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with respect to the center of mass the speed of each particle is (Gm/4R)1/2 as their is no external force acting on the system so velocity of the center of mass is zero velocity of each particle is (Gm/4R)1/2.
mv2/R=Gm2/(2R)2
The gravitational force of attraction between the two masses is
F = Gm2/(2R)2
This force provides the necessary centripetal force
mv2/R = Gm2/4R2
v2 = Gm/4R
v = √(Gm/4R)
The Speed will be equal to their velocities in inertial frame itself as there centre of mass will be stationary.
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