Radial distance, velocity and acceleration vs time along the chord

Solve[(R^2 - rf^2)/(rf^2 - rmin^2)^(1/2) const, rf] (* note to aid solving tgen for rf *)

{{rf -(R^2 + const^2 rmin^2)^(1/2)/(1 + const^2)^(1/2)}, {rf (R^2 + const^2 rmin^2)^(1/2)/(1 + const^2)^(1/2)}}

rf[t_, rmin_, R_, G_, M_] := (R^2 + (Tan[((G M)^(1/2) t)/R^(3/2)])^2rmin^2)/(1 + (Tan[((G M)^(1/2) t)/R^(3/2)])^2)^(1/2)

RowBox[{Plot, [, RowBox[{RowBox[{rf, [, RowBox[{t, ,, 6378140 - 1600, ,, 6378140, ,, RowBox[{6 ... e, ,, FrameLabel {"Seconds", "Meters"}, ,, AspectRatio1}], ]}]

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FullSimplify[D[rf[t, rmin, R, G, M], t]]

((G M)^(1/2) (-R^2 + rmin^2) Sin[(2 (G M)^(1/2) t)/R^(3/2)])/(2^(1/2) R^(3/2) (R^2 + rmin^2 + (R - rmin) (R + rmin) Cos[(2 (G M)^(1/2) t)/R^(3/2)])^(1/2))

vf[t_, rmin_, R_, G_, M_] := ((G M)^(1/2) (-R^2 + rmin^2) Sin[(2 (G M)^(1/2) t)/R^(3/2)])/(2^(1/2) R^(3/2) (R^2 + rmin^2 + (R - rmin) (R + rmin) Cos[(2 (G M)^(1/2) t)/R^(3/2)])^(1/2))

RowBox[{Plot, [, RowBox[{RowBox[{vf, [, RowBox[{t, ,, 6378140 - 1600, ,, 6378140, ,, RowBox[{6 ... rameLabel {"Seconds", "Meters/Second"}, ,, AspectRatio1}], ]}]

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vtot[t_, rmin_, R_, G_, M_] := v[rf[t, rmin, R, G, M], R, G, M]

FullSimplify[vtot[t, rmin, R, G, M]]

(G M (R - rmin) (R + rmin) Sin[((G M)^(1/2) t)/R^(3/2)]^2)/R^3^(1/2)

RowBox[{Plot, [, RowBox[{RowBox[{vtot, [, RowBox[{t, ,, 6378140 - 1600, ,, 6378140, ,, RowBox[ ... rameLabel {"Seconds", "Meters/Second"}, ,, AspectRatio1}], ]}]

[Graphics:../HTMLFiles/index_77.gif]

⁃Graphics⁃

FullSimplify[D[rf[t, rmin, R, G, M], {t, 2}]]

(G M (4 (-R^4 + rmin^4) Cos[(2 (G M)^(1/2) t)/R^(3/2)] - (R^2 - rmin^2)^2 (3 + Cos[(4 (G M)^(1 ... 2)])))/(2 2^(1/2) R^3 (R^2 + rmin^2 + (R - rmin) (R + rmin) Cos[(2 (G M)^(1/2) t)/R^(3/2)])^(3/2))

af[t_, rmin_, R_, G_, M_] := (G M (4 (-R^4 + rmin^4) Cos[(2 (G M)^(1/2) t)/R^(3/2)] - (R^2 - r ... 2)])))/(2 2^(1/2) R^3 (R^2 + rmin^2 + (R - rmin) (R + rmin) Cos[(2 (G M)^(1/2) t)/R^(3/2)])^(3/2))

... True, ,, FrameLabel {"Seconds", Meters/ Second }, ,, AspectRatio1}], ]}]

[Graphics:../HTMLFiles/index_83.gif]

⁃Graphics⁃

FullSimplify[D[vtot[t, rmin, R, G, M], t]]

((G M)^(3/2) (R - rmin) (R + rmin) Sin[(2 (G M)^(1/2) t)/R^(3/2)])/(2 R^(9/2) (G M (R - rmin) (R + rmin) Sin[((G M)^(1/2) t)/R^(3/2)]^2)/R^3^(1/2))

atot[t_, rmin_, R_, G_, M_] := ((G M)^(3/2) (R - rmin) (R + rmin) Sin[(2 (G M)^(1/2) t)/R^(3/2)])/(2 R^(9/2) (G M (R - rmin) (R + rmin) Sin[((G M)^(1/2) t)/R^(3/2)]^2)/R^3^(1/2))

... True, ,, FrameLabel {"Seconds", Meters/ Second }, ,, AspectRatio1}], ]}]

[Graphics:../HTMLFiles/index_89.gif]

⁃Graphics⁃

Created by Mathematica  (June 17, 2004)