36 minute layover, but what if the plane is delayed?

I'm flying from kansas city to san francisco but I have a 36 minute layover in salt lake city. I was just kind of wondering what would happen if the plane from kc to sl was delayed and I missed my flight to san Francisco? Would I be able to get on another plane? or would I have to buy a new ticket. My sister and I are flying together. I'm 15 and she is 14. So that's really the only thing I'm worried about. Also Will we be considered UM's? We will be flying delta on skyway airlines. I was an UM when I flew a few years ago and it sucked getting on the plane last, getting off the plane last, and those freaking people won't leave you alone! Anyway Thanks! =]]