Writing a letter of apology ?

Ok so I'm not the type of person to really ever talk smack. But not all people are like this so in some cases I get to a breaking point and snap on somebody. However it's not as if this in one word and I'm off. So recently I slammed a kid to the ground on my snowboard team because he was claiming he was going to mess with my board. This snowboard was the last thing my father bought me before he died last month. The thought of somebody touching that board makes me extremely mad. I do know the way I reacted was not the best but I still did it. Now my coach wants me to write a letter of apology to him. I already apologized to this kid and I ment it. But he turned around and threw it right back in my face. I really don't think he deserves any more of a apology but if I don't write this letter it can hurt any chances of riding with this team next season. I'm not a mean person in fact I got a big heart. So I just don't know if I should man up and write this letter. Because I really don't think it's going to fix any thing. The kid never touch the board but still that's low to even claim that. On top of this he's nothing but a spoiled rich brat. Doesn't have respect for me and alot of people. And I even waxed this kids board for him and didn't even get a thank you. More or less got a whatever screw you. Plus my coaches hold me to a higher standard than the other kids being I'm 18 and everyone else is 16 and younger. So does he really derserve any more of my time.