r=6;θ=π/2;M1=1;M2=10;

g00=(1/(1-2/r));

vPro=(Sqrt[(1/r)])(Sqrt[g00]); "vPro"->N[vPro]

ε=(1/((Sqrt[1-(vt^2)])(Sqrt[g00])));

vEsc=Quiet@Reduce[ε==1,vt][[2,2]]; "vEsc"->N[vEsc]

vDif=vd/.Solve[((vPro+vd)/(1+vPro vd))==vEsc,vd][[1]]; "vDif"->N[vDif]

v1=vDif;sol=Simplify[Reduce[
((M1/(Sqrt[1-(v1^2)]))-M1)+((M2/(Sqrt[1-(v2^2)]))-M2)==Ek&&(((M1 v1)/(Sqrt[1-(v1^2)])))+(((M2 v2)/(Sqrt[1-(v2^2)])))==0&&
Ek>0&&M1>0&&M2>M1&&v1>0&&v2<0, {Ek,v2}, Reals]];

"vRec"->N[sol[[2,2]]]

"Ek"->N[sol[[1,2]]]