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?

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

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

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

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

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

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

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