CRJ200 WARNING SYSTEM ACTIVATED ON ROTATION. A COLLECTIVE DECISION WAS MADE WITH THE COMPANY TO CONTINUE TO DEST. MAIN GEAR TIRE DAMAGE WAS DISCOVERED AFTER LNDG.

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WE DEPARTED RWY XXR AT ZZZ1. AT ROTATION; A 'WOW INPUT' CAUTION MESSAGE WAS RECEIVED. WE RAN THE QRH; CALLED DISPATCH; CALLED MAINT; AND MADE A COLLECTIVE DECISION TO CONTINUE TO ZZZ2. THE APCH AND LNDG IN ZZZ2 WERE NORMAL. AT FIRST; BRAKING WAS NORMAL; BUT BECAME MORE DIFFICULT AS THE ACFT WAS SLOWED. I INITIALLY ATTRIBUTED THIS TO THE WET RWY IN ZZZ2. WE CLRED THE RWY WITH ABOUT 1000 FT REMAINING AND THE TAXI-IN WAS UNEVENTFUL. ON THE POSTFLT INSPECTION; A FLAT AND DAMAGED TIRE WAS DISCOVERED ON THE R MAIN INBOARD. ABOUT A 4 INCH SECTION OF THE R INBOARD FLAP WAS ALSO DAMAGED. THIS WAS THE FIRST SIGN OF ANY DAMAGE KNOWN TO THE CREW. THE DAMAGE MAY HAVE OCCURRED ON EITHER TKOF OR LNDG. HOWEVER; I NOW ASSUME THAT IT OCCURRED ON TKOF BECAUSE OF THE 'WOW INPUT' CAUTION MESSAGE. NO FOD WAS OBSERVED ON TKOF. A CONTRIBUTING FACTOR IS THE ROUGH CONDITION OF RWY XXR IN ZZZ1. THIS ADDS TO THE ALREADY HIGH NOISE LEVEL IN THE COCKPIT AT ROTATION. I STATED TO THE FO AFTER TKOF THAT I MAY HAVE HEARD A POP. THE FO DOESN'T RECALL HEARING ANYTHING BECAUSE OF THE HIGH AMBIENT NOISE. ALSO; THE QRH FOCUSES ON THE POSSIBILITY OF A NOSEWHEEL STEERING PROB. THEREFORE; OUR ATTN WAS ON THE NOSE GEAR RATHER THAN THE MAIN GEAR. IN ADDITION; I CERTAINLY WOULD HAVE DECLARED AN EMER HAD I BEEN AWARE OF THE DAMAGE. SUPPLEMENTAL INFO FROM ACN 776315: I WOULD SUBMIT THAT EVEN IF WE DID KNOW THE TIRE HAD FAILED; AN IMMEDIATE LNDG WOULD HAVE BEEN UNDESIRABLE AS WE WOULD BE LNDG AT WELL OVER 5000 LBS ABOVE MAX LNDG WT WITH A DAMAGED TIRE. PERHAPS INCORPORATING A TIRE PRESSURE MONITORING SYS MAY BE BENEFICIAL IN THE FUTURE; THOUGH I DON'T THINK IT WOULD HAVE HELPED MUCH HERE. CALLBACK CONVERSATION WITH RPTR ACN 776480 REVEALED THE FOLLOWING INFO: THE 'WOW INPUT' WAS DISPLAYED ON EICAS AND APPEARS WHEN AN IMPROPER WEIGHT-ON-WHEELS INPUT IS DETECTED. NORMALLY THE GEAR AND FLAP EICAS INDICATIONS DISAPPEAR AFTER TKOF WHEN THE GEAR IS UP AND LOCKED AND THE FLAPS ARE RETRACTED.