im=480;

(* Droste *)

vp=Simplify[(Sqrt[2/r])(1-2/r),r>2]; vm=-vp;

{{"dr/dt"->vm},{"dr/dt"->vp}}

t=Assuming[2<r1<r2, Integrate[1/vp, {r,r1,r2}]]

p1=Show[ParallelTable[Plot[
t,{r1,2,10},Frame->True, PlotStyle->{Red,Thick},
ImageSize->im, AspectRatio->1,PlotRange->{{0,10},{0,10}}, 
GridLines->{{2},{}}, ImagePadding->{{60,2},{30,2}}],
{r2,Join[{2+1/5},Table[2+(3/2)n,{n,0,7}]]}]]

p2=Show[ParallelTable[Plot[
Block[{r1=0.001},10-t-n],{r2,0,2},
Frame->True, PlotStyle->{Red,Thick},ImageSize->im, 
AspectRatio->1,PlotRange->{{0,10},{0,10}}, 
GridLines->{{2},{}}, ImagePadding->{{60,2},{30,2}}],
{n,0,10,2}]]

p3=Show[ParallelTable[Plot[
t,{r1,0,1.999},
Frame->True, PlotStyle->{Red,Thick},ImageSize->im, 
AspectRatio->1,PlotRange->{{0,10},{0,10}}, 
GridLines->{{2},{}}, ImagePadding->{{60,2},{30,2}}],
{r2,{0,1,1.5,1.75}}]];

P1=p1
P2=Show[p1,p2]
P3=Show[p1,p3]

(* Raindrop *)

vM=dr/.Solve[dr/(dT-dr (Sqrt[2/r])/(1-2/r))==vm, dr][[1]];
vP=dr/.Solve[dr/(dT-dr (Sqrt[2/r])/(1-2/r))==vp, dr][[1]];
dT=1;

Vm=FullSimplify[vM, r>0]; Vp=FullSimplify[vP, r>2];
{{"dr/dt"->Vm},{"dr/dt"->Vp}}

t1=Assuming[2<r1<r2, Integrate[1/Vm, {r,r1,r2}]]

t2=Assuming[2<r1<r2, Integrate[1/Vp, {r,r1,r2}]]

p1=Show[ParallelTable[Plot[
t1,{r2,0,10},Frame->True, PlotStyle->{Red,Thick},
ImageSize->im, AspectRatio->1,PlotRange->{{0,10},{0,10}}, 
GridLines->{{2},{}}, ImagePadding->{{60,2},{30,2}}],
{r1,Join[{1/2},Table[2+(3/2)n,{n,0,7}]]}]]

p2=Show[ParallelTable[Plot[
t2,{r2,2.001,10},Frame->True, PlotStyle->{Blue,Thick},ImageSize->im, 
AspectRatio->1,PlotRange->{{0,10},{0,10}}, 
GridLines->{{2},{}}, ImagePadding->{{60,2},{30,2}}],
{r1,Join[{2.2,2.5},Table[2+(3/2)n,{n,0,7}]]}]]

p3=Show[ParallelTable[Plot[
t2,{r2,0,1.999},Frame->True, PlotStyle->{Blue,Thick},
ImageSize->im, AspectRatio->1,PlotRange->{{0,10},{0,10}}, 
GridLines->{{2},{}}, ImagePadding->{{60,2},{30,2}}],
{r1,{1.999,1.8,1.5,1,1/2}}]]

P4=Show[p1,p2,p3]

(* Finkelstein *)

vM=dr/.Solve[dr/(dT-dr (2/r)/(1-2/r))==vm, dr][[1]];
vP=dr/.Solve[dr/(dT-dr (2/r)/(1-2/r))==vp, dr][[1]];
dT=1;

Vm=FullSimplify[vM, r>0]; Vp=FullSimplify[vP, r>2];
{{"dr/dt"->Vm},{"dr/dt"->Vp}}

t1=Assuming[2<r1<r2, Integrate[1/Vm, {r,r1,r2}]]

t2=Assuming[2<r1<r2, Integrate[1/Vp, {r,r1,r2}]]

p1=Show[ParallelTable[Plot[
t1,{r2,0,10},Frame->True, PlotStyle->{Red,Thick},
ImageSize->im, AspectRatio->1,PlotRange->{{0,10},{0,10}}, 
GridLines->{{2},{}}, ImagePadding->{{60,2},{30,2}}],
{r1,Join[{1/2,2.001},Table[2+(3/2)n,{n,0,7}]]}]]

p2=Show[ParallelTable[Plot[
t2,{r2,2.001,10},Frame->True, PlotStyle->{Blue,Thick},
ImageSize->im, AspectRatio->1,PlotRange->{{0,10},{0,10}}, 
GridLines->{{2},{}}, ImagePadding->{{60,2},{30,2}}],
{r1,Join[{2.2,2.5},Table[2+(3/2)n,{n,0,7}]]}]]

p3=Show[ParallelTable[Plot[
t2,{r2,0,1.999},Frame->True, PlotStyle->{Blue,Thick},
ImageSize->im, AspectRatio->1,PlotRange->{{0,10},{0,10}}, 
GridLines->{{2},{}}, ImagePadding->{{60,2},{30,2}}],
{r1,{1.999,1.8,1.5,1,1/2}}]]

P5=Show[p1,p2,p3]

(* Newton *)

v=(Sqrt[2/r]);

{{"dr/dt"->±v}}

t=Assuming[2<r1<r2, Integrate[1/v, {r,r1,r2}]]

p1=Show[ParallelTable[Plot[
t,{r1,0,10},Frame->True, PlotStyle->{Blue,Thick},
ImageSize->im, AspectRatio->1,PlotRange->{{0,10},{0,10}}, 
GridLines->{{2},{}}, ImagePadding->{{60,2},{30,2}}],
{r2,Join[{1/2},Table[2+(3/2)n,{n,0,13}]]}]]

Quit[]