GraviTubes
Declarations
GraviTubes: A term, found in "The Eyre Affair" (I think) and "Lost in a Good Book" by Jasper Fforde, which refers to straight and nearly frictionless tunnels through the earth which have been 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).
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, and the integration is over any closed "Gaussian" 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...
When r > R
When r<R
yielding the general rule for gravitational acceleration...
![g[r_, R_, G_, M_] := If[r<R, (G M r)/R^3, (G M)/r^2]](HTMLFiles/index_9.gif){kind=small}
![... ot;, ,, FrameTrue, ,, FrameLabel {"Meters", Meters/ Second }}], ]}], }]](HTMLFiles/index_10.gif)
![[Graphics:HTMLFiles/index_11.gif]](HTMLFiles/index_11.gif)
Note from the figure above that 9.8 m/s^2 is only available at the earth's surface (at a radius of 6378140 meters), and that inside or outside of that surface radius the acceleration due to gravity decreases.
Energy gained on a trip to the center
Kinetic Energy as a function of distance from the center
![K[r_, R_, G_, M_, m_] := (G m M (R^2 - r^2))/(2 R^3)](HTMLFiles/index_17.gif){kind=small}
Time elapsed (or why all trips are 42.2415 minutes)...
On a trip to the center
Distance, velocity, and acceleration vs time on the way
Along a chord not through the center (42.2415 minutes derived)
Radial distance, velocity and acceleration vs time along the chord
Numerical Solutions and Plots
Created by Mathematica (June 17, 2004)