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Portfolio - Stopping Distances

Extracts from this document...

Introduction

image17.pngimage18.png

image00.png


Stopping Distancesimage04.pngimage01.png

When a driver stops her car, she must first think to apply the brakes. Then the brakes must actually stop the vehicle.

The table below lists the average times for these processes at various speeds.

Table 1. average times for these processes at various speeds

Speed (kmh-1)

Thinking distance (m)

Braking distance (m)

Stopping distance (m)

32

6

6

12

48

9

14

23

64

12

24

36

80

15

38

53

96

18

55

73

112

21

75

96

Using this table we can graph two data plots: (i) Speed versus Thinking distance, (ii) Speed versus Braking distance, and (iiI) Speed versus Stopping distance

  1. Speed versus Thinking distance

Table 2. Speed versus Thinking distance

Speed (kmh-1)

Thinking distance (km)

32

0.006

48

0.009

64

0.012

80

0.015

96

0.018

112

0.021

Graph 1. Speed versus Thinking distance

image19.png

This is clearly a linear graph since we can see a straight line. This shows us that the correlation between speed and thinking distance is directly proportional, meaning that as speed increases the thinking distance will also increase. In other words, as the speed of a car increases it takes a longer time for the driver to think about applying the breaks.

Since this graph is linear we can develop a model to fit the data using the equation y= mx+b where m stands for gradient and b stands for the y-intercept.

Steps taken:

  1. First we find the gradient m
...read more.

Middle

table 1 and the function
image47.png

Graph 4. Quadratic model for Speed versus Braking distanceimage02.png

image48.pngimage03.png

However, because it is a quadratic we have to evaluate whether the negatives will be a good fit to represent the data. Below is a graph showing the same graph as above but with an extended window frame.

Graph 5. Quadratic model for Speed versus Braking distance with enlarged window frameimage02.png

image49.pngimage50.pngimage03.png

Here we can see that the plots match well on the right side. However since we cannot have negative speed the model is not a good fit despite that it is a good fit to represent the data on the right.

Having that said, the other option is the power function and it was chosen because it is polynomial and we can eliminate all negative values since the domain is within positive values.

Steps taken to develop power model using GDC:

  1. Insert data into GDC table

image51.png

L1 – Speed
L2 – Braking distance

  1. Use implemented Power Regression for variables L1 and L2

image52.png

image53.png

  1. Insert the information into STAT PLOT

image54.png

  1. Plot data from table

image56.png

Graph 6. Speed versus Braking distance

image57.pngimage02.pngimage03.png

  1. Implement power function into the graph

Graph 7. Power model for Speed versus Braking distance

image58.pngimage03.pngimage02.png

In the end we get a function: image59.pngimage59.png

...read more.

Conclusion

graph 16-17 and graph 19-20. This is because it adds function1 and function2, instead of calculating it algebraically by using equations which depend on the coordinates taken. However, I would say Model B is the best model to represent the data since in graph 18 we can see the function go through every data plot since the data plot is filled by black, whereas in graph 15 the function misses two data plots.

Now that we have a good model to represent the data, it is time to test whether this model would fit further data.

Table 5. New data plots

Speed (kmh-1)

Stopping distance (km)

10

0.0025

40

0.017

90

0.065

160

0.18

Table 6. Speed versus Stopping distance with new data

Speed (kmh-1)

Stopping distance (km)

10

0.0025

32

0.012

40

0.017

48

0.023

64

0.036

80

0.053

90

0.065

96

0.073

112

0.096

160

0.18

Graph 21. Speed versus Stopping time using Model B

image26.pngimage02.pngimage03.png

As we can see the model is a very good fit to represent the data since the function passes through all data plots. However, a more accurate function could be developed by using more data plots.

We also have to consider anomaly results since some things are not accounted for. For example, the friction between the tires and the road, and weather conditions affecting driver’s reaction time which may not fit my model accurately.

With more data plots we can make modifications to the function since all the current data plots fit the function.

...read more.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

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