Why, after falling, is this iMac cycling on and off?

I agree with Sterling, this sounds like your heatsink could have come loose. I'm not sure if this is what you mean Forrest, but what we suspect is happening is that the heatsink is no longer making a good contact with your processor. The job of the heatsink is to remove as much heat as possible from the processor to prevent it from overheating.

Because the heatsink is likely no longer attached properly to the processor, the first time you boot your computer up, it takes a few minutes for it to get too hot, at which point it will auto-shut off the whole computer to stop itself from melting. The next time you start up the computer, the processor is already a bit warm, so it takes it less time to get to a critical temperature where it will shut itself down. Every time you turn on the computer, it will start off a bit warmer than the previous time.

I've had this problem on a PC of mine in the past, however the problem there was dust on the inside clogging the fan and preventing it from working properly.

I suggest you follow this guide to open your computer and re-seat the heatsink / cpu fan, applying arctic silver if necessary. Make sure you dust out the computer while you're in there, and you should be good to go. :)