placeholderSources:

-Brueningsen, Bower, Antinone, Brueningsen-Kerner. “Activity 30: Verifying Velocity.”Real-World Math with the CBL System. Texas Instruments Incorporated (1999).

-“Activity 4: Exploring Slope.”CBL Explorations in Algebra for the TI-82 and TI-83.Meridian Creative Group (1996).

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Introductionr

When you take a long trip, you are often interested in how far you went, how long it took, and how fast you went. When you buy a sports car, you may be interested in the time it takes to change speed, or accelerate, from zero to 60 miles per hour. In the following exploration, you will investigate the relationship between these concepts.er

You may recall that velocity is defined as the change in distance divided by change in time. It is common to use the Greek letter D to mean "change in". The equation for average velocity isder

der

which we read as "Velocity equals change in distance over change in time." This sounds very close to the definition of slope! In fact, on a distance-time graph, the slope gives the velocity.

Materialsr

1 Motion Cart 1 CBR Unit 1 TI-83 or TI-83 Plus Calculator with Unit-to-Unit Link Cable placeholder

SetupplaceholderConnect the CBR to the calculator using the Unit-to-Unit Link Cable. If necessary, download the Ranger program from the CBR. Run the Ranger program on your calculator. TI-83 users press [PRGM] and select Ranger. Hit [ENTER] to run. TI-83 Plus users press [APPS] and select CBL/CBR. Press any key, then select 3:RANGER to run. From the MAIN MENU, select 1:SETUP/SAMPLE. Set the options to match the screen below. Move the arrow to the START NOW command when done. Set up your cart as described in the picture below.

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Collecting Data

1. With the cart and CBR set up as indicated above, have one person prepare to release the cart while another prepares to start the CBR. (Make sure nothing is in the "clear zone" of the CBR)

2. Begin collecting data with the CBR and release the cart.CAUTION! Make sure the cart does not hit the CBR! (If it is close, move the CBR back to a safe distance).Ideally, you should not release the cart until shortly AFTER sampling begins (half a second), and the cart should come to rest BEFORE sampling stops. Your graph should have a flat region, then a downward-sloping region, and another flat region.

3. If you are not satisfied with your data, press [ENTER] to return to the PLOT MENU and select 5:REPEAT SAMPLE. Repeat steps 2 and 3 until you are satisfied with your results.

4. Return to the PLOT MENU and press 7:QUIT.laceholder

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1. Press the GRAPH key on the TI-83 to view the Distance-Time graph of the data you just collected. Sketch the graph in the space at right.

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2. Label the three regions on your graph: (1) the first flat region, (2) the downward sloping region, and (3) the second flat region. What part of the cart’s “journey” do these regions represent?

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Flat region (1): ________________________________________________

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Sloping region (2): ______________________________________________

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Flat region (3): ________________________________________________

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3. Recall that the slope of a Distance-Time graph gives us velocity. Based on the Distance-Time graph you sketched above, create a sketch of the Velocity-Time graph in the space below.(Hint: When an object is moving towards the CBR, its velocity is negative. Thus, as your cart slows down, its velocity becomesless negative.

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4. The CBR is called a “motion sensor,” but in reality it doesn’t sense motion at all. Instead, the CBR is capable ofcalculatingmotion (that is, velocity) based a collection of distance measurements. We can replicate this process to check the velocity-time graph you sketched above.

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To calculate the velocity of an object, the CBR computes the slope between each pair of points on the distance-time graph. The picture below demonstrates this process.

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The time data in your calculator is stored in the list L_{1}and the distance data is stored in L_{2}. To determine the velocity, we must apply the definition of average velocity to L_{1}and L_{2}. The definition is given at the beginning of this activity. Show your equation, in terms of DL_{1}and DL_{2}, below.(The picture above my help you.)placeholderv = ____________________

placeholder5. To create a list of velocity values, we will use the calculator's

DList(command. This command takes the difference between successive values of a list.

placeholderWe wish to compute DL

_{2 }divided by DL_{1}, and we will store the resulting velocity data in L_{4.}On your calculator, execute the following command to do just that: DList(L_{2}) / DList(L_{1}) L_{4}placeholderNote: To find theDList(command, press[2^{nd}] [LIST] [>] [7].

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When you press [ENTER], the average velocity values should be placed in L_{4}.

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6. To check the velocity graph that you sketched above, you can plot the average velocity values versus time. Let us copy the time data in L_{1}to L_{3}. To do so, execute the following command: L_{1 }arrowpointingright L_{3}.

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Now follow these steps:

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·Press [2^{nd}] [STATPLOT].

·Select1:Plot1.

·Move the cursor toOnand press [ENTER].

·Move the cursor down toXlist:and press [2^{nd}] [L_{3}]

·Move the cursor down toYlist:and press [2^{nd}] [L_{4}].

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This tells the calculator to plot the velocity data in L_{4}against the time data in L_{3}. Now press [GRAPH]. (Don’t worry! You)shouldget an error!

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The calculator reports an error due to a “dimension mismatch”. This means that L_{3}and L_{4}have different lengths. It turns out that L_{3}has exactly one more item than L_{4}. Explain why in the space below.(Hint: Think about howL_{4}was created. It may help to look at the picture above that shows the slope between each pair of points on the distance-time graph.)

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7. To fix the problem, we can delete one element of L_{3}.

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To do so, press [STAT] [ENTER] to enter the statistics editor. Use the arrow keys to highlight the 1^{st}row of the list L_{3}. Then press the [DEL] key. This will delete the first element of the list L_{3}making it the same length as L_{4}. That should make the calculator happy!

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Press [ZOOM] [9] to view the velocity-time graph you have created. Sketch the resulting plot at right. Look back at your prediction in Question 3. How well does your sketch in Question 3 match up with the "real" velocity-time graph computed by the calculator?

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8. Use the velocity-time graph above to predict the graph of acceleration-time. Sketch your prediction in the space at the right.(Hint: When an object is speeding up as it heads towards the CBR, its acceleration is negative. When an object is slowing down as it heads towards the CBR, its acceleration is positive. As always, the acceleration is zero when the velocity is not changing.)

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9. Follow the steps you used to compute the average velocity, above, to compute average acceleration. You’ll use the commands DList(L_{4}) / DList(L_{3}) L_{6 }and L_{3 }L_{5.}

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Don’t forget to use the statistics editor to delete one row of L_{5}or you’ll get another “Dimension Mismatch” error when you plot.

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Plot the acceleration-time plot by plotting L_{5}vs. L_{6}. (Press [STATPLOT] and select 1:Plot1. ChangeXlist:to L_{5}andYlist:to L_{6}.)

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Use [ZOOM] [9] to zoom out to fit your graph. Sketch a graph of the acceleration-time graph in the space at below.

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10. How well does your sketch in Question 9 match up with the "real" acceleration-time graph computed by the calculator?

Jon Hasenbank, 2000.

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