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Stopping Distances

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

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

Graph 1. Speed versus Thinking distance

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 by taking any two points of coordinates and calculating m using the following equation  

The two points taken: P1 (48,0.009)

           P2 (112,0.021)

      x1 = 0.009

        x2 = 0.021

        y1 = 48

        y2 = 112

  1. The value of b is the y-intercept but since the car is at rest the speed is 0 and therefore the breaking distance is also 0.

  1. Equation obtained:

 

Graph 2. Model of Speed versus Thinking distance

This model is a very good and accurate fit for the graph because the correlation coefficient, r, is 1 which means that it is a very strong correlation. The coefficient of determination, , is 1 which means that 100% of the total variation in  can be explained by the relationship between  and . However, since we cannot have negative values of speed or thinking distance we can eliminate all negative possibilities with this function.

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  1. Speed versus Braking distance

Table 3. Speed versus Braking distance

Graph 3. Speed versus Braking distance

This graph is, from observation, a quadratic or a semi-parabola in which the y-value (braking distance) increases exponentially.

Since it is a quadratic we can develop a model to fit a data using the equation  also known as, however, there is another model that would fit the data is by using a power function.  First I will develop a quadratic model.

Steps taken to develop quadratic model:

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