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When the data you collect is non-linear and looks like it might be parabolic, you will employ a powerful technique called linearizing the graph to determine whether or not the graph is a parabola with its vertex at the origin. Consider the data and graph below for an experiment in which a student releases a marble from rest allowing it to go down an inclined track. The student notes its position every 1.0 seconds along the track.
Since the graph is not linear for this experiment, you cannot determine the equation that fits the data using only the techniques shown for the previous graphs. The slope of this graph, for instance, is constantly changing. Notice, however, that the trend of the graph shows that as time increases in constant increments, position increases in greater and greater increments. This is why the graph bends away from the time axis. Since position increases at a greater rate than time, is there any mathematical manipulation of the data that could be performed which would allow us to plot a linear graph? Notice that the graph looks like it might be a parabola. Since quadratic equations yield parabolas that take the form y =ax² + bx + c, we might look to this form for a hint. First of all, if this is a parabola, it appears to have its vertex at the origin. When the vertex of a parabola is at the origin, the values of b and c are zero and the equation reduces to y = ax² . If we think that this graph is parabolic with a (0, 0) vertex, we might try to manipulate the data based on the form y = ax² . Think about it. If position is increasing at a greater rate than time, isn't it possible that squaring time would make it increase at a rate that keeps up with position? This reasoning is the basis of creating a test plot.
Basically, a test plot is a graph made with mathematically manipulated data for the purpose of testing whether or not our guess about a mathematical relation might hold true. Since we think that the graph above may represent a square relationship (parabola), we will make a new table in which we will square every time value. We square time because it is the variable that is not keeping up in this case. We will then plot a graph of position vs. time² to determine whether or not our hunch was right. If we are correct, our test plot will yield a straight-line position vs. time². graph. We can then follow the previously detailed process for linear relations to determine the mathematical model that describes the relationship between position and time for our rolling marble.
Note that squaring each of the time values and plotting a new graph of position vs. time² yields a linear graph that passes through the origin. This indicates that the position of the ball is proportional to the square of the time that it has rolled from rest. The mathematical analysis of such a graph is the same as for other linear relations.
The resulting equation is indeed that of a parabola.
The mathematical analysis of this experiment follows:
Our mathematical model that describes the relation between position and time for a marble rolling down an incline is:
position = 3.0 cm/s² · time²
It is important to understand that this equation is not only the equation of the linear graph shown above, but it is also the equation of the original parabolic graph. This idea eludes many students. We have arrived at an equation, which describes our original (curved) graph by manipulating the data (squaring time) to find the equation of a line. As in the direct proportion, the constant slope of 3.0 cm/s² may be replaced with a variable name that describes the nature of the rate of change represented by the slope of the straight-line graph, position vs. time squared. The meaning of this graph’s slope is a topic of study in your physics class this year, so we will not explore the physical meaning of the slope at this time.
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