BE23 STUDENT AND INSTRUCTOR EXPERIENCE RUNWAY EXCURSION AFTER AN UNSTABLE APCH IN TURBULENT WEATHER.

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I ARRIVED AT AUGUST MUNICIPAL ARPT AT XA35 AM. I LOOKED AT THE WX RADAR ON-LINE AND NOTED A LOT OF PRECIP W; E; AND N OF WICHITA. I PERFORMED A NORMAL PREFLT AT APPROX XA45 AM. I PERFORMED A NORMAL ENG START AND THE WX AT JABARA WAS CHKED AS WE TAXIED TO THE S END OF THE TXWY. I DON'T RECALL THE PRECISE WX NUMBERS BUT THE WIND WAS FROM ABOUT 50 DEGS AT 15-18 KTS. I TOOK OFF ON RWY 36 AND KEPT MY DEP LEG CLOSE TO THE ARPT PER MY INSTRUCTOR'S ADVICE TO STAY S OF THE WX. LIGHTNING WAS OBSERVED TO THE N BTWN THE ARPT AND EL DORADO. ON DOWNWIND I PERFORMED MY PRE-LNDG CHKLIST INCLUDING SEATBELTS; CARB HEAT ON; FUEL ON THE FULLEST TANK; BOOST PUMP ON; MIXTURE FULL RICH; PWR TO 2100 RPM. THE AIRPLANE WAS A LITTLE HIGH AND MY PATTERN WAS CLOSE IN TO THE ARPT SO MY INSTRUCTOR HAD ME EXTEND MY DOWNWIND LEG. I THEN PUT IN 1 NOTCH OF FLAPS AND BEGAN MY DSCNT. UPON TURNING BASE I WAS STILL HIGH AND MY INSTRUCTOR TURNED ON THE PAPI APCH SLOPE LIGHTS. THE LIGHTS WERE WHITE OVER WHITE INITIALLY. I PULLED THE PWR BACK TO ABOUT 1500 RPM AND ACHIEVED PROPER GS; RED OVER WHITE. MY AIRSPD WAS ABOUT 68 KTS. IT WAS VERY BUMPY ON FINAL WITH A LOT OF XWIND. I DID NOT EXTEND THE FLAPS ANY FURTHER MAINLY BECAUSE I WAS BUSY JUST TRYING TO KEEP THE AIRPLANE ALIGNED WITH THE RWY. AS I THOUGHT I HAD THE RWY MADE I PULLED PWR TO FAST IDLE; MAYBE 800 RPM. JUST AS I CROSSED THE END OF THE RWY THE AIRPLANE DROPPED SUDDENLY. MY INSTRUCTOR TOOK THE YOKE AND PUSHED. WE HIT THE RWY HARD AND BOUNCED. UPON HITTING THE RWY A SECOND TIME WE NO LONGER HAD DIRECTIONAL CTL AND VEERED OFF THE L SIDE OF THE RWY. AS WE CAME TO A STOP ON THE GRASS THE ENG WAS STILL RUNNING AND THE NOSEWHEEL COLLAPSED KILLING THE ENG. THIS HAPPENED AT ABOUT XB10 AND XB15 AM. AS I WALKED BACK TO THE ARPT OFFICE I NOTICED THE WINDSOCK CHANGING FROM LIMP TO STRAIGHT OUT REPEATEDLY AND VARYING IN DIRECTION BY ABOUT 45 DEGS. WHEN I GOT BACK TO THE OFFICE I NOTICED THE WX RPTING EQUIP SHOWING THE WIND VARYING FROM 13-20 KTS AND FROM 70-100 DEGS IN DIRECTION.