The sport pilot reported that he was landing on an airfield and overflew the runway multiple times searching for a wind sock. He watched another airplane land to the south, and he made his approach to the south also. He reported that his altitude was too high during his first approach, so he aborted the landing. He asserted that, "This should have alerted me that there was a tailwind." He flew one traffic pattern and made a second approach with the power at idle and full flaps. He established his approach, but the airplane sank quickly and touched down short of the runway and nosed over. The airplane sustained substantial damage to the canopy, the vertical stabilizer, and the rudder. The pilot reported that there were no preaccident mechanical malfunctions or failures with the airplane that would have precluded normal operation. According to the Airport Facility Directory, the airfield was equipped with a wind indicator. However, no wind indicator could be located on or near the airfield by the National Transportation Safety Board investigator-in-charge.