TI-Nspire™ Activities: Physics
April 22, 2008 through April 29, 2008
A Little Light Work
| Problem 1 – Using a regression to quantify the intensity-distance relationship | |
| Step 1: Tell students to use the directions on the student worksheet to guide them through the activity. They will first examine the distance/intensity dataset in a Lists & Spreadsheet application. Q1. Briefly describe how intensity changes as distance increases. Does the change appear to be linear (that is, do uniform increases in distance produce uniform change in intensity)? Justify your answer. A. Intensity decreases as distance increases, but the relationship does not appear to be linear. Each 0.02-unit increase in distance does not produce a uniform decrease in intensity. |
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| Step 2: Next, students are asked to produce a scatter plot of intensity vs. distance, as shown. Q2. Describe the shape of the graph of intensity vs. distance. A. The data appear to lie on a curved line. Q3. What types of mathematical functions produce graphs with shapes like this? A. Graphs of functions of the type  , also known as inverse-square functions, have shapes similar to that of this graph. |
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| Step 3: Students are next asked to use a power regression to find the best-fit equation for the data, as shown. Q4. Write the equation that best fits the data on your answer sheet. Round the calculated values to three significant figures. A. The power regression yields the following best-fit equation, where I is intensity and d is distance: I=(0.00172)d-1.96 Q5. Why do you think relationships like these are called “inverse-square laws”? A. Relationships like these are called “inverse-square laws” because one of the variables is equal to a constant times the inverse of the square of the other variable. |
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| Step 4: Next, students plot the regression equation on the graph of the data, as shown. Q6. Does the graph of the regression equation appear to fit the data well? A. Yes, the graph appears to fit the data very well. |
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| Problem 2 – Building your own model for the relationship | |
| Step 1: In this part of the activity, students attempt to fit an inverse-square curve to the data. They are first asked to plot the data on a scatter plot and use the Trace function to identify the coordinates of the leftmost point, as shown. Q7. Use substitution to find the value of k that makes the equation above true for the leftmost point on the graph. Show your work. A.  |
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| Step 2: Next, students graph their calculated functions on the graph of the data. They should vary the value of k to force the curve to fit the data. Q8. What value of k makes the curve fit the data best? A. The exact values that students will come up with will vary, but the calculated value of k (0.00152) produces a fairly good fit to the data. Q9. How does this value of k compare to the value of a calculated in problem 1? A. The best-fit value of k calculated here is significantly different from the value of a from problem 1. You may wish to discuss this result with students in more detail. Because they have not used the best-fit exponent (–1.96), they should expect that the best-fit coefficient for this equation will be different from that calculated in the power regression. |
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| Problem 3 – Another confirmation of the inverse-square relationship | |
Step 1: In the final part of this activity, students examine the relationship between intensity and  . They are first asked to define a new variable, lindist, which is equal to , as shown. Students may have difficulty understanding why the graph of intensity vs. lindist should be a straight line. If necessary, review the equations with them to help them grasp the concept before continuing. |
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| Step 2: Next, students plot intensity vs. lindist on a scatter plot, as shown. Q10. Describe the shape of the graph. A. The data appear to lie on a straight line. |
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Step 3: Finally, students use the Calculator application to determine the linear equation that best relates intensity to  . (Note that students could also calculate the linear regression on these variables using the Stat Calculations menu in the Lists & Spreadsheet application on page 3.1, as they did in problem 1.) Q11. Does the r2 value for the linear regression support the statement that intensity is directly related to  ? Explain your answer.A. The r2 value for the equation is very near 1, indicating a good fit between the equation and the data. Q12. Use each of the three equations relating I and d that you found in this activity to calculate the intensity of this light at a distance of 0.02 units from the source. Show your work. A. Using the equation from problem 1:  Using the equation from problem 2:  (Note that values calculated from this equation will vary based on the value of k the student selected.) Using the equation from problem 3:  Q13. How could you test your calculations from question 12 to determine which value is most accurate? A. One way to test the accuracy of the values would be to repeat the experiment that generated the values, but include an additional data point at 0.02 units away from the light sensor. Q14. Using the equation for the surface area of a sphere, explain why the relationship between intensity and distance follows an inverse-square law. A. The equation for the surface area of a sphere,  , shows that the surface area of a sphere is directly related to the square of the radius of the sphere. The intensity of the light is given by  , where E is the energy of the light. As this equation shows, the intensity of the light is directly proportional to the inverse square of the radius. |
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