The engine coolant temperature can't change that fast so it's doubtful you have an overheating problem. It's more likely the wire going to the coolant temperature sensor is rubbed through and grounding out against something metal. You can verify that by unplugging the sensor. Try to do that when it's pegged at 240 degrees. If the gauge stays there when the sensor is unplugged, the wire has to be grounded.
On an '02 model there should be two coolant temperature sensors. One will have two wires. That's for the Engine Computer. You want the one with a single wire for the gauge. Both of them are usually near the thermostat housing at the end of the upper radiator hose.
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Wednesday, June 5th, 2013 AT 9:05 PM