Careless Driving: Careless Driving for Tailing a Cop

My question involves a careless driving ticket from the State of: South Carolina
Hey I received a careless driving ticket in sc with a nc licenses. I'm only 16 years old and had my licenses for about 6 months. While I was driving in a speed limit of 25mph behind a cop the speed limit rose to 35mph as I was going to accelerate i noticed the cop didn't. So as i notice the cop kept speeding up and slowing down, so i backed all the way back like 2 cars. He pulled off (without a signal). I then proceeded through the light and thats when he flashed me. When he got to the car he said, "You were following me too close back there". I said what are you serious so I gave him my things and another cop came and stood on the other side of my car and waited. So as in drivers ed I was taught the 3 second rule (when the car in front of you passes something then 3 seconds later if you pass that same thing you are at a good following distance). I was a this small town and I wasn’t tailing him but he obviously thought I was when he pulled me over. He gave me a $240 ticket for something I don't even know that or what I do wrong. I looked up careless operation and it says putting others in harm. I didn't do this at all I didnt pull anyone in harms way. I’m planning on fighting this any tips on what to do? I getting a signed letter from my drivers ed teacher explaining the 3 second rule and also explaining what tailgating is. He said I violated section 2004-43 and offense code 94. Will this be thrown out because I don’t have that kind of money. I expect to pay nonthing because I didn't commit a crime going against my drivers ed hand book.