You may want to sit down with your mother and let her know that a lot more research has been done on vegetarian diets over the last few decades, and there is now "proof" that becoming vegetarian does no harm and in fact can definitely improve one's health.
You can ask her about what specific concerns she has. For example, of her concern is that you "won't get enough protein," you can get information on the internet about how much protein someone of your age needs, and find out how a vegetarian diet will meet this need, and show her.
Education is the key here. You can educate both her and you. Tell her that you and she can both explore and research this together, for example. That will bring her on your side.
Also, I suggest that as you are now 14, that you might be interested in starting to help cook dinner. That way, you can learn to prepare vegetarian foods and help her out at the same time. I am certain that she will appreciate it. (I was 14 when i began making meals for my family -- i loved Chinese food and my family never bought or made it, so i started cooking it at home).
Good luck and let us know how it goes.