The pilot reported he applied right rudder to correct for the airplane drifting to the left during his second touch and go landing. He stated that when he reduced the power to land, the nose of the airplane dropped, contacting the runway, and the airplane porpoised. The pilot regained control of the airplane and landed on the remaining runway. A postaccident examination revealed that the nose gear was pushed up resulting in substantial damage to the firewall. The pilot reported there were no failures/malfunction of the airplane.