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Stopping Distances

Extracts from this document...

Introduction

Stopping Distances

Portfolio

Mathematics SL Yr. 1

Period A

Speed (x) Vs. Thinking Distance (y)

Points plotted on the graph:

(32, 6)

(48, 9)

(64, 12)

(80, 15)

(96, 18)

(112, 21)

As you can see in the graph above all six points seem to line up in a fairly straight line. As speed increases, so does the thinking distance, always by a similar amount. This means that the speed to thinking distance ratio is constantly increasing as speed increases.

Functions:

Slope:

m= (y2 – y1) / (x2 – x1)

m= (12 – 6) / (64 – 32)

m= 6/32

m= 3/16

(32,6):

y – y1 = m(x – x1)

y – 6 = (3/16) (x – 32)

y – 6 = (3/16)x –6

y = (3/16)x

(64, 12):

y – y1 = m(x – x1)

y – 12 = (3/16) (x - 64)

y – 12 = (3/16)x – 12

y = (3/16)x

Percentage of error”

(64, 12)

y = (3/16)64

y = 12

(12 – 12) /12

≈ 0 %

The function, y = (3/16)x, matches the points, plotted on the graph, almost perfectly. You can see this by the calculated percentage of error which equals 0%. The function goes through every single point graphed. In the real life situation this means that the function represents the data recorded, while measuring

...read more.

Middle

y = 0.0039(80)2 + 0.188(80) – 4.0096

y = 24.96 + 15.04 – 4.0096

y = 35.9904

(35.9904 – 38) / 38

≈ - 5.288 %

The Function, y = 0.0039x2 + 0.188x – 4.0096, used to represent these data points is a parabolic function. It is not quite as accurate as the function representing speed vs. braking distance. I used the first three data points to come up with an appropriate function, thus it goes right through these points. The last three points however lay slightly above the function, with percentage errors of ≈ -12.04 %, ≈ - 9.124 % and ≈ -5.288.This function has several other limitations aswell. First of all it only works for lower speeds rather than when the car travels at higher speeds. Also, it represents a parabolic function however the data points only includes the positive numbers, meaning half of a parabola.

y = ax2 + bx + c

1. (80, 38)

2. (96, 55)

3. (112, 75)

  1. 38 = (80)2a + (80)b + c

38 = 6400a + 80b +c

  1. 55 = (96)2a + (96)b + c

55 = 9216a + 96b + c

  1. 75 = (112)2a + (112)b + c

75 = 12544a + 112b + c

  38 = 6400a + 80b + c

- 55 = 9216a + 96b + c

  17 = 2816a + 16b

   55 = 9216a + 96b + c

- 75 = 12544a + 112b + c

   20 = 3328a + 16b

  17 = 2816a + 16b

- 20 = 3328a + 16b

     3 = 512a

a ≈ 0.005900

17 = 2816 (0.0059) + 16b

17 = 16.6144 + 16b

0.3856 = 16b

b ≈ 0.02410

38 = 6400 (0.0059) + 80 (0.0241) + c

38 = 37.76 + 1.928 + c

c ≈ -1.688

...read more.

Conclusion

2 + 0.1709 + 0.2827, is very similar to the previous function, y = 0.006x2 – 0.0053x + 0.0256, representing speed vs. braking distance. Both are parabolic functions and intersect all points of the graph. IN addition, both have the same limitation which is the fact that the function represents an entire parabola, including both positive and negative points on the graph. The situation however only includes the positive numbers due to the real life situation in which it is physically impossible to have negative meters. This function is different to the function representing speed vs. thinking distance, y = (3/16)x, because it is a linear function representing one straight line. Also this function barely has any limitations when compared to the real life situation and the data recorded.

Percentage of error

(10, 2.5)

y = 0.0061(10)2 + 0.1709 (10) + 0.2827

y = 2.6017

(2.6017 – 2.5) / 2.5

≈ 4.068 %

(40, 17)

y = 0.0061(40)2 + 0.1709 (40) + 0.2827

y = 16.8787

(16.8787 – 17) / 17

≈ -2.376 %

(90, 65)

y = 0.0061(90)2 + 0.1709 (90) + 0.2827

y = 65.0739

(65.0739 – 65) / 65

≈ 0.1134%

(160, 180)

y = 0.0061(160)2 + 0.1709(160) + 0.2827

y = 183.7867

(183.7867 – 180) / 180

≈ 2.104 %

My model, y = 0.0061x2 + 0.1709x + 0.2827, does not fit the data of overall stopping distances for other speeds. It barely intersects the first point and is extremely off the next couple. Although the type of function (parabolic) fits the graph, several points

...read more.

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