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There is no such thing as a BIG emergency and a little emergency. If there is a problem, there is a problem.
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I'd have to disagree.....engine failure at 8000 AGL on a clear VMC day with plenty of nice, long big green fields or an airport right below me is not a "BIG" emergency. It definitely comes with a pucker factor and should not be taken lightly, but it's very survivable, and you know you're gonna live through it if you don't make a big mistake. I wouldn't even squawk 7700 in that situation. I might call out "pan pan" on frequency, but would not see a need to transmit mayday. I would consider letting a commercial student retain the flight controls for training and experiences sake, just to see how they would handle themselves, but definitely would take the controls immediately from a student pilot. I would take over from even the advanced student when it got to crunch time below 2000 AGL or so.
I once had a an engine cowl fly open on the takeoff roll in a twin.....my CFI gave me a few seconds to see if I would abort the takeoff before on my own he shouted "CLOSE THE THROTTLES". I learned a lot more that way than I would've had my CFI immediately reached over and closed the throttles on his own, and considering the 12,000 ft runway we were on, safety was not compromised....
An Engine Failure at less than 3000 AGL with oil on the windshiled, smoke in the cockpit, flames coming out of the cowling, and the engine rattling like it's about to come off its mounts is a "BIG" emergency. Losing pitch, roll, or yaw control is a "BIG" emergency. These are situations where I would immediately tell my student "I have the flight controls".