Distance, velocity, and acceleration vs time on the way

Inverting the tRadial equation above...

In[16]:=

r[t_, R_, G_, M_] := (R Tan[π/2 - (G M)^(1/2)/R^(3/2) t])/(1 + (Tan[π/2 - (G M)^(1/2)/R^(3/2) t])^2)^(1/2)

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FullSimplify[D[r[t, R, G, M], t]]

-(G M)^(1/2)/(R^(1/2) Csc[((G M)^(1/2) t)/R^(3/2)]^2^(1/2))

In[17]:=

vr[t_, R_, G_, M_] := -(G M)^(1/2)/(R^(1/2) Csc[((G M)^(1/2) t)/R^(3/2)]^2^(1/2))

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

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

In[18]:=

ar[t_, R_, G_, M_] := -(G M Csc[((G M)^(1/2) t)/R^(3/2)]^2^(1/2) Sin[(2 (G M)^(1/2) t)/R^(3/2)])/(2 R^2)

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