I think it'd be okay as long as you have your oil changed and the car checked out (pre-trip inspection) beforehand, have your up-to-date AAA card, and make sure to stop every couple-3 hours for stretching and a rest of about 10-15 minutes. Cell phone is a must, too. Also, check your oil every time you stop for gas.

Gas is a real issue in Atlanta area, which includes Chattanooga. It's very expensive and sometimes hard to find, so plan to fill up at half-tank; don't wait until you're empty. My mother and brother live in western NC and have had trouble finding gas, and when they do it's about $4.50 for regular (it's $3.50 here in central Virginia and there are no shortages). The gas has also been rationed over the past month, so they were not able to fill up when they did find a station (only allowed $30, which isn't very much at $4.50).

I have a friend who drove from Austin to North Carolina all by herself to see me years ago. It was the first time she had ever been out of Texas (she was 20)! I was so proud of her!

Have a good time!