James R
James R. Lee
Math 6161C
No Tailgating in Georgia!
NO TAILGATING IN GEORGIA!
This activity is designed to facilitate student’s understanding of uniform motion problems through the study of distance-rate-time relationships. Students will acquire experience in unit analysis which will give them a better understanding of the magnitude of speed. In the situations presented in this activity, speed is converted from miles per hour to feet per second in order to make comparisons that are more familiar to the student. Interest in driving an automobile is the motivating force that will captivate the student’s attention throughout this activity.
This activity is intended for early high school students who are or will be driving soon and is easy to moderately difficult so as to include most students.
NO TAILGATING IN GEORGIA!
Did you know that following too closely, otherwise known as tailgating, is the leading cause of accidents that involve rear-end collisions? The Georgia Department of Driver Services (DDS) has issued some guidelines for the proper distance between vehicles while traveling on the highways. The following is an excerpt from the DDS driver’s manual.
Following Too Closely
Most rear-end collisions are caused by following too closely. When following another vehicle on any street or highway, you must stay far enough behind to enable you to stop if the other vehicle suddenly slows down or stops. Watch the car ahead of you: when it passes some reference point, such as a telephone pole, count “one-thousand-one, one-thousand-two”. If you pass the same spot before you are through counting, you are following too closely. Always add additional seconds for other hazards such as inclement weather.
In other words, DDS suggests that you keep at least a two second interval between you and the car in front of you. We are accustomed to speeds in miles per hour, however, for this activity we will need to convert them into feet per second in order to determine the safe distance at varying speeds. Let’s begin our investigation with a slow speed, say fifteen miles per hour.
Exercise 1: To convert from miles per hour to feet per second, fill in the blanks below and then use unit analysis to divide out the common units.
[pic]=[pic]
=[pic]
=[pic]
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=___________ feet per second
Exercise 2: Repeat this process for increasing speeds in five mile per hour increments up to and including 75 miles per hour. Record your data in the table below. It may be necessary to round to the nearest whole number.
Miles Per Hour |15 |20 |25 |30 |35 |40 |45 |50 |55 |60 |65 |70 |75 | |Feet Per Second | | | | | | | | | | | | | | | | | | | | |TABLE 1 | | | | | | | |
Exercise 3: Determine the safe following distance for each of the speeds and record the data in Table 2. Hint: You have determined the speed in feet per second and you want a two-second interval.
Speed(mph) |15 |20 |25 |30 |35 |40 |45 |50 |55 |60 |65 |70 |75 | |Speed(fps) | | | | | | | | | | | | | | |Safe Distance(feet) | | | | | | | | | | | | | | | | | | | |TABLE 2 | | | | | | | | |
Exercise 4: Plot the data from Table 2 on the graph on the following page to show the safe following distance in feet at a given speed in miles per hour. Let the horizontal axis represent the speed (mph) and the vertical axis represent the safe following distance (feet). Make sure you label the axes accordingly.
220 | | | | | | | | | | | | | | | | |210 | | | | | | | | | | | | | | | | |200 | | | | | | | | | | | | | | | | |190 | | | | | | | | | | | | | | | | |180 | | | | | | | | | | | | | | | | |170 | | | | | | | | | | | | | | | | |160 | | | | | | | | | | | | | | | | |150 | | | | | | | | | | | | | | | | |140 | | | | | | | | | | | | | | | | |130 | | | | | | | | | | | | | | | | |120 | | | | | | | | | | | | | | | | |110 | | | | | | | | | | | | | | | | |100 | | | | | | | | | | | | | | | | |90 | | | | | | | | | | | | | | | | |80 | | | | | | | | | | | | | | | | |70 | | | | | | | | | | | | | | | | |60 | | | | | | | | | | | | | | | | |50 | | | | | | | | | | | | | | | | |40 | | | | | | | | | | | | | | | | |30 | | | | | | | | | | | | | | | | |20 | | | | | | | | | | | | | | | | |10 | | | | | | | | | | | | | | | | |0 |5 |10 |15 |20 |25 |30 |35 |40 |45 |50 |55 |60 |65 |70 |75 | | GRAPH 1
Exercise 5: After plotting the points on Graph 1, determine a best fit line to represent the data and draw it in Graph 1. Is it linear?
Exercise 6: Determine an equation to represent the relationship between the speed and the safe following distance.
According to the National Highway Traffic Safety Administration (NHTSA), it takes 133 feet for a 2500 pound automobile to stop if it is traveling at 55 miles per hour. It is imperative that you learn to follow at a safe distance behind another vehicle.
In May 2005, Danica Patrick qualified for the Indianapolis 500 in the fourth position and finished in the same place, the best ever for a female at Indy. She has since won a race in the Indy Racing League. Down the straightaways at Indy, Danica can reach speeds of 230 miles per hour or more.
Exercise 7: Using the equation that you formulated in Exercise 6, determine the safe distance to follow at this speed. Do you suppose Danica abides by this safe distance rule while on the race track? Why or why not?
REFERENCES
Perdew, P. R. (2002). Sports and distance-rate-time. Mathematics Teacher, 95, 192-199.
SOLUTIONS
Exercise 1 [pic]=[pic]
=[pic]
=[pic]
=[pic]
= 22 feet per second
Exercise 2
Miles Per Hour |15 |20 |25 |30 |35 |40 |45 |50 |55 |60 |65 |70 |75 | |Feet Per Second |22 |29 |37 |44 |51 |59 |66 |73 |81 |88 |95 |103 |110 | | | | | | |TABLE 1 | | | | | | | | |
Exercise 3
Speed(mph) |15 |20 |25 |30 |35 |40 |45 |50 |55 |60 |65 |70 |75 | |Speed(fps) |22 |29 |37 |44 |51 |59 |66 |73 |81 |88 |95 |103 |110 | |Safe Distance |44 |58 |74 |88 |102 |118 |132 |146 |162 |176 |190 |206 |220 | | | | | | |TABLE 2 | | | | | | | | |
Exercise 4 The points in GRAPH 1.
Exercise 5 The line in GRAPH 1. Yes, it is linear.
[pic]
Exercise 6 [pic] or something close to this slope.
Exercise 7 Danica’s safe following distance should be about 675 feet. No, she will not maintain this safe distance because she is in a race. Although racecar drivers take a high risk in following another car very closely, their cars are equipped with much more safety equipment than a regular passenger car.
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