I'm 50 years old and married with two sons only slightly older than you are. I can see the issue from both your mother's point of view and your own. My wife is a reluctant supporter of my bicycle touring with precisely the same concerns your mother has for your safety. If one of my sons told me he wanted to make a bike tour, I'd have the same concerns as your mother as well.

I understand there are some who are willing to strain or damage a relationship with their parents over the issue of (over) protectiveness. I'm certainly not willing to do that with my wife and I'm very happy that I've managed to work through various disagreements with my parents and my sons without damaging a long term relationship.

I'd suggest working to show your mother you understand her concerns and you'll do what you can to address them. You'd also like her to understand your goals/desires and you'd like her to be willing to work with you to address those as well. It's not productive to promise that you won't get hit by a car. You and your mom both know that it is, indeed, possible no matter what. You need to accept her concern and work with her on ways to mitigate it. Is there an interesting trip you can take where most of the riding is on bikeways? I ride 20 to 25 miles almost every day on lightly traveled roads close to my home. I'd want to assure my wife or my mom that I have no interest in a daredevil, high risk bike ride. I plan tour routes with safety in mind and my wife can see effort that goes into planning a route that is lightly populated with cars (and especially trucks). Over time, you build trust with the people who care about you that you're also concerned for your own safety and you make decent plans to see that you return home in one piece.

Are there some shorter trips you can take, beginning with day trips and single overnight trips, that she would feel better about? Sometimes, a parent's unstated concern is that you don't really know what you're getting into. Working up from shorter trips closer to home changes the request from "I want to do this really big thing I've never done before" to "I want to do this thing I've done successfully lots of times on a bit larger scale". As a parent, I'm much more comfortable with the latter approach.

My wife feels some reassurance that I always have my cell phone with me and I call her regularly (at least daily) to let her know where I'm at and how I'm doing. I encourage her to call me for even the smallest thing. Sometimes, the biggest fear of a parent or spouse is the thought of you laying injured beside the road and them not knowing about it (even if there is virtually nothing they can do to help). Frequent communication can help resolve that fear immensely.