During a full stop landing at an airport on the second leg of a multiple leg cross country flight, the student pilot landed, the airplane bounced, and then touched down again, hard on the nose landing gear. The student pilot then lost directional control, the airplane drifted towards the left side of the runway, and then came to rest. Examination of the airplane by a Federal Aviation Administration inspector revealed that the airplane had incurred substantial damage. The tip of one propeller blade was bent, the nose wheel landing gear tire was flat, the cabin floor was wrinkled, the belly skin of the fuselage was wrinkled, and the firewall was buckled. The student pilot was uninjured, and he advised that he had no mechanical problems prior to the accident.