The flight instructor reported that prior to his arrival, the student pilot conducted a preflight inspection of the airplane. The flight instructor then visually inspected the airplane and did not observe any anomalies. On the takeoff roll, with the student pilot at the controls, the left side of the engine cowling opened just after rotation. The flight instructor took control of the airplane and reduced the engine’s power setting to relieve stress on the cowling and abort the take off. However, the flight instructor determined that “was not the solution,” applied full power to the engine and elected to initiate a turn toward an alternate runway. The airplane continued to descend and was unable to reach the alternate runway. The flight instructor landed the airplane in muddy terrain off the runway. During the landing roll, the airplane’s right main landing gear separated, and the right wing impacted the surface, resulting in substantial damage to the fuselage and right wing. The flight instructor reported that there were no preaccident mechanical malfunctions or failures with the airplane that would have precluded normal operation.