Underloop
From New York City to Washington in under 20 min, without a motor, at zero energy cost
There is another form of renewable energy we tend to forget about: gravity. It can be tapped to power ultra-fast inter-city transportation. For example, a train traveling along a 6 km-deep evacuated subway tunnel can deliver passengers from New York City to Washington, D.C. in 17 min, at a very little energy cost.
Different propulsion systems for such “vactrains”, from conventional to compressed air to maglev have been proposed over the years. The most elegant solution, however, requires no propulsion system, no engines, and no brakes. The trains would use gravity to accelerate. Just plunge down the tunnel, like on a roller coaster. Gain high speed, coast along a horizontal tunnel at constant speed, then climb up and decelerate to stop at destination. The law of conservation of energy guarantees that the train will reach the ground level and stop right there.
Of course, there will be some energy loss due to imperfect vacuum, as well as wheel-on-rail rolling friction. It shall be somehow compensated. However, since the main energy loss at these speeds is due to air resistance, evacuation will dramatically reduce the energy consumption.
Better still, no fuel will be required to accelerate the train to ultra-high speed, and then to slow it down – this will be done by gravity.
Evacuated tube transport is not a novel idea. It was first proposed in early XX century and then pimped out by Robert Slater, in the seventies, mostly in conjunction with electromagnetic propulsion. Elon Musk has recently begun building suspense around his proposal, Hyperloop, soon to be announced. This article is not an attempt to guess what Mr. Musk will say; it is rather “If I had been given this physics problem in high school, how would I solve it?”
It is a fun mental exercise to see how much can be achieved without motors, on gravity alone.
There is a classic concept of “gravity train” that goes on a straight line from point A to point B. Counter-intuitively, the travel time for such train is always 42 min, regardless of the end points. However, if the end points are far away from each other it means digging very deep, sometimes through molten lava, solid core, etc. – not practical at all.
What we are talking about here is a “shallow evacuated gravity train”, whereby the main tunnel is horizontal (parallel to the Earth’s surface), and not very deep, no more than 12 km – the depth of the deepest Kola shaft near Murmansk, Russia. It does not sound completely outrageous.
Besides, for mid-range distances, the travel times will be substantially shorter than 42 min.
A straightforward application of the law of conservation of energy (at the high school freshman physics level) would tell us that plunging down 6 km (3.6 mi) will accelerate the train to 1.250 km/h or roughly the speed of sound, in less than 1 min. This speed is enough to cover 330 km (200 mi) between New York City and Washington, D.C. in 15 min. Add another 1 min to climb up and decelerate, to the total trip time of about 17 min, all with very little fuel required.
Let’s do the numbers. From the law of conservation of energy,
(mv^2)/2 = mgh, or
v = sqrt (2*g*h),
where h is the tunnel depth [1]. If h = 6 km then v= 1,250 km/h.
Then the train goes with the constant speed, and t = L/v. For L=330 km the travel time is about 15 min.
The acceleration and deceleration time will depend on the steepness and exact shape of the down and up connector tunnels. But it can be easily estimated by the time of a vertical plunge:
h = (gt^2)/2, or t = sqrt(2*h/g)
For h = 6 km we get about 35 sec. Adding generous 75% padding to account for increased distance on a slanted curve will give us a rough estimate of 60 sec.
One may think that a 6 km-high roller coaster would expose passengers to the g-forces no human would survive. In fact, the g-force doesn’t depend on the absolute scale of the roller coaster. It only depends on its shape. That is, since we survive Disneyland roller coasters we will survive its monster 6 km-high scaled-up copy.
Indeed, the g-force at the bottom of the curve (where it is maximal) is
N = mg + (mv^2)/R,
where R is the radius of curvature of the track at the bottom
with v^2 = 2*g*h, we get
N = mg*(1 + 2*h/R)
If the descending curve is a quarter-circle (i.e. R=h) we got a quite manageable 3g. It could be made even smaller by making the curve smoother near the bottom (making R larger).
Note that the g-force at the bottom, 3g, indeed does not depend on the roller coaster size. Likewise, for the circular path down, at angle phi, it will be 3*g*cos(phi) – no R again, or the same as on its Disneyland little cousin with R=30 m.
Also, like on a joy ride roller coaster, for the first few seconds of their way down the passengers will enjoy partial weightlessness, already a $5K value in today’s prices :-).
OK, we will survive the ride. How about the trains? Can we build a train that will travel at over 1,000 km/h? Well, the current world record, held by French TGV, is 575 km/h. Same order of magnitude – it should be doable :-)
And the last little detail: to make this possible we would have to bore a 330 km long tunnel in solid rock at a 6 km depth. Do we have the technology? And how much would it cost?
First, can we bore tunnels under 6 km of solid rock? Apparently, yes, we can. Mont Blanc tunnel in the Alps goes under, well, Mont Blanc, almost 5 km high. The tunnel is only 12 km long, but if we can do 12 km, we can do 300 km, too. It’s just a matter of time and money.
Can we build a 300 km long tunnel? Apparently, yes, we can. Moscow subway system is 300 km long. Though it is not 6 km deep, most of it is deeper than New York City subway and required boring, not digging. And a good portion of it was built with picks and shovels, back in the thirties. It would be easier today.
There are other long tunnel systems. E.g. railway tunnels under Cinque Terre in Italy are about 40 km long, and they were bored in solid rock. The famous Channel Tunnel is 40 km; the Seikan Tunnel in Japan is 54 km.
There are deep and long mines, too. The deepest mine, TauTona in S. Africa, is 4 km deep. It has 800 km of tunnels! 330 km looks doable.
How much would it cost? Unfortunately, the current subway construction costs in the US are astronomical, at around $1B/km. However, in China it’s about $60M/km, including stations, vestibules and everything. More intriguingly, in Spain, it’s also about $60M/km. If Spaniards, despite their EU wages, taxes and 30 day vacations, did it at that price we shall be able, too. At worst, we would have to waive visa restrictions and invite the Chinese :-).
And finally, 330 km * $60M/km = $20B. Add a few billion for air locks, trains, etc. and we end up with some $22B.
We can also add Philadelphia and Baltimore stations at marginal added cost: these two cities are almost on the straight line connecting New York City to Washington. We should only add descending and ascending tunnels – some extra 15 km or $1B per station. Of course, it shouldn’t add time to the New York City to Washington journey. Those trains will continue straight, without climbing up and down, or stopping. We would have a separate New York City – Philadelphia train.
$22B is not bad at all. Compare it with $6B we are paying for a lousy half of Bay Bridge. Likewise, the proposed surface San Francisco to Los Angeles chu-chu train is expected to cost $90B. And it would require gazillion tons of fuel to operate every year. An underground gravity train between San Francisco and Los Angeles – 550 km distance – would take 30 min to travel and about $35B to build.
For $35B one-time cost we will have the system that would be 3 times cheaper to build, 5 times faster, last forever, and require virtually no fuel or other energy to operate. Sounds like a good deal.
Can we go even faster? Yes, if we dig deeper. Oddly enough, since the horizontal tunnel is much longer than the up and down connectors, the construction cost won’t increase much if we dig deeper – until we hit molten magma and stuff :-). Building the San Francisco – LA tunnel at 12.5 km depth will increase the speed to 1,800 km/h (or ~1.5 M) and reduce the travel time to 20 min.
If we want to build a transatlantic tunnel, also 12 km – deep, the train will cover 5,550 km between New York City and London in about 3 hr (as Concorde did), but without fuel, on gravity alone.
The tunnel will, however, cost $300B to build. That bites. We shall probably start with shorter tunnels.
We will also need a propulsion system to compensate for minor energy loss due to imperfect vacuum and rolling resistance. The good thing, however, is that the loss will be so small that we won’t need a locomotive. The system can be localized at the departure or arrival terminals, and be outside the train. It will be enough to give the train a modest kick at departure or pull it up to the arrival terminal.
It can be estimated that the train will lose about 5% of its max energy on a 330 km NYC — Washington line; for 550 km SF — LA line, it will be 10% [2]. It will be enough to push the train from behind with a detachable conventional locomotive, a catapult or the like and let it go. All we need is to give it initial speed of 36 km/h (20 mph). That will be enough for the entire journey. Or alternatively we would have to pull it the last 300 m up at the arrival terminal. This can be done, for example, with cables, like a funicular or San Francisco cable car — the system will be needed only on the last 300m, and can be external — the train will only need a hook :-). This doesn’t look technically challenging or expensive. Just reuse 100 years old Swiss bergbahn technology :-)
Of course, many serious technical challenges remain, and perhaps are still unsolvable with the today’s technology.
We will need extra straight and smooth rails – a minor wobble at these speeds may result in derailment, the train smashing into the wall, and everybody instantly dead. Alternatively, we can use maglev instead of rails. The ride will be smoother, but lateral stability will likely be reduced. And the train will smash into the tunnel’s wall if the electric power to the magnets is lost. Ouch…
There are many more nagging questions. If one train derails, how can we stop the next one? What if the ground shifts and rails wobble? An earthquake occurs? A crack opens, gas seeps in and explodes? Or at least damages vacuum? A small rock or bolt falls and hits a speeding train? A passenger compartment loses pressure – do we need an emergency space suit for every passenger? Will the excessive temperature be a problem? At 6 km depth the temperature is 150C. Or, to the contrary, will we manage to exploit the thermal gradient (25C/km), as a side benefit? Any one of these “small” real-life problems may make the whole project unfeasible.
In other words, this essay is not a real engineering proposal. It is rather a mental exercise in elementary physics: how much speed we can get from gravitational energy alone, and how deep we shall dig to harness it. From that point of view at least, it doesn’t look completely unrealistic.
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[1] Technically, g decreases with depth – only the part of the Earth that is below the train attracts it. However, with the Earth radius of 6,400 km and the tunnel depth of 6 km the decrease in g is negligible.
[2] Assuming full vacuum, the only friction force will be rolling resistance:
F_rr = C*m*g
A rolling resistance coefficient for steel wheel on steel rail is typically about C = 0.001. From F_rr=ma, we have a = C*g and thus the decline in speed dv = a*t = C*g*L/v. For the NYC — Washington line (L=330 km) we have dv = 10 m/s. (The speed right after the descent is v = 350 m/s. I.e. the train would lose ~3% of its speed.) So, it’s enough to give the train the initial speed of 10 m/s for it to complete the journey.
Alternatively, we can let it go without initial speed but then grab and pull up at destination. The energy loss due to friction will be
dE = F_rr*L = C*m*g*L
The vertical distance dh the train would not clear due to this loss will be
dE = m*g*dh
Thus,
dh = C*L
With C=0.001, for L = 330 km, it will be dh=330m
For the relative energy loss dE/E we have
dE/E = C*m*g*L/m*g*h = C*L/h
for L = 330 km and h = 6 km, we’ll get 5.5%
for L = 550 km (SF — LA line), we’ll get 10%
