gravitubes.nb

Numerical Solutions and Plots

Here rf=final radial distance from earth center, rmin=trajectory minimum radial distance from earth center, R = radius of earth's surface, G = Gravitational Constant, and M = earth's mass

$tnum[rf_, rmin_, R_, G_, M_] := NIntegrate[-dxdr[r, rmin]/dxdt[r, R, G, M], {r, R, rf}]$

$RowBox[{{, RowBox[{6378140 Meter, ,, RowBox[{RowBox[{(, RowBox[{6.673*10^-11, , Meter^2, , Newton}], )}], /, Kilogram^2}], ,, RowBox[{5.9742*10^24, , Kilogram}]}], }}]$

$RowBox[{tnum, [, RowBox[{0, ,, 0, ,, 6378140, ,, RowBox[{6.673, *, 10^(-11)}], ,, RowBox[{5.9742, *, 10^24}]}], ]}]$

$RowBox[{RowBox[{1267.24, }], -, RowBox[{2.57658*10^-52, , }]}]$

[Graphics:../HTMLFiles/index_97.gif]

RowBox[{Plot, [, RowBox[{RowBox[{2, , RowBox[{tnum, [, RowBox[{rf, ,, rf, ,, 6378140, ,, RowB ... , FrameLabel {"Trip Depth [Meters]", "Trip Duration [Seconds]"}}], ]}]

[Graphics:../HTMLFiles/index_100.gif]

From the figure above, it appears that elapsed time for a frictionless dive is the same regardless of the distance of the trip!

[Graphics:../HTMLFiles/index_103.gif]

${{rmin -1/2 (-d^2 + 4 R^2)^(1/2)}, {rmin1/2 (-d^2 + 4 R^2)^(1/2)}}$

$RowBox[{RowBox[{Table, [, RowBox[{RowBox[{{, RowBox[{Convert[10^n Meter, Kilo Meter], ,, Conve ... (Pi/2 - ArcSin[rmn[10^n, 6378140]/6378140])}]}], , }}], ,, {n, 3, 6, .5}}], ]}], //, MatrixForm}]$

( Kilo Meter 0.019598189741373062` Meter 0.0044915743802474 ... 2` 1000.` Kilo Meter 19628.392744730227` Meter 4.496187515669029`

The above table lists Trip_Length, Maximum_Tube_Depth, and Angle_of_Starting_Incline in Degrees. As you can see, the slopes are pretty small for reasonable trips on earth, assuming that we don't have a practical way to build long straight tunnels penetrating 10 miles deep into the earth (bottom entry). If we did, however, the payoff might be worth 45 minutes of boring scenery at speeds into the 1000 km/hour range...

Plot[6378140 - rmn[d, 6378140], {d, 0, 300 1000}, AspectRatio1, PlotLabel->"Gr ... ;Run Length in Meters", "Max Depth in Meters"}] (* 1 mile deep for a 300 km trip *)

[Graphics:../HTMLFiles/index_115.gif]

Plot[6378140 - rmn[d, 6378140], {d, 0, 500 1000}, AspectRatio1, PlotLabel->"Gr ... Run Length in Meters", "Max Depth in Meters"}] (* 3 miles deep for a 500 km trip *)

[Graphics:../HTMLFiles/index_118.gif]

Plot[6378140 - rmn[d, 6378140], {d, 0, 6378140}, AspectRatio1, PlotLabel->"Gra ... th in Meters", "Max Depth in Meters"}] (* 800 km deep for an earth - radius trip *)

[Graphics:../HTMLFiles/index_121.gif]

Created by Mathematica (June 17, 2004)