7 ms·

To Everywhere in 42 Minutes

46 points by mariorz 18y ago

te_platt 18y ago
Fun idea, but didn't everyone here have to work this out in their college physics class?
- scorxn 18y ago
  Since Friday, Feb. 11, 1966 maybe ;)
  - eru 18y ago
    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?
    - madcaptenor 18y ago
      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...)
      - eru 18y ago
        > 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.
        jibiki 18y ago
        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.)
        eru 18y ago
        That differential equation only holds for positive x. Try something like the following: x'' = - signum(x) * (x^-2)
        madcaptenor 18y ago
        That's a good point. I don't know what I was thinking.
  - madcaptenor 18y ago
    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.
  - te_platt 18y ago
    Doh!
    - electromagnetic 18y ago
      You forgot the apostrophe, (Annoyed Grunt)!
jws 18y ago
This and atmosphere hopping space planes both have the same drawback: a significant number of people vomit on their first weightless trips. Do you really want to ride packed into coach in that kind of environment? I guess with enough frequent zero-g points you could upgrade to the "no hurling" section. (The not-quite-straight-through tunnels wouldn't be fully weightless, maybe we should concentrate on those. Then we just have to solve the insurmountable temperature, pressure, and vacuum problems.)
anigbrowl 18y ago
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.
- ramchip 18y ago
  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!
  - pmjordan 18y ago
    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.
    - madcaptenor 18y ago
      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.
- ars 18y ago
  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 problem. The pressure is a much bigger problem. You can't bore a tunnel in the conventional sense since the rock is a liquid. You'd need some sort of wall, but I don't know if we have anything strong (and stable) enough. Most metals will oxidize and destroy themself at those temperatures. Maybe a ceramic, or alumina coated metal. Perhaps a tungsten compound of some sort that is stable in oxygen at high temperature. Another problem is the speed (reaching 1000 MPH), you'd need magnetic bearings (mag-lev), but magnets don't like being hot. The curie point of iron is 768 C, which is too low. Cobalt might work - but barely.
deleted 18y ago
[deleted]
lewis 18y ago
I appreciate the concern which is been rose. The things need to be sorted out because it’s not about the individual but it can be with everyone. http://www.fastrealestate.net http://www.fastrealestate.net
d1rge 18y ago
Putting practical considerations aside, what might the effect on society be if we could be at any other place on earth in less than an hour? Would tourism explode? What would that mean for the environment? What about international business? Would it be the end of remote workers? Or would it encourage even more distributed workplace?
- ars 18y ago
  It would more or less mean the end of nations. In some parts of the world you can live on $5 a day, in others it's $100 - but with cheap travel, those earning lots would buy services in the cheaper places - increasing prices. And those earning little would seek employment elsewhere, reducing wages. 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.) Some places would of course "cut themself off the grid" (like North Korea), but the majority would not, and the world (or at least the connected places) would become much more similar. And some counties would put up barriers (like those separating the US and mexico - if not for those the wages in the US and mexico would tend to equalize, but they don't because of the barrier). All this assumes that the travel is cheap - if it's expensive, it doesn't much matter that it's fast.
  - gtufano 18y ago
    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. But may be it is only my european culture speaking...
    - dantheman 18y ago
      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.
lisper 18y ago
There's no such thing as a free lunch. Most of the energy required to go from A to B is needed to overcome friction, not provide kinetic energy. That remains true whether you are above or below the surface. So burrowing down buys you virtually nothing. The amount of energy you'd need to pull yourself up the other side of the tunnel would be almost exactly the same as you would need to make the same trip at (almost) the same speed on the surface.
- ars 18y ago
  Not completely true. You need energy to start, and then you waste it (or try to recover it) when you stop. With this you don't need to provide all that initial energy to get you going. You just have to handle the friction.
  - lisper 18y ago
    That's why I qualified with "most" and "virtually." I haven't actually done the math, but I'd be surprised if it didn't work out to something like >90% of the total energy for a long trip going into frictional losses. Consider any vehicle: the amount of the total energy that goes into generating actual motion is roughly proportional to the time it takes to accelerate to your final cruising speed, at which point all of the energy input goes into overcoming friction.
    - ars 18y ago
      For a car you are 100% right - it's nearly all friction. But to accelerate to over 1000 MPH you need a lot of energy just to get started, and you have little chance of recovering it. But an overland bullet train in a vacuum would be so much easier to build that the acceleration energy would be worth it.
      - lisper 18y ago
        > But to accelerate to over 1000 MPH you need a lot of energy You need exactly 4 times as much energy as you do accelerate to 500 MPH, which is the speed of a slow jet. Four times a small amount is still a pretty small amount.
david927 18y ago
Silly. It took us 24 years to get 12km straight down, and that's the best we've ever done. http://www.damninteresting.com/?p=567 http://www.damninteresting.com/?p=567
- david927 18y ago
  Actually, screw traveling -- if we could just drill a few hundred meters easily and cheaply, we could solve all our energy problems. Temperatures roughly increase about 1 degree C for every hundred meters you go down. The difference could power a Stirling Engine (some of which are powered on differences of as little as .5 degrees C). It seems to me you could use a captive bolt pistol to get down just a few hundred meters, carrying a plastic tube which would have two chambers. The engine would pump the water and generate additional electricity. What do you guys think?
patrickg-zill 18y ago
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.
- ph0rque 18y ago
  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 http://en.wikipedia.org/wiki/Plasma_Converter), then somehow use the pressure and heat of the earth, once you're deep enough?
  - electromagnetic 18y ago
    I wonder how Shaped Charges would perform. They're relatively inexpensive, would easily produce the temperature needed for glassing the rock and have excellent range penetration. A single shaped charge can easily penetrate beyond 10 times its diameter. Based on the Beach Pneumatic Transit diameter of around 2.5 meters, a single shaped charge designed to penetrate at this width (cone diameter of around 2.5 meters) would easily penetrate between 25 or 35 meters. Although on such an industrial scale, I wouldn't doubt some military contractor would go commercial with one that could penetrate up to 50 meters. The question would be, could such a destructive method (on the small scale) be more useful than current explosives used. There would likely be less shockwaves sent through the rock than traditional mining techniques, plus the potential glassing could help structural strength. The use of something so easily mass produced like a shaped charge could easily be used in vac-train mining like this. Although personally, I doubt any system like this would ever be used between continental plates.
kokofoo 18y ago
I would guess somebody or something need to catch the passengers from the other side of the hole; otherwise, it would fall back and forth forever like gigantic pendulums..
viggity 18y ago
I think Douglas Adams would be happy to hear this.
fgraham 17y ago
Does anyone know where Paul W Cooper, the author of Anywhere in 42 Minutes, is today? fgraham@kent.edu

46 points by mariorz 18y ago

te_platt 18y ago

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

scorxn 18y ago
Since Friday, Feb. 11, 1966 maybe ;)
- eru 18y ago
  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?
  - madcaptenor 18y ago
    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...)
    - eru 18y ago
      > 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.
      - jibiki 18y ago
        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.)
        eru 18y ago
        That differential equation only holds for positive x. Try something like the following: x'' = - signum(x) * (x^-2)
      - madcaptenor 18y ago
        That's a good point. I don't know what I was thinking.
- madcaptenor 18y ago
  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.
- te_platt 18y ago
  Doh!
  - electromagnetic 18y ago
    You forgot the apostrophe, (Annoyed Grunt)!

jws 18y ago

This and atmosphere hopping space planes both have the same drawback: a significant number of people vomit on their first weightless trips. Do you really want to ride packed into coach in that kind of environment? I guess with enough frequent zero-g points you could upgrade to the "no hurling" section. (The not-quite-straight-through tunnels wouldn't be fully weightless, maybe we should concentrate on those. Then we just have to solve the insurmountable temperature, pressure, and vacuum problems.)

anigbrowl 18y ago

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.

ramchip 18y ago
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!
- pmjordan 18y ago
  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.
  - madcaptenor 18y ago
    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.
ars 18y ago
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 problem. The pressure is a much bigger problem. You can't bore a tunnel in the conventional sense since the rock is a liquid. You'd need some sort of wall, but I don't know if we have anything strong (and stable) enough. Most metals will oxidize and destroy themself at those temperatures. Maybe a ceramic, or alumina coated metal. Perhaps a tungsten compound of some sort that is stable in oxygen at high temperature. Another problem is the speed (reaching 1000 MPH), you'd need magnetic bearings (mag-lev), but magnets don't like being hot. The curie point of iron is 768 C, which is too low. Cobalt might work - but barely.

deleted 18y ago

[deleted]

lewis 18y ago

I appreciate the concern which is been rose. The things need to be sorted out because it’s not about the individual but it can be with everyone. http://www.fastrealestate.net http://www.fastrealestate.net

d1rge 18y ago

Putting practical considerations aside, what might the effect on society be if we could be at any other place on earth in less than an hour? Would tourism explode? What would that mean for the environment? What about international business? Would it be the end of remote workers? Or would it encourage even more distributed workplace?

ars 18y ago
It would more or less mean the end of nations. In some parts of the world you can live on $5 a day, in others it's $100 - but with cheap travel, those earning lots would buy services in the cheaper places - increasing prices. And those earning little would seek employment elsewhere, reducing wages. 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.) Some places would of course "cut themself off the grid" (like North Korea), but the majority would not, and the world (or at least the connected places) would become much more similar. And some counties would put up barriers (like those separating the US and mexico - if not for those the wages in the US and mexico would tend to equalize, but they don't because of the barrier). All this assumes that the travel is cheap - if it's expensive, it doesn't much matter that it's fast.
- gtufano 18y ago
  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. But may be it is only my european culture speaking...
  - dantheman 18y ago
    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.

lisper 18y ago

There's no such thing as a free lunch. Most of the energy required to go from A to B is needed to overcome friction, not provide kinetic energy. That remains true whether you are above or below the surface. So burrowing down buys you virtually nothing. The amount of energy you'd need to pull yourself up the other side of the tunnel would be almost exactly the same as you would need to make the same trip at (almost) the same speed on the surface.

ars 18y ago
Not completely true. You need energy to start, and then you waste it (or try to recover it) when you stop. With this you don't need to provide all that initial energy to get you going. You just have to handle the friction.
- lisper 18y ago
  That's why I qualified with "most" and "virtually." I haven't actually done the math, but I'd be surprised if it didn't work out to something like >90% of the total energy for a long trip going into frictional losses. Consider any vehicle: the amount of the total energy that goes into generating actual motion is roughly proportional to the time it takes to accelerate to your final cruising speed, at which point all of the energy input goes into overcoming friction.
  - ars 18y ago
    For a car you are 100% right - it's nearly all friction. But to accelerate to over 1000 MPH you need a lot of energy just to get started, and you have little chance of recovering it. But an overland bullet train in a vacuum would be so much easier to build that the acceleration energy would be worth it.
    - lisper 18y ago
      > But to accelerate to over 1000 MPH you need a lot of energy You need exactly 4 times as much energy as you do accelerate to 500 MPH, which is the speed of a slow jet. Four times a small amount is still a pretty small amount.

david927 18y ago

Silly. It took us 24 years to get 12km straight down, and that's the best we've ever done. http://www.damninteresting.com/?p=567 http://www.damninteresting.com/?p=567

david927 18y ago
Actually, screw traveling -- if we could just drill a few hundred meters easily and cheaply, we could solve all our energy problems. Temperatures roughly increase about 1 degree C for every hundred meters you go down. The difference could power a Stirling Engine (some of which are powered on differences of as little as .5 degrees C). It seems to me you could use a captive bolt pistol to get down just a few hundred meters, carrying a plastic tube which would have two chambers. The engine would pump the water and generate additional electricity. What do you guys think?

patrickg-zill 18y ago

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.

ph0rque 18y ago
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 http://en.wikipedia.org/wiki/Plasma_Converter), then somehow use the pressure and heat of the earth, once you're deep enough?
- electromagnetic 18y ago
  I wonder how Shaped Charges would perform. They're relatively inexpensive, would easily produce the temperature needed for glassing the rock and have excellent range penetration. A single shaped charge can easily penetrate beyond 10 times its diameter. Based on the Beach Pneumatic Transit diameter of around 2.5 meters, a single shaped charge designed to penetrate at this width (cone diameter of around 2.5 meters) would easily penetrate between 25 or 35 meters. Although on such an industrial scale, I wouldn't doubt some military contractor would go commercial with one that could penetrate up to 50 meters. The question would be, could such a destructive method (on the small scale) be more useful than current explosives used. There would likely be less shockwaves sent through the rock than traditional mining techniques, plus the potential glassing could help structural strength. The use of something so easily mass produced like a shaped charge could easily be used in vac-train mining like this. Although personally, I doubt any system like this would ever be used between continental plates.

kokofoo 18y ago

I would guess somebody or something need to catch the passengers from the other side of the hole; otherwise, it would fall back and forth forever like gigantic pendulums..

viggity 18y ago

I think Douglas Adams would be happy to hear this.

fgraham 17y ago

Does anyone know where Paul W Cooper, the author of Anywhere in 42 Minutes, is today? fgraham@kent.edu