safeway is right, it is your responsibility to maintain the legal safe distance apart and if you plough into a vehicle in front, no matter how hard they brake or what they may be doing, legally, the collision is deemed your fault.
All of us drive too close, too fast and are therefore driving dangerously and yes, our roads would come to a standstill if everyone adhered to the rules but we are all taking risks by keeping up with the flow and 'protecting' our bit of road from someone jumping in front of us. Accept your fault and hopefully you will always be prepared for the unexpected in future. Main thing is that you and your child are ok