MatH II Unit 1 Project Full Stop Ahead
I think that the skid marks didn't double in length because there is still gravity pushing against the car. And even though the skids are the doubled in size, the traveling speed doesn't have to because the car could have enough velocity to keep going until it came to a complete stop. I think a safe stopping distance between cars would vary on how fast you are going and what the road conditions are.
I think you should be traveling around 64-65 miles per hour to leave a safe stopping distance between you and a car 150 feet apart.