Monday, May 11, 2009

A tunnel to China

Like every other little kid in the US, I once tried to dig a hole to China. Not having a globe and assuming at China was on the exact opposite side of the world, I tried to dig straight down through the center of the earth to China. Of course, recess always ended before I got more than a foot or two underground.

At a later age, I realized how incredibly useful such a hole would be if it could actually be built. Rather than flying or sailing around the world, you could just step into the hole and start falling. You would continuously accelerate until reaching the center of the earth, at which point you would decelerate until reaching a halt right at the surface of China. (Then you'd have to leave the hole quickly, lest you fall back down to America again.) Thus, there would be very quick transportation between the US and China, expending no energy whatsoever. Also, a hole like this would not have to go through the exact center of the earth. You could dig a hole from New York to Los Angeles and put train tracks in it. The train would accelerate "downhill" for 2000 miles, then decelerate "uphill" until stopping automatically at its destination. All extremely fast and totally free once the tunnel was built.

I wondered how long exactly it would take to make such a trip. But it took many years until I learned how to do this calculation. Since the acceleration of gravity varies with depth, the solution requires the use of calculus and differential equations. Not until my sophomore year of college did I learn how to solve the problem. And not until last night did I fully solve it, including the case when the tunnel does not go through the earth's center. I assumed of course that the earth is perfectly spherical, with uniform density, and that there's no friction.

It turns out that it takes 42 minutes to make the trip through such a tunnel. Amazingly, if my calculations are right, this time does not depend on the tunnel's path. Falling through the earth's center to China, or tunneling on a diagonal from NY to LA, or simply digging a perfectly horizontal and frictionless tunnel through the hill you live next to - no matter what tunnel you build, it will take 42 minutes for gravity to pull you from one end to the other.

Obviously you will fall faster when going to China. You will reach a speed of 18000 miles per hour at the center of the earth, while as for the hill next door, the 42 minutes spend sliding through that short frictionless tunnel would pass agonizingly slowly. If there was a slight breeze, it would overcome gravity and push you in the opposite direction.

Of course, the center of the earth is hot enough to melt rocks, not to mention metal, making a tunnel to China very short lasting even if it could somehow be built. And it's hard to imagine a vehicle surviving a trip at 18000mph. If you collided with someone at the center of the earth, both of you would instantly be vaporized. So really, this tunnel idea is totally impractical. But if it could somehow be built... well, the idea has given me something to think about now for roughly 20 years and counting.

[Technical note: 42min = pi*sqrt(g/R), 18000mph = sqrt(g*R). Equation of motion: X=R*cos(t*sqrt(g/R)). For tunnel not through the earth's center, replace the first R with half the tunnel length. R=earth's radius, g=acceleration at earth's surface. ]

6 comments:

bachrach44 said...

While there are clearly some technological problems (what do you do with all the displaced earth, how do you keep a tunnel from collapsing, how do you keep the passengers from boiling in the intense heat that far below the earth's surface, etc.), there is no reason that we couldn't have gravity trains in a few centuries. At least, so says a few other people:
http://www.time.com/time/magazine/article/0,9171,842469,00.html

http://www.docstoc.com/docs/566538/Gravity-Train-Project

D.C. said...

Are you sure about the hill next door? The very existence of such a hill contradicts your assumption of a perfectly spherical Earth with uniform density.

Beisrunner said...

AEB: In a few hundred years I think it will still be cheaper to send spaceships into orbit and/or the other side of the earth...

DC: If there were no hill, you would just need to cut a slight horizontal groove in the ground to overcome the earth's curvature and get to the house next door. For stylistic reasons I wanted to talk about tunnels rather than slight grooves, so I supposed that a hill would be there. Technically, this hill would have to be weightless to not contradict my assumptions. Of course, objects (like houses) on the earth's surface would exert their own gravitational field which might well be stronger than the horizontal part of the earth's field, I haven't calculated it.

bachrach44 said...

The gravity train does have the advantage of being fuel free (just about), while a spaceship flying at 9000 miles an hour (the speed you'd need to make a comparable trip from Washington DC to Shanghai in 42 minutes), would expend a lot of fuel. Depending on what happens with energy in the next few hundred year, there may be a place for the train.

Joshua Skootsky said...

There is a serious problem that you have not considered.

The earth rotates.

A thing free-falling down a tunnel... would hit the sides.

Beisrunner said...

The earth rotates rather predictably. You could make the tunnel slightly curvy/non-vertical to account for it.