GraviTubes

Declarations

GraviTubes:  A term found in "Lost in a Good Book" by Jasper Fforde, refers to straight and nearly frictionless tunnels through the earth designed to enable rapid travel from point A to B at minimal energy cost (i.e. only the energy needed to overcome the friction of straight-line motion is required).  Gravitubes thus allow travelers to conveniently "fall" between any two points at the same distance from earth's center.  As we'll show here for planets with a density of around 5.5 g/cc (like that of earth), the time for any trip between points at the same distance from planetary center is about 40 minutes.

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Acceleration g as a function of radius r from planetary center

outline and special cases

...where g is the local vector acceleration due to gravity, dA is the infinitesimal area vector pointing in the direction of the surface normal, the integration is over any close:d "Gaussian" surface, and is the mass contained inside that surface.  This gravitational flux is equated, on the right hand side, to the mass inside the closed surface times a suitable constant.  When the mass and the surface share spherical symmetry, two special cases are...

In the classical example of a spherically-symmetric mass (or charge) distribution, a concentric spherical Gaussian surface of radius R will have a constant radial g field whose magnitude on integration obeys g(4)=4, so that g = .  In the electric field case, this instead gives us the Coulomb field E = .  Thus a form of Gauss's Law may be implicated in all forces that fall off as one over radial distance squared.

Time-elapsed, or why all trips are around = 42.2415 minutes...

On a trip to the center

If you know the radial velocity = dr/dt as a function of radius, you can integrate its reciprocal (dt/dr) over a range of r values to get elapsed time, as shown  below...

Here's a function to calculate time to center...

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Here we note that R cubed over M is inversely proportional to density ρ, so that the time is also a function simply of ρ...

Here we check on earth's effective density in this context...

This gives time elapsed on a fall to the center in terms of density alone...

Here we test this latter function for the earth's effective density...

The time elapsed falling through the earth's center to the other side (this time using the t0 instead of t00) would be...

In minutes, this is...

It turns out (see the section on chords NOT through the center below) that = 42.2415 Minutes is also the elapsed time (in this uniform density approximation) for all gravitube "slides" between any two points on the earth at the same distance from earth's center.  To determine the time elapsed partway into a radial dive, one integrates only to rf...

Thus an analytic function for time-elapsed partway through a radial fall can also be defined...

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 Created by Mathematica  (February 13, 2007)