Name______________________________________ Date_______________
Lab 3: Creating Mathematical Models of Motion
Thus far we’ve managed to create mathematical models of position vs. time for constant velocity motion and velocity vs. time for constant acceleration. In both cases the graphs of the relevant quantities vs. time appeared to be linear, allowing us to base our models on the very familiar equation of a line, y(x) = mx + b. What about position vs. time for constant acceleration motion? You obtained such data last week in ME 2 Act 3.4. It turns out that a second-order polynomial (quadratic) equation, an old friend from your algebra class, can serve as a basis for creating such models. Our goal today is to learn how to create and understand mathematical models for position vs. time for constant acceleration motion.
But first, let’s see what we can discover about the graphs of second-order polynomials.
Investigation 1: Position vs. time graphs and the parabola: A connection?
In your high-school algebra course you probably learned that the graph of a second-order polynomial equation is a parabola. Polynomial means multiple terms, and second-order refers to the highest power of x that appears in the equation. Mathematicians may write the equation as follows: y(x) = A x 2 + Bx + C where x is the independent variable and y is the dependent variable (the notation y(x) reminds us of this). A, B, and C are parameters.
Although all parabolas have the same basic shape, some are skinny, some are fat, some start one place on a coordinate axis and some start another place. Some parabolas are even upside down! The magnitudes of the parameters and whether they are positive or negative determine how a given parabola looks. In the next few activities you are going to perform calculations and make graphs to demonstrate to yourself how the values of each of the parameters affect the shape of a parabola.
Some skinny, fat, and upside down parabolas in different locations on a graph (A) that uses typical mathematics notation of y vs. x, and (B) that uses typical physics notation of x vs. t.
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Here’s how to set up the computer to look at graphs of quadratic equations with various values of parameters:
1. Open LoggerPro. Select Analyze/Model, and click on “Quadratic”. Leave the values of the parameters at 1.000 for A, B, and C, and click “OK”.
2. Click on Options/Graph options/axes options, and select manual scaling for the y-axis: -
20 to 20, and manual scaling for the x-axis: 0 to 10
You should see the graph of the quadratic equation for which A, B, and C all equal 1.000.
How would we write this equation to include all of the information we wish to communicate? Beginning with the general form of the quadratic equation given above (y(x)
= A x 2 + Bx + C):
3. Think about the independent and dependent variables. In math class, the independent variable is usually denoted by x, and the dependent variable by y. But in this case, what physical quantities do the independent and dependent variables refer to? What symbols do we use for these quantities? Rewrite the general equation accordingly.
4. Think about the parameters A, B, and C. What are their values in this specific case?
Rewrite your equation accordingly.
5. Think about units. Use the fact that all of the terms in a polynomial must have the same units to determine the units corresponding to each of the variables and parameters.
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6. Now let’s explore the effect of the value of the parameter C on the parabola. a. Examine the equation for the parabola and predict what you think the effect of changing the value of C will be. Suppose it’s changed to +10? To –10? What will happen in each case?
Explain your reasons for your prediction. Using dashed lines sketch your predictions on the axes below. b. Test your prediction by changing the values of C from +1 to +10 and then to –10. Draw graphs for these three different values of C in the space below and mark the appropriate values of C on each sketch. c. Describe in words what C has to do with the y–intercept of the graph. Does the basic shape (curvature) of the parabola change in any way? d. Can you see that the y-intercept is the position of an object at time t = 0? Are the units of
C consistent with that interpretation?
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7. What’s the effect of B on the parabola? a. To explore this question, return the values of the parameters to +1 for all of them and then try changing B from +1 to say, +3 and then to –3. Sketch graphs for these three different values of B in the space below and mark the appropriate values of B on each sketch. b. Describe in words what B has to do with the slope of the graph at the y-intercept. Does the basic shape (curvature) of the parabola change in any way? (This is hard to see unless you change your time scale to encompass more of the graph. You might try -10 to +10 for the time scale.) c. Can you see that B determines the slope of the parabola when t = 0? Are the units consistent with this interpretation? Print the graph and attach it to the end of this activity. d. Check this by using the slope tool in LoggerPro to determine the slope of the graph at t =
0.
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8. Now let’s explore the effect of the value of A on the shape of the parabola. a. To do this, return the scale to 0-10, and return values of the parameters to +1 for all of them and then try changing A from +1 to, say, +2 and then to –2. Sketch graphs for these three different values of A in the space below and mark the appropriate values of A on each sketch. b. Describe in words what A has to do with the shape of the graph. Does the basic shape
(curvature) of the parabola change in any way? c. So, the greater the value of A, the ____________(greater or less) the rate that the slope of the graph changes, and thus the ____________(greater or less) the acceleration. d. And negative values of A correspond to parabolas that are _____________(right side up or upside down), in which the slope is____________(increasing or decreasing), corresponding to
____________(positive or negative) acceleration. e. It turns out that the parameter A is equal to half the acceleration. Are the units of A consistent with this interpretation?
9. Summarize your findings in the table below.
Variables x t
Parameters
C B A
Units
Physical interpretation
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Investigation 2: Mathematical modeling - Describing an observed motion with an equation
Can you create a mathematical model of position vs. time for this case of motion with constant acceleration by finding the appropriate values of the parameters C, B, and A in the parabola equation x(t) = A t 2 + Bt + C?
1. Display the data for a cart reversing direction that you collected in Activity 3.4 of the last lab. To obtain the x-t graph, select Insert/Graph, and click on the name of the dependent variable and change it to “position”. Change the scale of the axes so that the curve nearly fills your graph.
2. Model your position data, by adjusting the values of the parameters until the curve passes as close as possible to the greatest number of the experimental data points. Try to be systematic. For example, it may be easiest to start with C. Write down the values of the parameters which give the best fit, and attach a printout of your graph.
3. Write down the equation that describes the particular motion you observed in the way that you learned to do in the last activity.
4. How does the value of C compare with the initial position of the cart? (You can determine this by looking at your data table.)
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5. How does the value of B compare with the initial velocity of the cart, that is, the instantaneous velocity at t = 0 seconds? Check this in two ways: (1) Use the slope tool in
LoggerPro, and (2) Refer to your v-t graph.
6. Do your best to check how the value of A relates to the acceleration of the cart. Try at least two methods to determine the acceleration: (1) On your position graph, determine the instantaneous velocity at two widely different times using the slope tool, (2) Refer to the appropriate feature of your v-t graph. (As you learned, the value of A should correspond to half of the cart’s acceleration. Does this appear to be the case for your model?)
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