Sudden shutdowns are almost always due to overheating. Especially if it happens a consistent amount of time after turning on the computer. Get a can of compressed air, take the laptop outside, and blast the air into all the vents. Blast it carefully into the fan vent (don't let the fan spin up too fast). If you see huge dust plumes billow out of the laptop, keep blasting until no more dust comes out. Then take it back in and try running it again. it should work. The dust was blocking airflow, causing the laptop to overheat.
If you don't hear the fan spin up from the compressed air, turn on the laptop and put your hand over the fan vent. See if you can feel air coming out. If no air is coming out, either the fan is dead and...