused by the intelligence" a bit ingenuous.…

You misread his statement: > The overall result would be to flatten the income disparity about nations. (But it would have no effect on the disparity within nations, since that is caused by differences in intelligence, and that won't change.) He's saying that local effects of a nation would quickly be brought up to par, whereas the distribution in nations wouldn't likely change.

I think the misreading was you misreading gtufano.He disputed the unsupported statement that income differences within a nation are mainly because of intelligence differences

Re: To Everywhere in 42 Minutes

#24

Post by eru »

scorxn wrote: post #3
te_platt wrote: post #2

Fun idea, but didn't everyone here have to work this out in their college physics class?

Since Friday, Feb. 11, 1966 maybe ;)

I just wrote a simple python script to test whether this holds only when all of earth's mass is concentrated at its center, or in the slightly more realistic case of constant density.

My numerical experiments point to the latter.

Does anyone care to derive an analytic solution?

Re: To Everywhere in 42 Minutes

#25

Post by madcaptenor »

pmjordan wrote: post #20
ramchip wrote: post #8

Earlier quoted context omitted.

If you read the article carefully, he also talks about shorter tunnels which would only go a few miles below the surface. Not necessarily through the core, although I believe the journalist got a little confused himself with the Moscow-Washington bit (716 miles depth != Earth core). Anyway, it's nice to see he solved the pathfinding, but it's probably not the most challenging part of the problem!

Have a look at a globe. (or google earth or whatever) The direct line between DC and Moscow is surprisingly shallow. The exact number does seem too low by a factor of 1.5-2 though.

The exact number is right. The depth is R(1-cos(theta/2)), where theta is the angular distance between the two cities and R is the radius of the earth. (Draw a cross-section of the earth.) Let d = R*theta be the distance on land; then the depth is R(1 - cos(d/(2R)). (For short distances this is about d^2/(8R), so the depth varies quadratically with distance, which makes sense. But Moscow to DC isn't that short.) The radius of the earth is 3963 miles, and DC and Moscow are 4850 miles apart; the formula gives 719 miles for the depth of the tunnel, which is close enough to the claimed 716 that I blame rounding errors.

Re: To Everywhere in 42 Minutes

#27

Post by madcaptenor »

eru wrote: post #24
scorxn wrote: post #3

Earlier quoted context omitted.

Since Friday, Feb. 11, 1966 maybe ;)

I just wrote a simple python script to test whether this holds only when all of earth's mass is concentrated at its center, or in the slightly more realistic case of constant density. My numerical experiments point to the latter. Does anyone care to derive an analytic solution?

The derivation in the original article (Paul W. Cooper, Through the Earth in Forty Minutes, Am. J. Phys. vol. 34 (1966) p. 68) relies on the assumption of constant density. (It's hard to say this for sure because some of the details are left out, but Cooper at least states he's making this assumption.)

In the case where all the earth's mass is concentrated at the center, a point mass starting on the surface of the Earth would just go to the center and stay there.

In the opposite extreme where all the mass is concentrated on the surface, (that is, the Earth is a hollow shell) it actually turns out that the gravitational acceleration at any point inside the shell is zero, so it wouldn't work in that case either.

I suspect that for some reasonable class of spherically symmetric mass distributions (that is, the density only depends on the distance to the center of the Earth), tunnel systems like this are mathematically possible. But I'd be surprised to learn that there are mass distributions other than the uniform one for which the travel time doesn't depend on distance. But I'm not going to work this out because I have Real Work to do. (Now I wish I were teaching calculus so I would have an excuse to work out this problem...)

Re: To Everywhere in 42 Minutes

#28

Post by madcaptenor »

scorxn wrote: post #3
te_platt wrote: post #2

Fun idea, but didn't everyone here have to work this out in their college physics class?

Since Friday, Feb. 11, 1966 maybe ;)

I didn't notice that at first, and when the Time article mentioned that this was in the current issue of a journal I actually went and looked at the May 2009 issue. Of course it wasn't there.

Re: To Everywhere in 42 Minutes

#29

Post by eru »

madcaptenor wrote: post #27
eru wrote: post #24

Earlier quoted context omitted.

I just wrote a simple python script to test whether this holds only when all of earth's mass is concentrated at its center, or in the slightly more realistic case of constant density. My numerical experiments point to the latter. Does anyone care to derive an analytic solution?

The derivation in the original article (Paul W. Cooper, Through the Earth in Forty Minutes, Am. J. Phys. vol. 34 (1966) p. 68) relies on the assumption of constant density. (It's hard to say this for sure because some of the details are left out, but Cooper at least states he's making this assumption.) In the case where all the earth's mass is concentrated at the center, a point mass starting on the surface of the Ea…

> In the case where all the earth's mass is concentrated at the center, a point mass starting on the surface of the Earth would just go to the center and stay there.

Why? That would violate conservation of energy, wouldn't it?

In a naive model of a point mass you'd get a singularity at the center. But using standard techniques (e.g. numeric pertubation, or Lebesgue integration) one gets an objects that swings back and forth like in the other scenarios.

Re: To Everywhere in 42 Minutes

#30

Post by jibiki »

eru wrote: post #29
madcaptenor wrote: post #27

Earlier quoted context omitted.

The derivation in the original article (Paul W. Cooper, Through the Earth in Forty Minutes, Am. J. Phys. vol. 34 (1966) p. 68) relies on the assumption of constant density. (It's hard to say this for sure because some of the details are left out, but Cooper at least states he's making this assumption.) In the case where all the earth's mass is concentrated at the center, a point mass starting on the surface of the Ea…

> In the case where all the earth's mass is concentrated at the center, a point mass starting on the surface of the Earth would just go to the center and stay there. Why? That would violate conservation of energy, wouldn't it? In a naive model of a point mass you'd get a singularity at the center. But using standard techniques (e.g. numeric pertubation, or Lebesgue integration) one gets an objects that swings back an…

Hmm, unless I'm going crazy here, we get the differential equation

  x'' =  -(x^-2)
x = (kt)^(2/3)(with k = (2/9)^(-3/2), not that it matters) seems to be a solution?

I guess that doesn't help with the singularity, but neither does looking at energy (since you have infinite kinetic energy at the center and infinite potential energy everywhere else.)

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