The student pilot was on a solo cross country flight with the intention of landing at two different local airports before returning to the flight academy base. While landing at her first planned stop, the airplane porpoised twice after touchdown. The student pilot stated that her airspeed was low on final and that she did not flare. After the full stop landing, she taxied back to the runway and took off again to fly to her next planned stop. After she landed the airplane at her second stop, the flight academy sent two flight instructors to recover the airplane and student. While conducting a preflight of the airplane to return to the academy, the instructors discovered substantial damage to the firewall.