Why did our car's brakes fail (on black ice)?

sorry, but you're absolutely wrong. the brakes didn't fail at all, they did exactly what they're designed to do. black ice usually contains oil deposits and has no friction whatsoever. it doesn't matter whether you have ABS or not, the only thing that can help you on black ice is a set of studded tires(and even then you're still in bad shape). basically when your mom slammed on the brakes, it sent the car into a slide because the front wheels dug in slightly more than the rear, which caused the rear wheels to swing around and send the car sideways. in these situations, it's best to hold the wheel straight and coast across the ice, then apply the brakes after you've driven over the ice.