The student pilot reported that he was flying a solo traffic pattern flight when during the final approach to land and configured with 20° of flaps, the airplane began drifting left of the runway centerline. The student pilot applied power to extend the touchdown point while trying to reestablish the airplane over the runway centerline when the airplane “dropped suddenly” onto the runway and began to porpoise. Before the student pilot could recover, the airplane “dropped heavily” a second time, resulting in the nose wheel separating and the airplane departing the left side of the runway. The airplane came to rest on the lower fuselage in the grass. The airplane sustained substantial damage to the forward and bottom fuselage. The student pilot reported there were no preimpact mechanical malfunctions or failures with the airframe and engine that would have precluded normal operation. The student pilot reported that while in the traffic pattern, the air traffic controller tried to reach him three times with no response. When the student pilot did respond, the controller admonished the student pilot for not paying attention to the radio calls. The student pilot reported that he likely flared too high over the runway and that the admonishment from the controller was “unsettling” and that he should have executed a go-around rather than attempt to land on that approach.