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Portfolio Type II: Stopping Distances

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Introduction

Anousha Manji

Maths Portfolio Type II- Stopping Distances

The table below shows the average stopping and thinking distances when a person is driving a car and needs to apply the brakes at various speeds.

Speed (kmh-1)

Thinking Distance (m)

Braking Distance (m)

32

6

6

48

9

14

64

12

24

80

15

39

96

18

55

112

21

75

From this data we can graph two data plots: one showing the relationship between speed versus and the other between speed versus braking distance.

image01.png Figure 1

image02.png Figure 2

Figure 1 shows a straight line, and therefore it can be said that it is a linear graph. It shows that the correlation between speed and thinking distance is directly proportional and shows the trend that as the speed increases the thinking distance also increases.
The equation of this graph is in the form
image06.pngimage06.png, where m is the gradient, and c is the y intercept. The equation can be worked by following these steps:

  • Find m (the gradient) using the following equation: image11.pngimage11.png

image12.png

  • b is the y intercept but as the graph passes through the origin this value is 0 because while the car is moving at 0 kmh-1 the braking distance is 0 m.
  • The final linear equation is image03.pngimage03.png for the domain 32<x<112 because we do not know how long the thinking distance will be for any higher speeds. We could guess from the graph but in a real life situation this would be putting the drivers’ life at risk.
...read more.

Middle

15

96

18

18

112

21

21

This shows that the line, image03.pngimage03.png, fits exactly with the values given.

Figure 2 has a graph with an increasing gradient and looks like a part of a quadratic curve suggesting that the braking distance increases with speed. The equation of the curve above almost follows the curve, image04.pngimage04.png. This can be found using a graphic calculator.

To see how well the curve fits mathematically, below is a table showing the output values from the equation image04.pngimage04.png and comparing them to the values already given.

Speed (kmh-1)/x

Braking Distance already given (m)

Braking Distance from function (m)/y

32

6

6.144

48

14

13.824

64

24

24.576

80

39

38.400

96

55

55.296

112

75

75.264

We can see from looking at the table that the output values lie very close to the original values so it can be concluded that the equation is image04.pngimage04.png

...read more.

Conclusion

The overall stopping distances for other speeds are shown below:

Speed (kmh-1)

Overall Distance (m)

10

2.5

40

17

90

65

160

180

image10.png

The graph above shows the values for the different overall stopping distances. The points in the red show the values from the last table and the points in blue show the values that were obtained from adding the initial thinking and braking distance values.

The table below shows how the new values for the stopping distance fits into the equation image09.pngimage09.png

Speed (kmh-1)/x

Overall Distance already given (m)

Overall Distance from function (m)

10

2.5

0.89

40

17

14.24

90

65

72.09

160

180

227.84

The first and last values to do not fit the curve well but the middle 2 values do. From looking at the graph above we can see that all the overall stopping values sit on a curve so it is evident that the problem is in the curve function itself.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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