If you haven't seen it already, this article is worth a read if you're going to be commuting by bike, although if you've been cycling since the accident, you may already have found some of it out.

As for convincing your family, most of the arguments they're coming up with are understandable, as they're not commuters themselves. However, I have to commute in the worst weather England can throw at me, as I don't have a car, and I'm never too tired to work after my commute, so as someone who's actually tried it, I can refute both those points and say that it's perfectly possible to ride to work when it's raining at 45 degrees to the ground and then do a day's work afterwards without being too tired.