2 ACR LTTS CHOSE TO SEPARATE THEMSELVES USING TCASII.

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WE WERE TOLD TO MAINTAIN VISUAL SEPARATION BEHIND AN LTT Y WHICH WAS JUST DEPARTING. WE WERE CLRED FOR TKOF AT WHAT I WOULD ESTIMATE WAS THE MINIMUM REQUIRED SEPARATION OF 6000 FT BTWN US AND THE LTT Y. I WAS NOT FAMILIAR WITH THE PERFORMANCE CHARACTERISTICS OF THE LTT Y; BUT BELIEVED IT WOULD EASILY OUTRUN OUR LTT X. I CLBED OUT AT THE RECOMMENDED PROFILE SPDS; BUT WAS SURPRISED TO FIND THAT WE WERE RAPIDLY OVERTAKING THE LTT Y. WE HAD BOTH BEEN ASSIGNED A 270 DEG HDG; AND I BEGAN A TURN TO THE R TO OVERTAKE THE LTT Y ON THE R SIDE. I ALSO STARTED TO CLB FASTER WHEN THE LTT Y SUDDENLY BEGAN A R TURN. THE LTT Y PLT TOLD ATC THAT HIS TCASII HAD SHOWN A TARGET CLOSING RAPIDLY FROM THE REAR; 200 FT BELOW AND 'CLBING AT AN EXTRAORDINARY RATE;' SO HE ELECTED TO TURN R AND CLR OUR PATH. HOWEVER; AT THE TIME HE TURNED; I WAS ALREADY AT LEAST 200 FT ABOVE HIM AND ON HIS R SIDE. THE LTT Y PLT STATED THAT HE DID NOT BELIEVE THAT I EVER HAD HIM IN SIGHT; BUT IT WAS APPARENT THAT ONCE HE WAS CLRED FOR TKOF; HE LOST ALL SITUATIONAL AWARENESS OF WHAT WAS HAPPENING BEHIND HIM. CONTRIBUTING FACTORS: 1) ATC'S FAILURE TO ASSIGN DIVERGING HDGS; 2) FAILURE OF THE LTT Y PLT TO REALIZE THAT WE HAD HIM IN SIGHT; 3) INADEQUATE OR MISINTERPRETED TCASII RA; 4) MY ASSUMPTION THAT THE BIGGER AIRPLANE WOULD BE FASTER.