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2 the basis of student explorations which require them to make comparisons of various kinds. Speed (km/h) Table 1. Alert reaction distance (ARD), unexpected reaction distance (URD) and braking distances for five types of car at different initial speeds. Speed (m/s) ARD (m) URD (m) Interpreting the data Ferrari 550 Maranello Lexus LS400 Braking distance (m) Mercedes C36 Nissan 200SX Toyota Corolla Have the students plot the URD, the ARD and the braking distance for the Ferrari on the same axes and join the points with either a straight line or a smooth curve, whichever seems appropriate (see Figure 1). 1 This may be the first time students have seen a non-linear graph and helps give the idea that such things are not in any way strange but occur all the time. Typical questions might include: Which reaction distance should be larger for a given speed? Why? Find the gradient of the two lines. What do you notice? Can you explain? What is more significant at 8.3 m/s (30 km/h), reaction distance or braking distance? Explain. Is the braking distance at 22.2 m/s (80 km/h) double the braking distance at 11.1 m/s (40 km/h)? Explain. Is the alert reaction distance at 22.2 m/s (80 km/h) double the braking distance at 11.1 m/s (40 km/h)? Explain. 1. For consistency of units the speeds are plotted in m/s; but it is useful for students to get a sense of how fast these speeds are in terms they understand and this is the reason that the speed in km/h is also given in the table. The mathematics of stopping your car Australian Senior Mathematics Journal 19 (1) 45

3 Australian Senior Mathematics Journal 19 (1) Woods Figure 1. Graph of the unexpected reaction distance (dotted line), the alert reaction distance (solid line), and the braking distance (solid curve), against time for a Ferrari. The gradients of the straight lines will give the two reaction times a fact that can be confirmed by noting that the gradient is a ratio of quantities measured in m and m/s which results in seconds for the units of the gradient. The steeper the line, the longer the reaction time and therefore the greater the reaction distance. Furthermore, because the relationship between speed and reaction distance is linear, doubling one will cause the other to double and so on. This leads naturally to the equation or Distance (m) Speed (km/h) Speed (m/s) reaction distance = reaction time speed y = mx. In this case the equations are y = 0.7x and y = 1.5x for the lines representing ARD and URD respectively. The relationship between braking distance and speed does not show this same pattern, however, and so it is not surprising that the graph is not linear. In fact, doubling the speed results in a quadrupling of the braking distance. So how are the two quantities related? A spreadsheet is a useful way of examining this data numerically to see if we can establish the mathematical relationship. One way of looking at the data numerically is to calculate the differences between successive data values. Students often use this technique to establish missing terms in sequences. For example in the sequence 3, 7, 11, 15 it is easy to see that the difference between successive terms is 4 and we can see that the functional rule which generates this sequence x 4x 1 is which is indeed linear. With the sequence 1, 4, 9, 16, 25 the differences are 3, 5, 7, 9 and so on. Clearly the relationship is not linear but if we find the differences of the differences (second differences) we get 2, 2, 2, 2 In this case we can see that the functional rule to generate the sequence is x x 2 and this 46

4 suggests that when the second differences are constant the relationship is quadratic. This same technique can be used on data sets where the x-values (in this case speed) are evenly spaced. If a spreadsheet is used to calculate the first differences for the ARD data we can see (subject to rounding error) that they are reasonably constant (~1.9) and this confirms the linear relationship we found by graphing. For the braking distance data, the first differences are clearly not constant but the second differences are reasonably constant (~1) and this suggests that the relationship between braking distance and speed may be quadratic. Now that a numerical approach has given us a clue about the relationship between speed and braking distance we can go back to geometric and algebraic techniques to establish what the actual relationship is. Have students draw a graph similar to the previous one but this time graph distance versus the square of the speed (see Figure 2). The points should be collinear, and if a straight line is drawn though them students should find the gradient to be approximately So the straight line has equation Y = 0.054X and this tells us that the relationship between braking distance and speed is, in fact, y = 0.54x 2 since in this case Y y and X x 2. The value is related to the braking efficiency of a car; smaller values will mean that a car takes less distance to stop. 2 Students could verify that this equation is correct by graphing the data and the newly found quadratic function with either Graphmatica or a graphing calculator to see if it passes through the points. One advantage of this method is that it requires more analysis on the part of the students than merely using a blackbox curve fitting strategy; and, it provides for numerical and geometrical confirmation of the algebraic form of the equation. Distance (m) Square of Speed (m 2 /s 2 ) Figure 2. Braking distance against the square of speed. 2. Actually it is proportional to the reciprocal of the deceleration so a large deceleration means a small value. The mathematics of stopping your car Australian Senior Mathematics Journal 19 (1) 47

5 Australian Senior Mathematics Journal 19 (1) Woods Putting it all together stopping distance As a final exercise students could be asked to now consider the idea of stopping distance which is braking distance + reaction distance. Adding together the alert reaction distances and the braking distances and then plotting this data versus speed along with the equation y = 0.054x x with Graphmatica or a graphing calculator will allow them to verify that the equation does fit (see Figure 3). Distance (m) Speed (m/s) Figure 3. Stopping distance against speed. In this graph a traditional x-axis and y-axis have be added to show the more traditional view of the parabola. Of course it is important to remember that whenever we create a mathematical model for something such as stopping distance we must realise that although the mathematical function may be defined for negative values, it may not have any value as a model for such values. 48

6 A challenge After this discussion students could be challenged to answer the following question: A police officer is investigating a traffic violation. The car, a Ferrari 550 Maranello, has left skid marks 37 metres long on the roadway in its attempt to stop at a red light but has ended up in the intersection when spotted by the police officer. The officer charges the driver with running a red light but is also concerned about how fast the car was travelling. The driver claims he was only going about 70 km/h when he applied the brakes. (a) (b) Is he telling the truth? Justify your answer. If you think he is lying, how fast would you estimate he was travelling at the time his brakes were applied? Since skid marks are often made when brakes are applied, we could use braking distance data to get a rough idea of speed 3. Either the graph or the formula can be used to get a speed estimate of somewhere between 90 and 100 km/h. 3. The police do use a slightly more complicated formula than this simplified version although the principle is exactly the same. Anti-lock brakes do still leave skid marks but they are not as obvious as those left by cars without them. The mathematics of stopping your car Australian Senior Mathematics Journal 19 (1) 49

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