PA28 CFI AND CFI STUDENT PLT EXPERIENCE ENG PWR LOSS ON TKOF FROM 6TH PWR OFF APCH AND RUN OFF THE RWY END IN AN ATTEMPT TO LAND ON THE REMAINING RWY.

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WE REFUELED THE ACFT AND TOOK OFF AT ABOUT XA20 ON JUL/THU/04. WE COMPLETED ALL CHKLISTS AND HEADED OUT TO THE NW PRACTICE AREA. WE SWITCHED FUEL TANKS UPON ARR IN THE AREA AND BEGAN PRACTICING COMMERCIAL MANEUVERS FOR ABOUT 45 MINS. WE PICKED UP THE ATIS AND HEADED BACK TO ZZZ FOR SOME TOUCH-AND-GOES ON THE RWY. WE COMPLETED THE DSCNT CHKLIST; WHICH REQUIRES THE FUEL SELECTOR TO BE SWITCHED TO THE OTHER TANK; AND ENTERED A R DOWNWIND. WE DID 3 PWR OFF 180 DEG TURNS AND A SHORT-FIELD BEFORE SWITCHING TANKS AGAIN. WE DID 3 MORE PWR-OFF 180'S CLRING THE ENG EACH TIME. ON THE THIRD ONE; WE TOUCHED DOWN OK AND BEGAN A NORMAL CLBOUT. I APPLIED FULL PWR AND RETRACTED THE GEAR AND FLAPS AT POSITIVE RATE ACCORDING TO THE POH. AT ABOUT 200 FT AGL; I HEARD THE ENG SPOOL DOWN AND FELT THE ACFT LOSING THRUST AND SINKING. I LOWERED THE NOSE A LITTLE TO KEEP THE SPD UP IN ORDER TO AVOID A STALL AND INFORMED MY INSTRUCTOR OF THE PROB. HE TOOK THE CTLS AND VERIFIED THE PROB. AT THIS POINT WE WERE ABOUT 1/2 WAY OVER THE RWY. WE HAD BRIEFED EARLIER THAT WE WOULD LAND STRAIGHT AHEAD IN THIS SIT. MY INSTRUCTOR MADE A QUICK EMER CALL TO THE TWR NOTIFYING THEM THAT WE HAD LOST PWR. MY INSTRUCTOR THEN LOWERED THE LNDG GEAR AND TOUCHED DOWN ON RWY CTRLINE ABOUT 3/4 OF THE WAY DOWN. WE CONTINUED ROLLING PAST THE RWY AND ONTO THE DIRT. IT WAS SOFT DUE TO RAIN THE PREVIOUS DAYS. A LITTLE WAY INTO THE DIRT; THE NOSE GEAR COLLAPSED. A SHORT DISTANCE LATER THE R MAIN GEAR COLLAPSED CAUSING US TO SPIN TO THE R WHERE WE CAME TO REST. MY INSTRUCTOR SHUT DOWN THE AIRPLANE WHILE I OPENED THE DOOR. WE BOTH EXITED THE ACFT AND WAITED FOR THE TRUCKS. IT WAS ABOUT XC00. WE BELIEVE THE CIRCUMSTANCES LEADING TO THE ACCIDENT ARE THE TYPE OF TOUCH-AND-GOES WE WERE PERFORMING COMBINED WITH THE HIGH TEMP. THE TEMP AT THE TIME OF THE ACCIDENT WAS ABOUT 95 DEGS F (35 DEGS C); THE FIELD ELEVATION WAS APPROX 1500 FT MSL. GLIDING DOWN FROM TFC PATTERN ALT TO RWY TOUCHDOWN WITH PWR AT IDLE MAY HAVE CAUSED VAPOR LOCK. WHEN PWR WAS INCREASED TO FULL AVAILABLE AFTER TOUCHING DOWN; THE HT AND HORIZ DISTANCE OVER THE RWY AT WHICH WE IDENTED THE PROB PREVENTED THE ACFT FROM STOPPING IN THE AVAILABLE RWY RESULTING IN A CONTINUED GND ROLL PAST THE RWY. WE BELIEVE THIS CAN BE PREVENTED BY LIMITING THE NUMBER OF APCHS TO LNDGS WITH PWR AT IDLE.