this is what happened, is this what I should expect?

traveling at 45mph @45 ft behind car in front. I react quickly to brake lights, plenty of time to stop. my screeching tires keep on screeching,though, and soon i am watching my front end crunching in front of me. I realize that the car behind me was on my bumper all the way in to the car i eventually impacted

there was relatively minor damage to the car i hit (maybe a replaced bumper and minor ding repair) as well as my rear bumper, and the car behind's front end. pretty much destroyed my front end. Is this what i should expect if I am essentially braking for two cars, ultimately resulting in a quite low speed impact
(no one was hurt)

I'm puzzled and don't feel i was at fault, therefore wanted some educated opinions.
Any help would be GREATLY appreciated!

thanks
ted putnam