PA-28 Cherokee/Archer/Dakota/Pillan/Warrior
Flight Crew Flight Instructor
Troubleshooting; Time Pressure
Aircraft Equipment Problem Critical; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Fuel Issue
Air Traffic Control Provided Assistance; Flight Crew Returned To Departure Airport; Flight Crew Requested ATC Assistance / Clarification; Flight Crew Landed in Emergency Condition
Aircraft; Human Factors
After taxiing to the ramp; I have instructed my student to switch tank from L to R as it is in his side (and asked my student if he did it according to the procedures of feeling the click while doing so and I checked the Fuel flow gauge which was good). After that we performed Run-Up according to checklist. Run-Up check was successful. We obtained a clearance from ZZZ Ground for a full stop taxi back and we got taxi instructions for Runway XXL. While taxiing; engine operation was good; all engine gauges in the green. As we were holding short on XXL we performed Before Takeoff Checklist and advised ZZZ Tower that we are ready for departure. Tower gave us 'cleared for takeoff Runway XXL; make right traffic for XXR' and we read back. While on ground roll; we applied full power; engine gauges in the green; airspeed alive and waiting for rotation speed. After rotation speed reached we pitched up for Vy climb attitude and got 79KTS of airspeed. At initial climb around 150 feet around Taxiway XX (confirmed later with Tower) we had a large drop of RPM; reaction - pitched down and prepared to land on little available runway or field grass straight ahead while maintaining control. I tried to troubleshoot; checked the mixture full rich and Fuel Pump on while adjusting the throttle. After a few seconds engine power came back and then I thought I can try find a better place to land or continue climb and perform additional troubleshoot but then we have experienced another big drop of RPM. I declared in the cockpit 'My Controls' and turned 45 degrees to the right [to] try to land on a more suitable field as I no longer had the option to land straight ahead safely. As I turned right I saw I had the option to make it to XXR so I [advised Tower]; 'We have engine roughness; want to land XXR'. And then two seconds later tower issued us a clearance to land on XXR. As I maneuver myself above XXR I applied full flaps and landed safely; vacated at the end of the runway. After vacating the runway I noticed that the Fuel Selector was a little bit above the right tank between R and L; 2/5 towards the right tank.
CFI reported landing shortly after takeoff due to a large RPM drop and engine roughness on initial climb out.