According to the flight instructor (CFI), the student pilot completed one touch-and- go landing and proceeded around the pattern for a second landing. On the approach, the airplane was left of centerline and the CFI asked the student pilot to correct back to runway center line. The second landing was on centerline until touchdown. Upon touchdown the airplane veered to the right and the CFI directed the student pilot to not "fight" the airplane, and attempt to regain control. The CFI stated that the airplane was equipped with flight controls on the right side, but no rudder or brake pedals. The student pilot applied full power to the engine while heading to the right side of the runway. He attempted to takeoff before departing the right side of the runway; however; the airspeed was too slow to maintain flight. The airplane departed the runway on the right, collapsed the nose gear, and came to rest inverted in the grass. The student pilot stated that he should have used more rudder when lining up on the runway. Postaccident examination of the airplane by a Federal Aviation Administration inspector revealed bending of the left wing struts, flaperons, vertical stabilizer, and buckling of the fuselage behind the main cabin area. The CFI did not report any mechanical malfunctions or failures with the airplane that would have precluded normal operation. The CFI reported the wind at the airport, about the time of the accident, to be light and variable with no gusts.