# 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

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. thinking/braking distance. Finally, we will evaluate how the additional set of data relates to the graph in the previous step and the possible modifications to be made.

Initial Data Set

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

Speed vs. Thinking distance

As it is visible from the table below when speed doubles, thinking speed doubles, and when speed triples, thinking speed triples. There is a clear linear relationship which can be found from any 2 points, like (64, 12) and (96, 18). Then if we consider m as a stable ratio, so the following relation is found to be true:

then    and  = 0.1875

We can present the above mentioned result as a result of the following equation:

y - y1 = m(x - x1)

So for the chosen points (64, 12) and (96, 18) the equation has the following  view:

y - 12 = (x - 64)

y = x, where x is the thinking distance, and  y is the speed.

So we may deduce that there is a simple linear relationship from the type:

y = ax + b, where a = 0.1875, and b = 0.

...