Q: Macbook pro energy problem

Hi

i have a macbook pro a1278

and today i was working normally but then my magsafe turned off (the light on the cable)

and i tried to unplug it and plug it back but didnt work , so i disconected from the wall and waited a few secods and then i pluged it back and it started charging but just for like 10 seconds and went off again, i thought it was a battery problem so i unpluged the battery and conected the magsafe again to my macbook and it worked nice ,then i turned of my macbook and plug the battery back , i turned it on and happend the same thing the charger only works like for 10 seconds and turns off ,  so i think its a battery problem , what you think? should i change my battery?