total mechanical energy = KE + GP E

R_{A} = (\frac{M_{B}}{M_{A} + M_{B}} ) a

R_{B} = (\frac{M_{A}}{M_{A} + M_{B}} ) a

M_{A} (\frac{v_{A}^{2}}{R_{A}} ) = \frac{GM_{A}M_{B}}{a^{2}}

v_{A} = \sqrt{\frac{GM_{B}}{a} (\frac{M_{B}}{M_{A} + M_{B}} )}

v_{B} = \sqrt{\frac{GM_{A}}{a} (\frac{M_{A}}{M_{A} + M_{B}} )}