Introduction: The following exercises build upon your intuitive ideas and previous experience in labs about motion. Remember that Calculus is the mathematical language of motion and change.

Elements of this lab were adapted from TI's "Getting Started with CBR" and Ostebee & Zorn's "Calculus".

A Prepare for data collection.

1. Connect one TI-85 from your group to the CBR with the long calculator-to-CBR cable. Later, all members may link to this calculator to receive the data.
2. Use the keystrokes (2nd-Link), F2 to prepare the calculator to receive a program.
3. Open the pivoting head on the CBR. Press the (85/86) button to send the program. The CBR should beep once, and the calculator screen should display DONE.
4. You have just loaded the "RANGER" program from the CBR to your calculator.

B. Try walking a velocity graph.

• This is not easy. Do the best you can. The main idea is to interpret the graph physically.

1. There are several ways you may try this, as you probably recall from the other walking lab. I suggest Method 1. Remember that the distance from the wall should always be between 0.5 meters (1.5 feet) and 4 meters (12 feet).

Method 1: Hold the CBR in one hand and the calculator in the other. The walker aims the sensor directly at the wall while walking. In this set-up, it is helpful for another group member to press the buttons on the calculator, although sometimes it is difficult to synchronize movements.

Method 2:

• Attach the CBR to some fixed object, like a chair.
• Have a group member hold it; (it must be steady).

• The walker may hold the calculator. However, sometimes the CBR detects the long cord. Also, the cord limits the walker's motion.
• The walker may be completely free if someone else views the graph and directs the walker's movements.

1. Use the keystrokes PRGM, NAMES. Select "RANGE" for "RANGER"; (you may need to use MORE to see it). Press ENTER to begin running the program.
2. Press ENTER again to access the menu.
3. From the MAIN MENU, choose APPLICATIONS. Choose METERS.
4. From the APPLICATIONS menu, choose "Vel Match". Brief instructions are given.
5. Press ENTER to display the graph to match. Take a moment to study the graph.
6. Press ENTER to begin data collection. There are a clicking sound and a green light as the data is collected. The walker's velocity is plotted on the screen.
7. When the sample is finished, examine how well your "walk" matched the graph.
8. Press ENTER to display the OPTIONS menu. Using the SAME MATCH several times, begin from different distances.

What effect does this have on the velocity graph?

 

Wonder why this is so. Your group may spend about 10 minutes attempting to walk graphs.

9. Select QUIT from the OPTIONS menu. You will see the lists containing your data:

L1 TIME

L2 DISTANCE

L3 VELOCITY

L4 ACCELERATION

10. So that we all have the same data, RECEIVE L3 from my calculator and distribute it to your group.

C. Process the data.

1. Choose EDIT. Set the lists appropriately. To avoid any complications, retype the first element of the xList.
2. In the DRAW menu, select SCAT to see your data. Think of an appropriate regression function.
3. In the CALC menu, set the lists. Perform a regression. Store the regression equation with STREG in y1. Remember to use lower case y. Print your regression equation from this screen with "Get LCD" from the RECEIVE menu of TI-GRAPH LINK.
4. In the DRAW menu, choose DRREG. Reselect SCAT if necessary.
5. Print this picture using TI-GRAPH LINK.
6. EXIT the STAT menu.

D. Analyze and interpret the results.

1. Describe y1 in words: y1 represents the ___________ velocity (in meters/second) of an object moving ... (complete the sentence)

2. Write an expression using y1 for each of the following.

E. Here is a simple example to illustrate the ideas involved in the next part.

1. Suppose you travel for 4 hours at 60 miles per hour. How far did you drive? Show your calculation.

2. You drive another 2 hours at 40 mph, then 1 hour at 50 mph in the opposite direction.

Fill in the table. For leg 2, indicate where the numbers came from with annotations such as "given data", "velocity for leg 2 - velocity for leg 3", etc.

Let t be the elapsed time since the start and v(t) the velocity at time t.

Sketch the velocity and speed functions for this example.

5. What is the net distance from your starting point?

Sketch a picture to geometrically illustrate the calculation.

6. What is the total distance from your starting point?

Sketch a picture to geometrically illustrate the calculation.

F. Now let’s return our attention to the velocity data we collected.

1. Examine the L1 and L3 data. Use the first several data points to fill in the table. For leg 2, indicate where the numbers came from.

Let t be the elapsed time since the start. Let v(t) be the velocity at time t where velocity is assumed constant on a single leg.

2. What is the net distance from the starting point for the first three legs, using v(t)? Indicate the data values used in the calculation with something like 3*5 + (4-9) (an erroneous calculation) and label the data values (time 1, velocity 2, etc.)

Note: There are 2 small variations on this calculation. One takes advantage of the constant increment in time; this one is easier for a later calculation.

3. What is the total distance from the starting point for the first three legs, using v(t)?

4. Set the GRAPH RANGE to show the regression function for the velocity data, y1, on the first three legs of this journey; allow for the fourth time data point. Be sure to use an xScl and yScl in small enough units to place tick marks on the axes. Use the keystrokes STAT, DRAW, SCAT to show the actual data points. Remember, if a data point lies on an axis, you will not see it. Press EXIT, then CLEAR to remove the menus. Print the graph with GET LCD from GRAPH LINK. On this graph, provide a sketch that illustrates the geometry of your net distance calculation.

5. Define a function y2 from which to calculate total distance traveled. Record here.

Reset the GRAPH RANGE as necessary, still for the first three legs. Print the graph of y2 with GET LCD from GRAPH LINK. On this graph, provide a sketch that illustrates the geometry of your total distance calculation.

6. Apply an appropriate function to L3 to enable total distance calculations; store this in L5. Record your command here.

7. Calculate the net distance traveled for the entire trip. Refer to #2. Hint: In part, you will use the keystrokes (2nd-LIST), OPS, MORE, sum list name, ENTER. Give the value and indicate the calculation.

Repeat for total distance traveled.

7. The velocity actually changes continuously, so it is more accurate to use continuous functions to calculate net distance and total distance. Give expressions similar to those in D2, using y1 and y2, and values for each item below. Unless otherwise specified, use the entire interval.

********

Note: There are 2 ways to obtain an automatic (approximate) value of an integral on the calculator.

a) GRAPH the function you wish to integrate on an appropriate RANGE. It may be best to widen the range from the interval of integration. Find MATH in the GRAPH menu (MORE after the "y(x)="portion of the main menu). From the MATH menu, select the integral. Cursor left, near the left endpoint; press ENTER. Cursor right, close to the right endpoint; press ENTER. You will see triangles on the screen indicating the interval over which you are integrating. Press ENTER. The calculator will return a numeric value to approximate the integral.

b) Here’s another way to get the integral value. From the home screen, use the keystrokes (2nd-CALC) and select fnInt. Type the expression or function name you wish to integrate, the variable of integration, the right endpoint, and the left endpoint. Your command should look something like

fnInt(y3,x,0,10)

or fnInt(x^2+1,x,0,10)

The x tells the calculator to integrate with respect to x (like the dx that we write). In this example, the interval is 0 to 10.

********

1. the net distance traveled
2. the average velocity of the walker
3. the instantaneous velocity at 3 seconds
4. the speed of the walker at 3 seconds
5. the total distance traveled
6. the average speed
7. the net speed
8. the average acceleration

8. Compare the calculation for net distance from this last part with your calculation from the data points. Are they close?

Repeat for total distance.

3. Save and print the lists.

• SEND the lists L1 and L3 from the calculator to TI-GRAPH LINK.
• RECEIVE from GRAPH LINK as a GROUP FILE to my class disk. To keep the file names straight, choose one from my list and sign up on the sheet at the desk with your group members' names. Make a note of the filename.
• In TI-GRAPH LINK, choose FILE - OPEN your file to view the lists.
• In TI-GRAPH LINK, choose FILE-PRINT-GROUP. Then "add" your filename, select DONE. Print the file.