A common laptop fail is the power jack being snapped off the MB (e.g. by someone kicking the power cord while it is plugged in). If the jack is loose but not completely snapped it can short/overheat or cause the type of power-down you are seeing.
With the laptop powered down and the power supply unplugged insert the charging cable. Is there any play in the connection that wasn't there previously?
If so there are places that resolder the jacks for around $100. Take your HDD out before you send.
good luck.
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Other possibilities - the power brick has failed. If you have another brick try it. If not buying an adjustable voltage car kit for $40-$60 is not a bad investment. Try it with this PC, if this laptop is dead and you end up with a different one the card kits can be used with the new laptop. Make sure you get one that puts out enough wattage, don't go cheap.