Well pump shuts off on it's own

It I turn on the sprinklers and just let them run so it is in a constant flow mode for the well, when the pump comes on it runs for 15 minutes and shuts off. It stays off for exactly one minute, then turns back on. The control box is new and it did the same thing with the old box. I have no idea the condition of the pump as it was in the ground when we bought the place. I do know it is about 15 years old. Do the pumps have thermal shutdown capability in them and maybe that is what this one is doing? Has good pressure and always has plenty of water as far as I can tell.

I expect that the only smart thing to do is replace it before it dies in the middle of winter.... It is 300' down, water table is at about 100'. 6 inch casing. Looking at the specs of most pumps, I think I'm looking at 1 1/2 HP.

Just before it shuts off, what is the water pressure gauge showing?

Yes the majority of pumps have a thermal overload protection. And with full flow, the water level in the well may be drawn down far enough to make the pump overheat and shut off.

The cure for that is to reduce the flow.

Pumps come in two parts, wet end (rated by gpm) and motor (rated as hp). You size a pump by calculating the max gpm it has to deliver and then the horses to get the job done.