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

4. Rule Time:
Salute to Brakes!

6. 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

7. 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

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

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

10. 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

11. 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

12. 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!

13. D=399.31
D=221.64

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

17. 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:

18. 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

19. Quadratic Functions

20. 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

21. 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

22. 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 MPG/MPH for each increase of 1 MPH

23. 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

24. Thinking about Mike’s Sentence Covariationally
2.5
4.5
6.5
9.5

25. Quadratic Formula

26. Quadratic Formula

27. 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 (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

28. 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.