Rule Time: Salute to Brakes! Scott Adamson, Ph.D.

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Presentation transcript:

Rule Time: Salute to Brakes! Scott Adamson, Ph.D. sadamson23@gmail.com getrealmath.wordpress.com

Rule Time: Salute to Brakes!

Reaction Distance (feet) Braking Distance (feet) The data: Speed (MPH) Reaction Distance (feet) Braking Distance (feet) 20 22.0 22.2 25 27.5 34.7 30 33.0 50.0 35 38.5 68.0 40 44.0 88.8 45 49.5 112.4 50 55.0 138.8

Total Stopping Distance Speed (MPH) Reaction Distance (feet) Braking Distance (feet) Total Stopping Distance (feet) 20 22.0 22.2 25 27.5 34.7 30 33.0 50.0 35 38.5 68.0 40 44.0 88.8 45 49.5 112.4 50 55.0 138.8 + 44.2 = + = 62.2 + = 83.0 + = 106.5 + = 132.8 + = 161.9 + = 193.8

Stopping Distance (feet) Linear Prediction How much distance is needed to stop a car traveling at a speed of 100 MPH? Speed (MPH) Stopping Distance (feet) Difference (feet) 20 44.2 25 62.2 30 83.0 35 106.5 40 132.8 45 161.9 50 193.8 18.0 20.8 23.5 26.3 29.1 31.9

Stopping Distance (feet) Linear Prediction How much distance is needed to stop a car traveling at a speed of 100 MPH? Speed (MPH) Stopping Distance (feet) Difference (feet) 20 44.2 25 62.2 30 83.0 35 106.5 40 132.8 45 161.9 50 193.8 18.0 20.8 23.5 26.3 29.1 31.9

Stopping Distance (feet) Quadratic Prediction How much distance is needed to stop a car traveling at a speed of 100 MPH? Speed (MPH) Stopping Distance (feet) Difference (feet) 20 44.2 25 62.2 30 83.0 35 106.5 40 132.8 45 161.9 50 193.8 Difference (feet) 18.0 2.8 20.8 2.7 23.5 2.8 26.3 2.8 29.1 2.8 31.9

Stopping Distance (feet) Quadratic Prediction How much distance is needed to stop a car traveling at a speed of 100 MPH? Speed (MPH) Stopping Distance (feet) Difference (feet) 20 44.2 25 62.2 30 83.0 35 106.5 40 132.8 45 161.9 50 193.8 Difference (feet) 18.0 2.8 20.8 2.7 23.5 2.8 26.3 2.8 29.1 2.8 31.9

Constant Rate of Change Prediction: 425 feet Compare Predictions How much distance is needed to stop a car traveling at a speed of 100 MPH? Constant Rate of Change Prediction: 425 feet Constant Rate of Change in the Rate of Change Prediction: 668 feet With the “flannel factor” added in, Jim is standing 790 feet away!

D=399.31 D=221.64

Is this the end of Rule Time? Will Hal stop in time? Is this the end of Rule Time?

Quadratic Functions In times of increasing gas prices, many drivers become concerned about the fuel efficiency of their vehicles. The United States Department of Energy reports that although each vehicle reaches its optimal fuel economy at a different speed, gas mileage usually increases up to speeds near 45 mph and then decreases rapidly at speeds above 60 mph (Source: www.fueleconomy.gov).

Quadratic Functions Speed (MPH) 10 20 30 40 50 60 70 80 Fuel Economy (MPG) 17.6 24.1 28.6 31.1 31.7 30.2 26.7 21.3 Imagine a continuous function that models these data. With your right index finger, track the increasing speed in MPH. With your left index finger, track the behavior of the fuel economy in MPG. Then track them both simultaneously. Explain the behavior of the function in the context of the situation. 18

Quadratic Functions

Fuel Economy vs Speed 6.5 -2.0 4.5 -2.0 2.5 -1.9 0.6 -2.1 -1.5 -2.0 Speed (S ) (MPH) Fuel Economy Model (F ) (MPG) F  (F) 10 17.6 20 24.1 30 28.6 40 31.1 50 31.7 60 30.2 70 26.7 80 21.3 6.5 -2.0 4.5 -2.0 2.5 -1.9 0.6 -2.1 -1.5 -2.0 -3.5 -1.9 -5.4 20

Fuel Economy vs Speed 9.1 10.5 -2.0 8.5 -2.0 6.5 -2.0 4.5 -2.0 2.5 Speed (S ) (MPH) Fuel Economy Model (F ) (MPG) F  (F) 10 17.6 20 24.1 30 28.6 40 31.1 50 31.7 60 30.2 70 26.7 80 21.3 9.1 10.5 -2.0 8.5 -2.0 6.5 -2.0 4.5 -2.0 2.5 -2.0 0.6 -1.9 -1.5 -2.1 -3.5 -2.0 -5.4 -1.9 21

Fuel Economy vs Speed “Starting” at 9.1 MPG when speed is (hypothetically) 0 MPH Initial rate of change is somewhere between 10.5 and 8.5 MPG for an increase of 10 MPH. Let’s say 9.5 MPG per 10 MPH or 0.95 MPG/MPH This initial rate of change is changing (decreasing) by half of 2 MPG per 10 MPH for each increase of 10 MPH in speed. Let’s say -0.01 MPG/MPH for each increase of 1 MPH

Fuel Economy vs Speed Mike’s Sentence The hypothetical initial fuel economy is 9.1 MPG for a car that travels at 0 MPH. This value initially changes at a rate of 0.95 MPG per 1 MPH increase in speed. This rate of change decreases by a constant 0.02 MPG per MPH for each increase in 1 MPH. 23

Thinking about Mike’s Sentence Covariationally 2.5 4.5 6.5 9.5

Quadratic Formula

Quadratic Formula

Major League Baseball Batted Ball Speed Quadratic Functions The table below is data modeled of the batted ball speed (in MPH) for various bat weights (in ounces) for Major League Baseball players in 2007. (Source: Popular Mechanics, June 2007). Major League Baseball Batted Ball Speed Bat Weight (ounces) Batted Ball Speed (miles per hour) 20 68.12 25 72.75 30 76.10 35 78.16 40 78.94 45 78.43 50 76.64

Quadratic Functions First, thinking picture…describe the relationship between the ball weight and the batted ball speed. Next, compute first and second differences. Third, find “initial amount.” Fourth, estimate “initial rate of change.” Finally, estimate value of a in B = aw2 + bw + c Find where B = 0 and interpret the meaning of these values.