1. Vehicular Stopping Distance
Scenario Consider the following rule often given in driver education classes:
Allow one car length for every 10 miles of speed under normal driving conditions. One way to accomplish this is to use the 2-second rule for measuring the correct following distance no matter what your speed.
Question: How good is this rule?

2. Ultimate goal: test this rule and suggest another rule if it fails
Idea: Predict the vehicle’s total stopping distance as a function of its speed

3. Are the two rules the same?
Rule 1: Allow one car length for every 10 mph of speed
Rule 2: Allow 2 seconds between cars

4. Rule 1: One-car-length rule
The distance is proportional to the speed.
Assume a car is 15 ft long, then the equation of the distance D and the speed v, is
D= k v,
where k = 15 / 10 = 1.5.
Geometrical interpretation of the one-car-length rule

5. Simplify the model
Total stopping distance = reaction distance + braking distance
Data: Observed reaction and braking distances

6. Reaction distance Data: reaction distance vs. speed
Conclusion: Reaction distance is proportional to speed

7. Braking distance
Recall: kinetic energy = ½mv² where m: mass and v: velocity.
W = F d where W: work done by the brake, F: braking force, and d: braking distance.
From Newton’s second law, F = m a
Bottom line: d = kv²
See the data and graph

8. Model of the problem d = 1.1 v + 0.05 v² d: stopping distance v: speed
A graph of comparison