You cannot "completely" understand your family's concerns until after you become an officer. You are about to be exposed to people who live on the margins of society, and no amount of training can fully prepare you for this. It's not simply a belief that you will get physically hurt, but also about how it changes you as a person.
But, since it is your choice, they should be supportive of you, even if they do not agree. They should no longer be trying to talk you out of it, but rather, encouraging you. In fact, many departments will talk to friends, family, neighbors, school teachers, etc as part of the background investigation. If they learn that your immediate family is not supportive of your decision, it could adversely influence their decision to hire you - especially since you live in the same house with those who are not supportive, and will be seeing them every day. I suggest you have this discussion with your family right now, in a mature way. You need their support!
As to conflict at home after you start working - you will need to learn to "compartmentalize." Think about work when at work, and home when at home. You cannot be preoccupied with personal problems while on patrol - it will cause you to make mistakes, not see/hear something important, make bad decisions, etc. If you have children, you will need to have it set up to where someone else can pick them up from daycare/school if they are sick. It's all about having a plan for how to deal with things at home so they don't require your attention. Don't get me wrong - if your Mom has a heart attack and goes to the hospital, obviously you are going to go. It's the little things that need to be dealt with - and some people have lots more little things than others!
During the applicant processing stage, you may be asked how you deal with stress. This is an important question. People who do not recognize when they are stressed, or who internalize it rather than dealing with it, are not considered good police candidates. Since you are engaged in weekly training program right now, that sort of activity is something you should continue - physical activity helps to reduce stress. Alcohol is not the answer, and, they will want to know about your drinking habits as well. (Do not lie about anything - remember, they are going to talk to others about you.) Exercise, meditation, gardening, building model railroad layouts, (hobbies) etc are all good stress relievers.
It's cool that you want to be a cop. I wish you the best in your career! Feel free to contact me if you have additional questions.