Along a chord not through the center (42.2415 minutes derived)

In[19]:=

x[r_] := (r^2 - rmin^2)^(1/2)

D[x[r], r]

r/(r^2 - rmin^2)^(1/2)

In[20]:=

dtrav[rmin_, R_] := 2 R Sin[ArcCos[rmin/R]]

In[21]:=

dxdr[r_, rmin_] := r/(r^2 - rmin^2)^(1/2)

In[22]:=

dxdt[r_, R_, G_, M_] := v[r, R, G, M]

General :: spell1 : Possible spelling error: new symbol name \"dxdt\" is similar to existing symbol \"dxdr\". More…

dxdt[r, R, G, M]

(G M (-r^2 + R^2))/R^3^(1/2)

Integrate[-dxdr[r, rmin]/dxdt[r, R, G, M], {r, R, rmin}, Assumptions-> {R>rmin && rmin>0 && G>0 && M>0}]

(π R^(3/2))/(2 (G M)^(1/2))

Integrate[-dxdr[r, rmin]/dxdt[r, R, G, M], {r, R, rf}, Assumptions→ {R>rf, rf>rmin, rmin>0, G>0, M>0}]

(R^(3/2) ArcTan[(R^2 - rf^2)/(rf^2 - rmin^2)^(1/2)])/(G M)^(1/2)

In[23]:=

tgen[rf_, rmin_, R_, G_, M_] := (R^(3/2) ArcTan[(R^2 - rf^2)/(rf^2 - rmin^2)^(1/2)])/(G M)^(1/2)

Limit[tgen[rf, rmin, R, G, M], rf→rmin]

(π R^(3/2))/(2 (G M)^(1/2))

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

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