The private pilot reported that after the airplane rotated for takeoff, the nose pitched downward and the airplane bounced on the runway. He stated that after the first bounce, the flight instructor took control and the airplane bounced several more times before coming to rest on its nose. The flight instructor reported that pre-flight and pre-takeoff checks were performed, no airplane deficiencies were noted, and the elevator trim was set for takeoff. He added that he took control after the first bounce and attempted to abort the takeoff. The airplane bounced violently four or five more times as he attempted to stop the airplane by retarding the throttle, applying up elevator for aerodynamic braking, and trying to keep the nosewheel off of the ground. Examination of the airplane after the accident revealed that the elevator trim was in the full nose down position. The airplane’s nose landing gear had collapsed and the firewall and fuselage had sustained damage. The lower portion of the fuselage/firewall was pushed rearward into the pilot compartment displacing the rudder pedals. Tests of the elevator trim system revealed no anomalies. Examination of the airplane's control system revealed no anomalies that existed prior to the accident. Federal Aviation Administration publication FAA-H-8083-3A, “Airplane Flying Handbook” stated the following in regard to recovery from a bounced landing: “The corrective action for a bounce is the same as for ballooning and similarly depends on its severity. When it is very slight and there is no extreme change in the airplane’s pitch attitude, a follow-up landing may be executed by applying sufficient power to cushion the subsequent touchdown, and smoothly adjusting the pitch to the proper touchdown attitude.”