On June 7, 2002, at 0718 Pacific daylight time, a Piper PA-38-112, N49672, veered off the runway during a touch-and-go landing at Henderson (L15), Nevada. DSA, Inc., was operating the airplane under the provisions of 14 CFR Part 91. The certified flight instructor (CFI) pilot was not injured, and the student pilot sustained minor injuries; the airplane sustained substantial damage. The local instructional flight departed about 0700. Day visual meteorological conditions prevailed, and no flight plan had been filed. The CFI reported that the airplane was in the landing pattern practicing touch-and-go landings. On the third approach to landing, the student flared too high; the airplane contacted the ground hard and bounced. The CFI instructed the student to go around by applying full power and reducing the flaps to the first notch. The student applied full power, and raised the flaps to the full up position. The CFI then lowered the nose to gain airspeed. The airplane contacted the ground two more times before coming to rest on the left side of the runway. The right wing dug into dirt and rocks in the uneven terrain beside the runway. The wing and fuselage sustained substantial damage.