Is water pump going bad

i put coolant in resivoir(plastic jug) it goes empty notice that coolant sprew everywhere close to raditor, and on other parts under the hood, is this a sign water pump going bad, temp signal, go up but never overheating position. need answer asap. my car 2004 cherolet impala, oh yes after turn off car i can feel the heat.

It means that you have a coolant leak.

I have no idea what you mean about feeling after you turn the car off. When you run the car it gets hot. When you shut it off you can feel heat in any number of places.

Try starting the engine from cold with the hood up.
Then watch for where the coolant comes out as it warms up.

Bad hose, cap or radiator. Like circuitsmith said. Fill it up and run it until it at normal operating temp. Look for leaks. Also, if you do not see them while car is running, turn it off and look some more. Pressures will build after the car turns off and you my find your leak then.

I would inspect all conections and put a pressure tester on the raditor to check for leaks

I’d start with a visual while the engine is operating, followed by a pressure test if the visual isn’t productive.
I’m betting your water pump is history.