Dangerous Weather SUMMARY CURRICULAR PLACEMENT OBJECTIVES MATERIALS NEEDED CLASSROOM ACTIVITY. Mathematics: Algebra

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2 Mathematics: Algebra Dangerous Weather SUMMARY Students investigate safe driving speeds. They derive equations for reaction distance, braking distance, and stopping distance. They graph stopping distance as a function of speed. They use this graph as well as the quadratic formula to calculate safe driving speeds in fog. CURRICULAR PLACEMENT This activity can be used when studying applications of quadratics in Algebra I or Algebra II. Prerequisite knowledge and skills include an understanding of kinematics (physics), graphing, and the quadratic formula. OBJECTIVES Apply knowledge of algebra and graphing to calculate safe driving speeds. Write equations to describe real world scenarios involving distance, time, velocity and acceleration. Graph functions and interpret their graphs. Solve quadratic equations. MATERIALS NEEDED Copies of Reproducible Master 1: Crash! Copies of Reproducible Master 2: Safe Speed Graph paper and rulers OR a computer with graphing software OR a programmable calculator Chart paper/markers CLASSROOM ACTIVITY 1) Ask students to imagine themselves driving on a highway where the speed limit is 55 miles per hour. Ask, What do you think the posted speed limit is based on safety in perfect driving conditions, the worst driving conditions, or some other criteria? 2) Distribute Reproducible Master 1: Crash! to students and have them read it. 3) Ask students, What would be a safe driving speed in heavy fog? How fast should you go? Record their answers. Explain that they will calculate the actual safe speed and then compare that number to their intuitive estimates of a safe speed. Education Development Center, Inc. Page 1

3 Dangerous Weather 4) Ask students what information they would need in order to determine a safe driving speed in foggy weather. Possible answers: Visibility (how far ahead the driver can see) Reaction time (how long it takes a driver to begin braking after seeing an obstacle) Braking distance (how far the car travels from the time the brake is applied until the car stops moving) 5) Distribute Reproducible Master 2: Safe Speed. You may have the whole class work together to complete the handout, have students complete it in small groups, or assign it for homework. Alternatively, you may conduct the activity without giving students the handout, eliciting from students at each step what the next step needs to be. Depending on your students knowledge of physics, they may need some guidance to complete question 3 on RM 2. 6) After students complete RM 2, ask them to think about what they considered a safe driving speed during foggy weather before they did this activity. Ask, Did your instincts about how fast you could drive prove correct? Ask students to list some other conditions under which visibility would be reduced. Possible answers: Rain Snow Darkness A dirty windshield 7) Discuss the variables in the braking distance equation, including reaction time and the initial velocity of the car. Ask, What factors might affect your reaction time? Possible answers: Distractions, including cell phones, music players, and friends Alcohol and drugs 8) Remind students that the equation for the braking distance was based on typical conditions. Ask students to list some conditions under which braking distance might be increased. Possible answers: Rain, snow, or ice that makes the road slippery Worn tires that result in less friction between the car and the road Problems with the brakes Developed by Education Development Center, Inc. Page 2 of 9

4 Dangerous Weather EXTENSION Discuss students thoughts on the role that legislation could and should play in promoting highway safety in dangerous weather. Ask students to come up with ideas for policies and practices for encouraging safe driving in poor weather. Present the following scenario to students: Imagine that this bill has been proposed in Congress: In any weather condition requiring the use of windshield wipers, the legal speed limit will be 10 miles per hour below the posted speed limit. This will be true for all roads and highways in all states in the United States. Have students work in teams to come up with arguments for or against this proposal. Have teams revise this proposal or come up with a new one to promote safe driving in poor weather. Have teams present their arguments and new proposals to the class. DID YOU KNOW... A highway in Texas with the highest speed limit in the United States 85 mph is set to open in November Read about it at the following Web sites: Manny Fernandez. (September 10, 2012). Reclaiming the Title of Fastest in the Land. The New York Times. Available at CBS News. (September 8, 2012). 85 mph Speed Limit in Texas a Potential Killer? CBS This Morning. Available at Developed by Education Development Center, Inc. Page 3 of 9

5 Reproducible Master 1 Crash! Every year, car accidents happen when drivers go too fast in conditions where visibility is reduced. Posted speed limits are usually based on how fast it is safe to drive in good weather conditions. How much slower do you need to drive to be safe when visibility is poor? Your challenge: Determine the safe driving speed in heavy fog. At least 50 vehicles collide in foggy Texas pileup CBS News, January 5, 2012 Fog and slick roads lead to crash KATU, December 22, 2011 Developed by Education Development Center, Inc. Page 4 of 9

6 Reproducible Master 2 Safe Speed To figure out safe driving speeds during heavy fog, you need two pieces of information: Visibility: The distance you are from an obstacle when you can first see it. If the visibility under given weather conditions is 50 feet, you will not see an obstacle until you are 50 feet away from it or even less, if you are not paying careful attention! Stopping distance: How far your car travels from the time you see the obstacle to the time your car stops. Total stopping distance for a car depends on the reaction distance and the braking distance: Reaction distance: The distance your car travels from the time you see the obstacle to the time you begin braking. Braking distance: The distance your car travels from the time you apply the brake to the time your car tops. Conversion factors: 1 mi = 5,280 ft 1 mi/hr = 1.47 ft/s 1. Write an equation expressing stopping distance (ds) as a function of the reaction distance (dr) and the braking distance (db). 2. In driving, reaction time is the amount of time from when you notice an obstacle to when you apply the brake. tr = reaction time in seconds vi = initial velocity of car in miles per hour dr = reaction distance in feet Use these variables to write an equation that expresses dr in terms of tr and vi. Note the units the variables are expressed in, and be sure to include any conversions needed in your equation. 3. Write an equation for the braking distance (db). Keep in mind: The deceleration rate of the car is the rate at which its speed decreases when you apply the brake. Assume the deceleration rate for your car is constant at 17 ft/s 2. When a car decelerates at a constant rate, the average velocity equals the average of the initial velocity and the final velocity. 4. Express the total stopping distance (ds) as a function of tr and vi. Developed by Education Development Center, Inc. Page 5 of 9

7 Reproducible Master 2 5. Typical reaction time is around 1.5 seconds. Using this value of tr, plot stopping distance as a function of velocity on graph paper or using graphing software or a graphing calculator. 6. For you to avoid crashing into an obstacle, what should be the relationship between your car s stopping distance and the visibility? 7. To determine a safe speed: a) How would you use the graph of stopping distance versus velocity to determine the safe speed at a given visibility? b) How would you use the equation derived in question 4 to calculate the safe speed at a given visibility? 8. In heavy fog, visibility is reduced to about 50 feet. a) If you are on a straight section of highway traveling 60 miles per hour, how does your stopping distance compare to the visibility? b) Assume that the car in front of you has stopped completely by the time you see it. What is the maximum safe speed for you to travel? Developed by Education Development Center, Inc. Page 6 of 9

8 Reproducible Master 2 answers Safe Speed (answers) To figure out safe driving speeds during heavy fog, you need two pieces of information: Visibility: The distance you are from an obstacle when you can first see it. If the visibility under given weather conditions is 50 feet, you will not see an obstacle until you are 50 feet away from it or even less, if you are not paying careful attention! Stopping distance: How far your car travels from the time you see the obstacle to the time your car stops. Total stopping distance for a car depends on the reaction distance and the braking distance: Reaction distance: The distance your car travels from the time you see the obstacle to the time you begin braking. Braking distance: The distance your car travels from the time you apply the brake to the time your car tops. Conversion factors: 1 mi = 5,280 ft 1 mi/hr = 1.47 ft/s 1. Write an equation expressing stopping distance (ds) as a function of the reaction distance (dr) and the braking distance (db). 2. In driving, reaction time is the amount of time from when you notice an obstacle to when you apply the brake. tr = reaction time in seconds vi = initial velocity of car in miles per hour dr = reaction distance in feet Use these variables to write an equation that expresses dr in terms of tr and vi. Note the units the variables are expressed in, and be sure to include any conversions needed in your equation. 3. Write an equation for the braking distance (db). Keep in mind: The deceleration rate of the car is the rate at which its speed decreases when you apply the brake. Assume the deceleration rate for your car is constant at 17 ft/s 2. When a car decelerates at a constant rate, the average velocity equals the average of the initial velocity and the final velocity. First, calculate the amount of time it takes the car to come to a stop: Developed by Education Development Center, Inc. Page 7 of 9

9 Reproducible Master 2 answers Then, calculate the distance traveled during this time: x 4. Express the total stopping distance (ds) as a function of tr and vi. 5. Typical reaction time is around 1.5 seconds. Using this value of tr, plot stopping distance as a function of velocity on graph paper or using graphing software or a graphing calculator. Developed by Education Development Center, Inc. Page 8 of 9

10 Reproducible Master 2 answers 6. For you to avoid crashing into an obstacle, what should be the relationship between your car s stopping distance and the visibility? The total stopping distance must be less than or equal to the visibility. 7. To determine a safe speed: a) How would you use the graph of stopping distance versus velocity to determine the safe speed at a given visibility? Find the visibility (in feet) on the vertical axis, and look for the corresponding velocity. The car should not exceed that velocity. b) How would you use the equation derived in question 4 to calculate the safe speed at a given visibility? c) The equation derived in question 4 is a quadratic equation. Rearrange it into the standard form for a quadratic equation. Use the quadratic formula to solve for vi. Substitute in a value for tr. Set ds equal to the visibility, and solve for vi. 8. In heavy fog, visibility is reduced to about 50 feet. a) If you are on a straight section of highway traveling 60 miles per hour, how does your stopping distance compare to the visibility? At 60 mph, the stopping distance of the car is about 350 feet, far exceeding the visibility b) Assume that the car in front of you has stopped completely by the time you see it. What is the maximum safe speed for you to travel? Using the graph or the quadratic formula, the speed at which the stopping distance is 50 feet is 16 miles per hour. To be safe, a car should not exceed this speed when driving in fog. Developed by Education Development Center, Inc. Page 9 of 9