On July 8, 1994, about 1530 eastern daylight time, N63021, a Cessna 172P airplane, a training flight, overran the side of the runway during landing at Leesburg Airport, Leesburg, Virginia. Visual meteorological conditions existed. The student pilot, the sole occupant, received no injuries. The airplane was destroyed. The flight was operated under 14 CFR Part 91. The student pilot had 9 hours of solo time, and was returning from a solo cross country flight to Charlottesville, Virginia. After landing at Charlottseville, he had lunch and departed for Leesburg Airport. He reported no problems on the return flight to Leesburg. On final approach to runway 17, the student pilot saw the aircraft in front of him still on the runway. He initiated a go- around, and went around the pattern again with no difficulties. According to the student pilot, as he was on the landing roll out the plane veered left. "Once I touched down the plane turned to the left. Thinking I was still in the air flying, I added power to go around. In the process the plane went off the runway, across the grass and hit the taxiway. So I pulled back on the wheel, I stalled the plane and I ended up in the grass nose down. The pilot reported no mechanical problems.