2008 Hyundai Matrix unpredictable loss of power

Car starts and drives perfectly. Several kilometres into the journey (50 km plus), the car starts to lose power. The car then starts to give a periodic (frequent ) jerk as if the engine momentarily cuts out. This sometimes accompanied by back firing. The car continues to lose power until we are forced to pull over. The engine may or may not cut out! After waiting for several minutes the car will restart and one can proceed for a few kilometres before the problem recurs. This repeats over and over. Waiting for an extended time seems to clear the problem, and one can then drive for a fairly lengthy distance. Just to clarify I will describe our experience in the last three days. Day 1 - drove 100km , car was parked for about 6 hours, was driven for 40 km, parked for 1 hour and then driven for 30 km before the problem occurred. After sitting for about 20 minutes the car was driven for another 30 km with no problems. Day 2 - Car was driven 100 km with no problems, parked for 5 hours, driven for 70 km when problem started. After sitting for about 20 minutes the car was started and driven for 30 km with no further problems. Day three - car was driven for 60 km when problem started. Limped on for 20 km with 2 short stops. After a stop of about 5 minutes the car was driven for 25 km with no further problems.