According to the student pilot, the first landing attempt of the solo cross country flight resulted in a go-around. During his second attempt, the student pilot landed the airplane about one-third of the way down the 5,188-foot runway, "hard but smooth (not a bounce);" however, the airplane subsequently became airborne. The student pilot landed the airplane once more, and although he believed the “plane had safely and completely landed,” it became airborne yet again. At that point, the student pilot felt a go-around was not possible, and "pulled back" on the flight controls. The airplane then "probably stalled and dropped to the runway," the nose landing gear collapsed, and damage to the engine firewall and left wing resulted. Winds at the time were variable at 4 knots. No preaccident mechanical anomalies were noted by the student pilot.