Who is asking: Student
Below is a diagram that shows the two cars, at time t = 0, just as the light changes, and at time t a short time later (your car is green, the other is red).
The length of your car is c miles. At time t the red car is travelling v miles per hour, you are travelling at v1 miles per hour, the red car's acceleration is r miles per hour squared and your acceleration is r1 miles per hours squared . Also the red car has travelled s miles and you have travelled s1 miles.
Since you must be "one car length away from the car in front of you for every 10 miles per hour", the distance from you to the red car is c v1/10 miles. Hence, from the diagram
Differentiating both sides with respect to t gives