Post by
DocPlez »
https://forums.nicoclub.com/docplez-u45972.html
Sun Jan 07, 2007 6:23 am
TASK. Some drivers learn the rule of thumb, "Follow two car lengths behind for every 10 miles per hour." Others learn, "Stay two seconds behind the car ahead." Do these two rules give the same results? Is one safer than the other? Is one better for roads with speed limits of 45 or 50 miles per hour and another for highways on which the speed limit is 65 or 70 miles per hour?
COMMENTARY. Obtaining a driver's license has become one of the "rites of passage" in the U.S. On almost every written driver's test, applicants are asked how closely one driver should follow another on the highway. We all appreciate the dangers of tailgating--not enough stopping time and not enough space to avoid an accident. However, it is not clear that there is agreement about what actually constitutes tailgating--how far apart cars should be.
Rules of thumb are helpful guidelines--sometimes derived from experience--that are calculated using easily available measurements. Often they are developed under particular conditions and may be extremely inaccurate if those conditions are not fulfilled. The existence of two rules of thumb for the same situation suggests a natural question: Are the two rules simply two different ways of saying the same thing or are they offering different advice? As stated, the rules may provide visual images of how far to stay behind another car, but translating that understanding into practice on the road may be quite a different matter. The exercise of interpreting rules of thumb and comparing their results with real data could help students realize that the rules they use have implications for their actions. Also, there is the reality of high incidences of automobile accidents among new drivers. This exercise may help students examine and improve their driving habits.
In order to do the task, students need to know what it means to make a comparison. They have to identify the quantities needed in order to calculate the following distances given by the two rules and represent the rules mathematically. There are many ways to do this--written descriptions, tables, equations, or graphs, all basic tools of mathematical literacy. A comparison requires that the two representations use the same units of measurement--hence some conversions are necessary from the units used in the original rules of thumb. Such conversions are an essential part of many everyday situations, both at work and at home.
MATHEMATICAL ANALYSIS. To begin, students might be well advised to consider the case in which two automobiles are traveling at a steady rate. The information presented is not complete and students will find that they have to seek out missing data. Naturally, what students seek will depend on their interpretation of the task. One necessary piece of information may be average car length.
The units for the car-length rule are miles per hour and car lengths, and the units for the two-second rule are miles per hour and seconds. To compare the two rules, both need to be written in the same units. A typical sedan is about 14 feet, so the car-length rule might be translated as "follow about 28 feet behind for every 10 miles per hour" or as the equation y = 28(x/10), where x is the speed of the car in miles per hour and y is the following distance in feet.
If a car is traveling at x mph, then it travels x miles in one hour--in other words, x/3600 miles in one second. The two-second rule is then "if your speed is x mph, follow about 2x/3600 miles behind." As an equation, it is z = 2(x/3600), where x is again the speed of the car in miles per hour, but this time z is the following distance in miles (not feet as in the previous equation), and we use a different letter to distinguish it from y above.
Now the rules are both in terms of miles per hour and units of distance but not the same units of distance. The car-length rule is as follows:
y = 28(x/10),
where y is the following distance in feet. The two-second rule is
z = 2(x/3600),
where z is the following distance in miles. Simplifying the car-length rule gives
y = 2.8x,
where y is the following distance in feet. Simplifying the two-second rule gives
z = x/1800,
where z is the following distance in miles.
Now it's a matter of converting z to feet (or y to miles). There are 5,280 feet in a mile, so x/1800 miles is 5280(x/1800) feet. That's about 2.93x feet--very close to the distance given by the car-length rule!
Some driver's manuals give data on the distance cars travel before they are able to come to a complete stop. Often the distance is broken into two components, the reaction distance and the braking distance. The reaction distance is the distance traveled while the driver reacts to a situation and hits the brakes. The braking distance is the distance traveled from the time the brakes are applied until the car comes to a stop. A simplified version is given in Table 1.
--------------------------------------------------------------------------------
Table 1: Reaction and braking distances for various speeds Speed Reaction Distance Braking Distance 20 mph 20 feet 20 feet 30 mph 30 feet 45 feet 40 mph 40 feet 80 feet 50 mph 50 feet 125 feet 60 mph 60 feet 180 feet Bottomline ... too fast for road conditions and/or following too close. Sorry for the simple high school math but come on people ... calculus wouldn't have helped prevent this. If I recall correctly you were following 4 - 6 car lengths behind?? That's 80 to 120 feet if your consider and average car to be 20 feet ... in the case of our Versa that's what 52 to 78 feet?? On wet roads? Yes, Nissan may need to work the brakes a bit and the tires may not be Z rated for 200 miles per hour but know your cars limitations and drive accordingly and we'd be complaining about something else. I failt o see how the Versa is responsible for this at all.
Doc