Métodos Numéricos

Métodos Numéricos
16 de Noviembre de 2012
The physics of motion pictures, or why science-fiction is all about fiction
The plot of a recently-released science-fiction film reads:
At the end of 21st Century, the Earth is divided into two territories — the United Federation
of Britain (UFB) and the Colony (Australia) — after World War 3 devastates the remainder
of the planet. Many residents ofthe Colony travel to the UFB to work in their factories via
"the Fall", a gravity elevator, which travels through the Earth. Habitable space is at a
minimum in both the UFB and the Colony…
“The Fall” in the film is simply a straight tunnel passing through the center of the Earth.
Payloads are dropped into one end of the tunnel and they emerge at the other side after a 17
minutes trip. Thebeauty of this system is that it is powered entirely by the gravitational field
of the planet. It is, literally, a free ride. But the question remains:
Should you trust movie scripts?
Anyone with one or more partially functional neurons knows the answer is a definite NO.
For a start, no known material can endure the temperatures and pressures found on Earth’s
mantle, much less those reigning inthe core. Next, Australia is not the antipode of the
United Kingdom, but this can be considered an artistic license. Let us be fair and give the
movie some credit. We shall assume some Tritanium-GTi-24V® hyperalloy has been
discovered in the future making the construction of such a tunnel a possibility. Would it
work as depicted in the film?
We shall use the combined power of mechanics andnumerical methods to make an educated
assessment of the film’s plot. Follow the steps below and you will gain a better
understanding of the physics of a “gravity elevator” (in the process, you may also achieve a
nice grade):
1) Write down the equation governing the motion of the elevator. Make the following
assumptions:
 Earth’s mass is uniformly distributed inside a sphere of radius
R 6371 km . The gravitational parameter of our planet is
  GM  398600 km3·s 2 . Consider gravity as the only body force acting
on the elevator1
 The tunnel is a straight line which goes through the center of the sphere
 The gravity lift is subject to some form of atmospheric drag (in the movie
the characters are able to breathe normally when standing outside the
1
elevator) which can becomputed as2 D  V 2 SCD
2
1

The centrifugal force due to Earth’s rotation can be accounted for with a slight change of the gravitational
parameter, while the Coriolis acceleration is perpendicular to the tunnel shaft. Thus, we shall ignore these two
effects.
2
In a real tunnel the air pressure would increase with depth, due to the weight of the air column. Therefore, the
air densitywould not be uniform either. In fact, the extreme pressure would make the gas mixture lethal (see
for example “nitrogen narcosis”). To give the writers some (more) slack we shall ignore this fact and assume a
uniform density across the tunnel, close to the value at sea-level. If the increase in air density were accounted
for the aerodynamic drag would rise to intolerable levels (not that this isgoing to make any difference in the
end).

1

Métodos Numéricos
16 de Noviembre de 2012


There are no frictional loses apart from aerodynamic drag

2) To make the problem easier to manage, convert the equations into non-dimensional form.
In this way the magnitude of most terms will be close to one (except the drag term, which
varies depending on the shape of the vehicle and theconditions inside the tunnel).
3) Assuming there is no drag, obtain the analytical solution of the problem. Determine the
period of motion. Compare the half-period (one way trip) with the value given in the movie
(17 minutes).

By now, you probably suspect there is something awfully wrong with the film script. Again,
let’s be flexible and assume we can give the elevator a little push at the...