roller-coaster cars

[Kragen]

Date: Mon, 9 Nov 1998 13:56:25 -0500 (EST)
From: Kragen <kragen@pobox.com>

How fast can I get to work?

I live about ten miles away from work, and it takes me about half an
hour and about fifty cents worth of energy to get there.  That's
because I spend most of my time stopped, speeding up after stopping, or
slowing down before stopping.  If I could drive 60 MPH from my front
door all the way to work, it'd be about three times faster.  (I
approximated this once when I was almost late for an appointment.)

But why should 60 MPH be the limit?  We adhere to it currently for
safety reasons --- with our current highways, driving 100 MPH or 200
MPH is rather risky, because what lies in front of us is
unpredictable.  We might have to turn, we might have to stop, we might
have a tire blowout and spin off the road into the lake.

Suppose we were limited only by what our bodies could handle --- about
4 G.  How fast could I get to work?  And how much would it cost?

4 G is 39.2 m/s^2.  I could accelerate uniformly at 4 G until I was
halfway there, and then decelerate uniformly at 4 G until I was there.
How long would that take?

Well, 10 mi is about 17 km.  Half of this is about 9 km; how quickly do
you cover 9 km at 4 G?  at^2/2 = d says that 2d/a = t^2, so 18 km s^2 /
39.2 m  = 460 s^2 = t^2.  So t is about 21 seconds.  I'd get halfway to
work in 21 seconds, and the rest of the way in 21 more seconds!

Better yet, it wouldn't have to cost any energy --- although it would
use quite a bit of energy to get me accelerated, all that energy would
be recoverable (in theory) when I was decelerating.

It would, of course, be necessary to get everything out of my way as I
sped on my way to work.  After 21 seconds, I'd be going 800 meters per
second -- about 2,880 kph, or 1700 MPH.  That's faster than most
bullets.  Air resistance would be very significant --- I'd have to
travel in an air-free environment in order to avoid dissipating
megawatts to move the air out of my way --- and any flying insects I
encountered would penetrate deep into my body.

What would the power consumption be?  I mass about 110 kg; accelerating
me at 4 G would require a force of about 4000 N.  At 800 m/s, when I'd
be "using" the most power, the power consumption would be 4000 * 800 =
3.2 megawatts.  Of course, a moment later, I'd be *generating* 3.2
megawatts as I decelerated, and the power consumption would vary over
the course of the trip --- for example, ten seconds after I left, I'd
only be using 1.5 megawatts or so.

Such a system could be used for much-longer-range travel as well as
short-range travel.  Traveling 20,000 km --- the distance from one side
of the Earth to the other --- would take considerably longer: 2 *
10,000 km s^2 / 39.2 m = 510,000 s^2, so 714 s for each half of the
trip.  The trip would total 1400 s or so --- about 23 minutes.

It might prove more difficult to operate such a system.  Your peak
speed would be 28000 m/s (100,000,000 kph, or about 60 million miles an
hour) and you'd be using 110 megawatts at that point (assuming you
weigh the same as I do.)

(I think you might have to slow down a bit to take care of the
curvature of the Earth --- you obviously wouldn't want to send people
through the core!  And if your moving passenger is following a path
near the surface of the earth at 60 million miles an hour, they're
going to have to contend with significant centrifugal force.  How
much?  Well, the Earth is about 40 Mm in circumference, which means
it's about 13 Mm in diameter, which means it's about 6 Mm in radius.
Centripetal acceleration is v^2 / r, so when does v^2/6 Mm reach 39.2
m/s?  At about 15000 m/s.  So the trip would actually take
significantly longer, since you'd speed up more and more slowly as you
approached 15000 m/s.)

So these are upper limits on Earth transportation speed, unless we
strengthen our bodies somehow.  

We obviously can't reach them yet.  Safety concerns, at present,
prohibit accelerating people to such high speeds while on the ground
--- we never know what'll happen in front of them, and of course, any
mistake would be instantly fatal at thousands of meters per second.
Also, people may not enjoy accelerating at 4 G for minutes on end.

Airplanes had the capacity to tremendously speed transportation, but
currently, they don't speed it up that much --- even a short plane ride
takes half an hour to an hour to drive to the airport, then another
half-hour or hour to board the plane, then another 15-30 minutes to get
off the ground, plus 10-40 minutes claiming your baggage on the other
end and getting out the door, and another half-hour to hour to drive
from the destination airport to where you're going.  You can easily
spend four hours on the ground for a half-hour plane trip.

As a result, I don't fly to work.

Even cars don't speed some things up much.  I still have to spend a
couple of minutes walking to the car.  Even if I decide to take the car
to, say, McDonald's, the difference is that it takes me four minutes
instead of six --- not the order-of-magnitude improvement the car is
capable of.

So even if we did come up with such a means of transportation, there's
no guarantee it'd have a significant effect, except on long-distance
travel.

But if it were taken to its limits, every forest glade, every
mountaintop would become accessible within minutes, anywhere on earth.

No doubt such transportation would lead to enormous economic prosperity
(on average --- it'd also lead to bigger crashes.)  I'm not sure we'd
want it, though.

Kragen

-- 
<kragen@pobox.com>       Kragen Sitaker     <http://www.pobox.com/~kragen/>
Irony and sarcasm deflate seriousness, and when your seriousness becomes detum-
escent, you're not held responsible for your thoughts. Irony beats thinking like
rock beats scissors. -- http://www.hyperorg.com/backissues/joho-june2-98.html