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Stopping Distances

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Introduction

Maths portfolio: Stopping Distances (Standard Level)

Stopping Distances

Before a vehicle stops, the driver has to think before applying the brakes and the brakes take time to actually stop the vehicle. These two processes vary at different speeds as shown the table below:

Speed (kmhˉ¹)

Thinking distance (km)

Braking distance (m)

32

0.006

0.006

48

0.009

0.014

64

0.012

0.024

80

0.015

0.038

96

0.018

0.055

112

0.021

0.075

So by adding the thinking distance and braking distance together we are able to find the total average stopping distance:

Stopping distance (km)

0.012

0.023

0.036

0.053

0.073

0.096

Using the records above, our task is to:

  • Develop individual functions that model the relationship between speed and thinking distance as well as speed and braking distance
  • Develop a model for the relationship between speed and over all stopping distance
  1. Using a graphing software I have created two data plots:

image00.png

Speed versus Thinking distance graph

The values between the thinking distances have a fixed interval and a set value for the speed making it a straight line with a positive constant gradient of 16/3.

...read more.

Middle

2- x1
   y
2-y1

by substituting theses values to find the gradient from these two co-ordinate points (32,0.006) and (48,0.009):

m=0.009- 0.006=1.875 x 10^-4
48-32

the y-intercept of the linear is 0 because as the speed is 0 there is no need to think to brake the vehicle so:

c= 0

the final function we get from using the equation is:

y=1.875*10 ^-4x+0

image01.pngimage02.png

Since there cannot be any negative thinking distances or speed, the equation for the linear of speed versus thinking distance has to be:

y=|1.875*10 ^-4x+0|

Speed versus Braking distance

image03.png

From the values of the braking distance, there isn’t a common difference and the braking distance rising exponentially creates a curve suggesting a parabola.

Therefore the quadratic equation, y=a(x+ a)² also known as  y=ax²+bx+c can be applied to model its behavior.

As the speed is 0 the braking distance would be 0 as well therefore the y intercept will be a repeated root, this will mean that a

...read more.

Conclusion

y=function1+function2

Speed vs. Overall Average Stopping distance= (|1.875*10 ^-4x+0|) + (|5.859375*10^-6 x²|)

Here is another set of data which can be used to see if the model fits

Speed (kmhˉ¹)

Stopping distance (km)

10

0.0025

40

0.017

90

0.065

160

0.18

image05.png

                                        Speed versus Stopping distance        

As we can see, the points are forming a curve which is alike the previous quadratic graphs however, the equation of the graph:

image06.pngimage07.png

May still be a quadratic equation but it is not exactly the same as the other stopping distance equation because different factors such as the tire friction, the weather/conditions, the driver’s age and the road’s surface all affect the overall stopping distance. Anomalies have to be taken into account as well as accidents can happen which affect the outcome. With more data, the accuracy of the graph can be improved so that the function can also be modified and become more precise.

Tisha Yap 6C2

...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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