This is the original graph of the ‘Car Test Run Data’; I will be using this graph for this investigation. From this graph I will need to model three linear functions, to represent an estimate speed of the car for the whole journey. To do this at first I will need to draw three different straight lines on this graph, which will represent an estimate of this graph, an example of such models: V=mt + c. Then I will need to evaluate how well these functions fit the original speed, which will be carried out by calculating the percentage error of each functions.
Following this I will also need to model the speed of the car using a quadratic function. This should represent an estimate for the whole journey of the car. The quadratic function will be modelled by choosing three points from the original graph.
Then solving them as simultaneous equations to find the values of ‘a’ and ‘b’ of this quadratic: V=at2 + bt + c. Then I will need to evaluate how well the quadratic function fits the original speed of the test run car.Method & Calculations Information from the ‘car test run line graph’ Q1. The maximum speed reached by the car is 67.5mph. It took 14.8s for the car to reach this speed. I have worked the maximum speed of the car by looking at the peaks of the graph (maximum speed is the highest peak of the graph), then I read the time and speed from the corresponding axis V= speed and t = time.
For the car to reach 60mph it took 10.7s. I found this by taking reading from the V-axis at 60mph.
During that time the car accelerated at 5.61 mph (acceleration = speed/time).Q2. The car changed gear at least three times. I have worked this out by looking at the graph, where the line is flat only for few seconds. I know this because when a gear change occurs the car will drop its acceleration slightly and then pick up acceleration.
Each small square on the graph is worth 0.2s, because 10 small squares are equal to 2s. So 2s divided by 10 = 0.2s.
The gear change took place at the following time: 1st gear change took placed at: 4.8s-5.2s (it lasted 0.4s) 2nd gear change took placed at: 8s-8.6s (it lasted 0.6s) 3rd gear change took placed at: 12.4s-13s (it lasted 0.
6s)Modelling the data with linear functions Q3. The time interval for the linear functions I will use going to be between the gear changes that took place. I could just do one linear model for the whole journey of the car, but this will be too misleading as the car sometimes slows down (particularly towards the end of the journey) and the graph isn’t a straight line. For the 1st linear function the time interval will be: 0.00s-4.8s (where 1st gear changed occurred) For the 2nd linear function the time interval will be: 5.
2s-7.8s (where 2nd gear changed occurred).For the 3rd linear function the time interval will be: 8.6s-16.0s (where 3rd gear changed occurred) The method that I used to find the linear functions equations: I used the same method that is applied to find the equation of a straight line (y = mx + c).
But first I will need to draw three straight lines graph on the original graph along side the actual speed of the car and then I will need to find the gradient and the value of the constant ‘C’ in order to display these linear models using spreadsheet in the computer on the original data.In my equation V = speed (mps) t = time (in seconds) m = gradient 1st linear Function: To get the values for the gradient I chose two points on the first model which are (0, 0) and (4, 29). I chose these two points because they cover the range of speed of the car before the first gear change occurs. I have included the graph which I have drawn on the original graph, and it also shows the two points that I have used to work out the gradient of each linear functions.The gradient = change in y/ change in x m = (0-29) / (0-4) = -29/-4 m = 7.
25 The equation is: V= 7.25t + C On the original graph the value of C = 0 To find the value of ‘C’ (constant) you have to rearrange the equation and insert one point (0,0) from the graph one from t-axis and it’s corresponding speed on the v-axis, which gives: From this sets of values you could see the quadratic function has very high percentage errors in the beginning and through out the model the percentage errors vary, also that the percentage errors becomes lower towards the end.The model does not fit very well in the first gear of the original speed of the car, but from three minutes onwards in its first gear the model start to fit in well with the original speed of the car.
The model fits the original speed of the car better towards the end of the second gear in comparison to the first gear of the car, but the model seems to fit even better through out the third gear in comparison to both the first and the second gear.