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# Stopping distances portfolio. In this task, we may develop individual functions that model the relationship between speed and thinking distance, as well as speed and braking distance. We could also develop a model for the relationship between speed and ov

Extracts from this document...

Introduction

Section: 11/8                                    Type: II                                                     Date: February 22, 2011

IB Diploma

Standard Level Mathematics

Portfolio: Type II

Stopping Distances

American college of Sofia

STOPPING DISTANCES

Introduction

Stopping a moving car requires to apply the brakes which must actually stop the vehicle. There must be a special mathematical relationship between the speed of a car and the thinking distance, braking distance and overall stopping distance. In this task, we may develop individual functions that model the relationship between speed and thinking distance, as well as speed and braking distance. We could also develop a model for the relationship between speed and overall distance. This will be achieved by using Excel Graphing Software to create two data plots: speed versus thinking distance and speed versus braking distance, for which the results will be evaluated and described. Subsequently we can develop a function that will model the relationship and behavior of the graphs based on basic knowledge of functions. After all that we can add the thinking and breaking distance to obtain an overall stopping distance, from which we will once again graph the data and describe it’s results, whilst evaluating the correlation between the graph obtained from these results and the graphs obtained from the comparison between the speed vs.

Middle

Speed vs. Braking distance

 Speed (km/h) Braking distance (m) 32 6 48 14 64 24 80 38 96 55 112 75

Excel models

For speed vs. braking distance, the formula for a linear equation through (48,14) and (96,55) is found by:

The equation is found by
y - y1 = m(x - x1)

y – 14 =  (x – 48)
y - 14 =
x - 41
y =
x - 27 (x = braking distance & y = speed)

y = 0.8541x – 27

 Speed [km/h] Braking distance [m] y = 0.8541x – 27 32 6 0.3312 48 14 13.9968 64 24 27.6624 80 38 41.328 96 55 54.9936 112 75 68.6592

From chart two it is clear that the points represent a curve which is suitable for quadratic or cubic equation. For the quadratic equation of the type: y = ax2 + bx + c, the graphic software calculates the relationship in the following way: y = 0.0061x2 – 0.0232x + 0.6

 Speed [km/h] Braking distance [m] Quadratic relation 32 6 6.104 48 14 13.5408 64 24 24.1008 80 38 37.784 96 55 54.5904 112 75 74.52

Cubic Relationship

y = ax3 + bx2 + cx + d

The software calculates for the quadratic relationship:

a = 1.1118 – 6; b = 0.0059; c = 1.4724 – 4; d =3.0093 – 6

Description

In the graphs above, the plotted points are in a curve. This is possible in two cases: if there is a quadratic relationship or a cubic relationship. As it’s seen from the equation for the quadratic and cubic relationships, they both fit the data properly. However, the cubic function is only suitable for this data,

Conclusion

Clearly, the quadratic equation is a better predictor of the total stopping distances.

How does it fit the model and what modifications can be made?

The additional points have been plotted in bold in collaboration with the stopping distance data created from the initial data. As we can see on the graph, the additional information is fairly accurate, and fits the model perfectly. There are no major modifications needed, as the data seems to fit on to the line rather perfectly. One minor addition rather that modification that could be made is to try the vehicle’s highest speed to see whether or not this model is valid through all the speed of the vehicle.

Conclusion

In conclusion, it is clear that there is a linear relationship between the speed and thinking distance, whilst there is a quadratic relationship between the speed and breaking and the stopping distances. This seem to be an obvious outcome because it makes sense that when the car is at a low speed, the break will be pushed down in a controlled manner, hence making the distance increase quicker than the time, where as, if the vehicle is going at a much quicker speed, the driver will tend to push down on the brake more abruptly and will stop over a smaller distance but quicker.

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