Show your mom that you are taking her concerns seriously and that you have thought about it. If you haven't really thought about it, then you haven't planned sufficiently.

Make her part of the process. Write down what her specific concerns are and then figure out what it will take to convince her that you will be safe.

If it's about being hit by a car, show her the statistics regarding car-bike accidents and how they generally happen (at night with no lights, bicycle riding on wrong side of road or on sidewalk) and show her that you know how to ride aroud cars in a safe manner. Also, show her that you've done research on where you can find safe bike routes that are low traffic.

If it's that she doesn't want you that far from home, maybe you could plan a shorter route that is essentially a big circle around your house so that you are never so far from home that she couldn't come get you if something went wrong. Remember, involve her in the process.

If it's about camping by yourself, find out what it will take for her to feel ok about camping - go on a car-camping trip with her so she will see that there aren't any wackos out there, bike to a relative's house and camp in the backyard one weekend, or something like that. Maybe you could do some research about crime in state parks.

The main thing is to show her that you appreciate her position and that you value her input. I worry that it may be too late because it sounds like this is already an adversarial topic. If you value your familial relationship at all, you shouldn't just disappear one day. There's no way to know how that will turn out, but it doesn't sound like it will end with a healthy relationship.