During his cross country flight, the student pilot diverted to an unfamiliar airport due to deteriorating weather conditions. The pilot was attempting to land and elected to touch down long because of the presence of a road near the approach end of the runway. The pilot reported that he flared too aggressively and the airplane bounced after touching down and yawed to the right. The pilot applied left rudder to try and correct the airplanes yaw and the nose gear collapsed when the airplane touched down the second time. Subsequently the airplane slid off the runway to the right. A post accident inspection of the elevator showed that it had been substantially damaged. The pilot reported that there were no preimpact mechanical malfunctions or failures with the airframe and engine that would have precluded normal flight.