[opendtv] Re: Mag-lev

  • To: <opendtv@xxxxxxxxxxxxx>
  • Date: Tue, 31 Oct 2006 15:18:16 -0500

Morris Jones wrote:

> I recall a school physics puzzle in which we were asked to prove
> that if you drilled a straight line through the earth between
> any two points, you could operate a frictionless train using
> only gravity. The tunnels wouldn't actually be U-shaped, but
> straight lines bypassing the spherical surface geometry. Cute
> thought.

Interesting concept.

A simple case is to go straight through the center of the earth. In that
case, when you're at the surface, acceleration is 9.8 m/sec^2, but then
acceleration reduces to 0 as you reach the center, and then slows you
down as you approach the opposite side.

Since acceleration is a function of gravitational attraction, it must be
proportional to the amount of mass pulling you to the center as opposed
to earth's mass pulling you away from the center, as your depth
increases. So it should be 9.8 * [(r-d)^3]/r^3, where r is the radius of
the earth and d is your depth at a given moment. (Volume is proportional
to r^3.)

If there's no friction, you start accelerating fast at first, like a
free fall. You achieve max velocity and constant speed (0 acceleration)
at the core. Then you start decelerating on the way up to the surface,
and you should come to a complete stop at the surface on the other side,
with no extra braking required.

I didn't do the somewhat tedious calculus, but integrating that
expression between 0 depth and 6,378,206 meters (equatorial) or
6,356,583 meters (polar) should give the total acceleration experienced
on the way down. And t = SQR(2r / a) should provide the time required
for the trip.

> My intuition also tells me that the trip time for such a train
> would be identical between any origin and destination. But I've
> not done the math.

Tricky. I guess you're contemplating shallow tunnels through a small
section of the globe. So you wouldn't perhaps ever experience free fall.
A much slower trip, but the same principles should apply. You should
still reach 0 acceleration at the deepest point of the trip, then
deceleration as you approach the opposite end of the tunnel.

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