INSTRUCTOR PLT AND STUDENT TOUCH DOWN AFTER TKOF IN JQF; BLOWING TIRE.

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I WAS CONDUCTING IOE FOR A NEW CAPT. ON A ROUTINE TKOF; WITH NEW CAPT FLYING; WE ROTATED AT VR. A FEW SECONDS LATER AS I WAS LOOKING OUTSIDE FOR A POSITIVE RATE THE CAPT RELAXED BACK PRESSURE AND WE BEGAN TO SINK. AS I GRABBED THE YOKE THE AIRPLANE HAD ALREADY RECONTACTED THE RWY ACCOMPANIED BY A VIBRATION. AFTER I GOT THE ACFT AIRBORNE AND PASSING 500 FT AGL I GAVE THE ACFT BACK TO THE PF (NEW CAPT) TO MONITOR AS WE RETURNED FOR LNDG. I KNEW WE BLEW A TIRE SO I DID NOT RETRACT GEAR. LNDG WAS UNEVENTFUL. EQUIP WAS CALLED OUT AS A PRECAUTION. NEW CAPT DOES NOT USE GAR BARS FOR TKOF ROTATION. OUR COMPANY IS NEW AND HAS NO SOP. WE TRAIN OUR CAPTS AS SINGLE-PLTS ALTHOUGH WE FLY ON THE LINE AS A 2 PLT CREW EXCLUSIVELY. CAPT TRAINING IS ONLY 2 DAYS LONG MIXED IN ACTUAL REVENUE TRIPS. NO SIMULATOR EXISTS FOR P180. TRAINING CAPT DOES NOT HAVE INSTRUCTOR RATING OF MEI. IOE CAPTS RECEIVE NO ADDITIONAL TRAINING OR GND TRAINING. P180 HAS NO WHEEL OR BRAKE SNUBBERS TO STOP WHEEL ROTATION ON RETRACTION; SO BRAKES MUST BE APPLIED BEFORE RETRACTION. STALL HORNS GO OFF WITH AUDIBLE AND VISUAL (RED CAUTION) WHEN THEY SHOULDN'T ON SOME OF OUR ACFT. CAPT EXPLAINED THAT WAS WHY HE ROTATED A MORE SHALLOW TKOF. ONLY 2 COMPANY MECHS HAVE FACTORY TRAINING FOR P180. CORRECTIVE ACTIONS: FIRST CORRECT ABOVE OBVIOUSLY AND IS ALREADY IN PROGRESS AS I WRITE THIS I AM TOLD BY NEWLY PLACED MGMNT STAFF. ONLY HAVE 'PNF' APPLY BRAKES. MORE THOROUGH TRAINING. IOE TRAINING. CRM TRAINING. CAPT SAYS HE WAS TAUGHT TO BRING UP GEAR IMMEDIATELY AFTER TKOF BECAUSE ACFT DOES NOT PERFORM ON 1 ENG WITH GEAR DOWN.