GETTING STARTED WITH CBR

INCLUDING

STUDENT ACTIVITIES
Calculator-Based Ranger (CBR)
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calculator-to-CBR cable

4 AA batteries

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Table of contents

INTRODUCTION
What is CBR? Getting started with CBR Its as easy as 1, 2, 3 Hints for effective data collection Activities with teacher notes and student activity sheets
Activity 1 Match the graph Activity 2 Toy car Activity 3 Pendulum Activity 4 Bouncing ball Activity 5 Rolling ball
Teacher information Technical information CBR data is stored in lists RANGER settings Using CBR with CBL or with CBL programs Programming commands Service information Batteries In case of difficulty TI service and warranty RANGER menu map
COPYING PERMITTED PROVIDED TI COPYRIGHT NOTICE IS INCLUDED 1997 TEXAS INSTRUMENTS INCORPORATED
linear linear sinusoidal parabolic parabolic

44 inside back cover

What is CBR?
CBR CBR (Calculator-Based Ranger ) Ranger
sonic motion detector use with TI-82, TI-83, TI-85/CBL, TI-86, and TI-92 bring real-world data collection and analysis into the classroom easy-to-use, self-contained no programming required
Includes the RANGER program
the versatile RANGER program is one button away MATCH and BOUNCING BALL programs are built into RANGER primary sampling parameters are easy to set

What does CBR do?

With CBR and a TI graphing calculator, students can collect, view, and analyze motion data without tedious measurements and manual plotting.
CBR lets students explore the mathematical and scientific relationships between distance,
velocity, acceleration, and time using data collected from activities they perform. Students can explore math and science concepts such as:
motion: distance, velocity, acceleration graphing: coordinate axes, slope, intercepts functions: linear, quadratic, exponential, sinusoidal calculus: derivatives, integrals statistics and data analysis: data collection methods, statistical analysis

Whats in this guide?

Getting Started with CBR is designed to be a guide for teachers who dont have extensive calculator or programming experience. It includes quick-start instructions for using CBR, hints on effective data collection, and five classroom activities to explore basic functions and properties of motion. The activities (see pages 1332) include:
teacher notes for each activity, plus general teacher information step-by-step instructions a basic data collection activity appropriate for all levels explorations that examine the data more closely, including what-if scenarios suggestions for advanced topics appropriate for precalculus and calculus students a reproducible student activity sheet with open-ended questions appropriate for a wide range of grade levels

(cont.)

green light to indicate when data collection is occurring (sound also available)
red light to indicate special conditions

button

to initiate sampling

battery door (on bottom)

sonic sensor to record up to 200 samples per second with a range between 0.5 meters and 6 meters (1.5 feet and 18 feet)
port to connect to CBL (if desired)
port to connect to TI graphing calculators using the included 2.25-meter (7.5-foot) cable
standard threaded socket to attach a tripod or the included mounting clamp (on back)
CBR includes everything you need to begin classroom activities easily and quickly just add
TI graphing calculators (and readily available props for some activities).
sonic motion detector RANGER program in the CBR
calculator-to-CBR cable 4 AA batteries
pivoting head to aim sensor accurately
buttons to transfer RANGER program to calculators
mounting clamp 5 fun classroom activities
Getting started with CBRIts as easy as 1, 2, 3
With CBR, youre just three simple steps from the first data sample!

Connect

Connect CBR to a TI graphing calculator using the calculator-to-CBR cable. Push in firmly at both ends to make the connection. Note: The short calculator-to-calculator cable that comes with the calculator also works.

Transfer

RANGER, a customized program for each calculator, is in the CBR. Its easy to transfer the appropriate program from the CBR to a calculator.
First, prepare the calculator to receive the program (see keystrokes below).
TI-82 or TI-83 TI-85/CBL or TI-86 TI-92 Go to the Home screen.

[LINK]

Next, open the pivoting head on the CBR, and then press the appropriate program-transfer button on the CBR.
During transfer, the calculator displays RECEIVING (except TI-92). When the transfer is complete, the green light on CBR flashes once, CBR beeps once, and the calculator screen displays DONE. If there is a problem, the red light on CBR flashes twice and CBR beeps twice. Once youve transferred the RANGER program from CBR to a calculator, you wont need to transfer it to that calculator again unless you delete it from the calculators memory. Note: The program and data require approximately 17,500 bytes of memory. You may need to delete programs and data from the calculator. You can save the programs and data first by transferring them to a computer using TI-Graph Link or to another calculator using a calculator-to-calculator cable or the calculator-to-CBR cable (see calculator guidebook).

Run the RANGER program (see keystrokes below).
TI-82 or TI-83 Press ^. Choose RANGER. Press. TI-85/CBL or TI-86 Press ^ A. Choose RANGER. Press. TI-92 Press L [VAR-LINK]. Choose RANGER. Press .
The opening screen is displayed. Press. The MAIN MENU is displayed.
MAIN MENU SETUPSAMPLE SET DEFAULTS APPLICATIONS PLOT MENU TOOLS QUIT
& & & & &
view/change the settings before sampling change the settings to the default settings DISTANCE MATCH, VELOCITY MATCH, BALL BOUNCE plot options GET CBR DATA, GET CALC DATA, STATUS, STOPCLEAR
From the MAIN MENU choose SET DEFAULTS. The SETUP screen is displayed. Press to choose START NOW. Set up the activity, and then press to begin data collection. Its that easy!
For quick results, try one of the classroomready activities in this guide!

Important information

This guide applies to all TI graphing calculators that can be used with CBR, so you may find that some of the menu names do not match exactly those on your calculator. When setting up activities, ensure that the CBR is securely anchored and that the cord cannot be tripped over. Always exit the RANGER program using the QUIT option. The RANGER program performs a proper shutdown of CBR when you choose QUIT. This ensures that CBR is properly initialized for the next time you use it. Always disconnect CBR from the calculator before storing it.
Hints for effective data collection

Getting better samples

How does CBR work?
Understanding how a sonic motion detector works can help you get better data plots. The motion detector sends out an ultrasonic pulse and then measures how long it takes for that pulse to return after bouncing off the closest object.
CBR, like any sonic motion detector, measures the time interval between transmitting the ultrasonic pulse and the first returned echo, but CBR has a built-in microprocessor that does much more. When the data is collected, CBR calculates the distance of the object from the CBR using a speed-of-sound calculation. Then it computes the first and second derivatives of

the distance data with respect to time to obtain velocity and acceleration data. It stores these measurements in lists L1, L2, L3, and L4. Performing the same calculations as CBR is an interesting classroom activity.
Collect sample data in REALTIME=NO mode. Exit the RANGER program. Use the sample times in L1 in conjunction with the distance data in L2 to calculate the
velocity of the object at each sample time. Then compare the results to the velocity data in L3.
(L2n+1 + L2n)2 N (L2n + L2n-1)2

L1n+1 N L1n

Use the velocity data in L3 (or the student-calculated values) in conjunction with the
sample times in L1 to calculate the acceleration of the object at each sample time. Then compare the results to the acceleration data in L4.

Object size

Using a small object at a far distance from the CBR decreases the chances of an accurate reading. For example, at 5 meters, you are much more likely to detect a soccer ball than a ping-pong ball.

Minimum range

When the CBR sends out a pulse, the pulse hits the object, bounces back, and is received by the CBR. If an object is closer than 0.5 meters (1.5 feet), consecutive pulses may overlap and be misidentified by CBR. The plot would be inaccurate, so position CBR at least 0.5 meters away from the object.

Maximum range

As the pulse travels through the air, it loses its strength. After about 12 meters (6 meters on the trip to the object and 6 meters on the trip back to the CBR), the return echo may be too weak to be reliably detected by the CBR. This limits the typical reliably effective distance from the CBR to the object to less than 6 meters (19 feet).

The clear zone

The path of the CBR beam is not a narrow, pencil-like beam, but fans out in all directions up to 10 in a cone-shaped beam. To avoid interference from other objects in the vicinity, try to establish a clear zone in the path of the CBR beam. This helps ensure that objects other than the target do not get recorded by CBR. CBR records the closest object in the clear zone.

Reflective surfaces

Some surfaces reflect pulses better than others. For example, you might see better results with a relatively hard, smooth surfaced ball than with a tennis ball. Conversely, samples taken in a room filled with hard, reflective surfaces are more likely to show stray data points. Measurements of irregular surfaces (such as a toy car or a student holding a calculator while walking) may appear uneven. A Distance-Time plot of a nonmoving object may have small differences in the calculated distance values. If any of these values map to a different pixel, the expected flat line may show occasional blips. The Velocity-Time plot may appear even more jagged, because the change in distance between any two points over time is, by definition, velocity. You may wish to apply an appropriate degree of smoothing to the data.

RANGER settings

Sample times
TIME is the total time in seconds to complete all sampling. Enter an integer between 1
second (for fast moving objects) and 99 seconds (for slow moving objects). For
REALTIME=YES, TIME is always 15 seconds.
When TIME is a lower number, the object must be closer to the CBR. For example, when TIME=1 SECOND, the object can be no more than 1.75 meters (5.5 feet) from the CBR.

Starting and stopping

The SETUP screen in the RANGER program provides several options for starting and stopping sampling.
BEGIN ON: [ENTER]. Starts sampling with the calculators key when the person initiating the sampling is closest to the calculator. BEGIN ON: [TRIGGER]. Starts and stops sampling with the CBR button when the person initiating the sampling is closest to the CBR.
In this option, you also can choose to detach the CBR. This lets you set up the sample, disconnect the cord from the CBR, take the CBR where the action is, press , sample, reattach the CBR, and press to transfer the data. Use BEGIN ON: [TRIGGER] when the cord is not long enough or would interfere with data collection. This is not available in REALTIME=YES mode (such as the MATCH application). BEGIN ON: DELAY. Starts sampling after a 10-second delay from the time you press. It is especially useful when only one person is doing an activity.

Trigger button

The effect of varies depending on the settings.
starts sampling, even if BEGIN ON: [ENTER] or BEGIN ON: DELAY is selected. It also
stops sampling, but usually you will want to let a sample complete.
In REALTIME=NO, after sampling has stopped, automatically repeats the most recent sample, but does not transfer the data to the calculator. To transfer this data, from the MAIN MENU choose TOOLS, and then choose GET CBR DATA. (You also can repeat a sample by choosing REPEAT SAMPLE from the PLOT MENU or START NOW from the SETUP screen.)

Smoothing

Smoothing capabilities built into the RANGER program can reduce the effect of stray signals or variations in the distance measurements. Avoid excessive smoothing. Begin with no smoothing or LIGHT smoothing. Increase the degree of smoothing until you obtain satisfactory results.
For an activity with a higher-than-average likelihood of stray signals, you may wish to increase the smoothing on the SETUP screen before sampling (see page 38). For already-collected REALTIME=NO data, you can apply smoothing to the data. The calculator must be connected to the CBR. Choose PLOT TOOLS from the PLOT MENU, choose SMOOTH DATA, and then choose the degree of smoothing.
Noisewhat is it and how do you get rid of it?
When the CBR receives signals reflected from objects other than the primary target, the plot shows erratic data points (noise spikes) that do not conform to the general pattern of the plot. To minimize noise:

Each time you change smoothing, the CBR applies the new smoothing factor, transfers the adjusted data to the calculator, and stores the smoothed values in the lists. Choosing a domain changes the lists stored in the calculator. If you need to, you can recover the original data from the CBR. From the MAIN MENU in the RANGER program, choose TOOLS. From the TOOLS menu, choose GET CBR DATA. You also can share the same data with many students, even if they are using different types of TI graphing calculators. This allows all students to participate in data analysis activities using the same data (see page 11).
10 GETTING STARTED WITH CBR
Using CBR in detached mode
Because the CBR cannot send data to the calculator immediately in detached mode, certain settings are required. On the SETUP screen:
Set REALTIME=NO. Set BEGIN ON=[TRIGGER].
The RANGER program prompts you when to detach the CBR and when to reattach it. No special procedures are required.

Sharing data

What if you want the entire class to analyze the same data at the same time? With CBR you can disseminate REALTIME=NO data quickly within a classroom.
Transfer the RANGER program to all students calculators prior to data collection. Collect the data with the CBR in REALTIME=NO mode. Have the first student attach his or her calculator to the CBR using either the calculatorto-CBR cable or the calculator-to-calculator cable.
From the MAIN MENU in the RANGER program, choose TOOLS. From the TOOLS menu,
choose GET CBR DATA. TRANSFERRING. is displayed and the plot appears.
Press to return to the PLOT MENU, and then choose QUIT. Detach the cable. Connect another calculator (of the same type) to the calculator with the data. On the
receiving calculator, from the MAIN MENU in the RANGER program, choose TOOLS. From the TOOLS menu, choose GET CALC DATA. Instructions are displayed telling you how to set the sending calculator. When it is ready, press , and lists L1, L2, L3, L4, and L5 are transferred automatically.
Transfer the data to another students calculator from CBR while other students
continue the calculator-to-calculator transfers. Once all students have the same data, they can analyze the data in RANGER using the PLOT MENU or outside RANGER using the calculators list and graphing features. To share data on the TI-85, use the LINK feature outside of RANGER to transfer the lists.
GETTING STARTED WITH CBR 11
Beyond simple data collection

Materials

calculator CBR calculator-to-calculator cable A TI ViewScreen allows other students to watch and provides much of the fun of this activity.
Students really enjoy this activity. Plan adequate time because everybody will want to try it! This activity works best when the student who is walking (and the entire class) can view his or her motion projected on a wall or screen using the TI ViewScreen. Guide the students to walk in-line with the CBR; they sometimes try to walk sideways (perpendicular to the line to the CBR) or even to jump up! Instructions suggest that the activity be done in meters, which matches the questions on the student activity sheet. See pages 612 for hints on effective data collection.

Typical plots

GETTING STARTED WITH CBR 13

Data collection

linear
Hold the CBR in one hand, and the calculator in the other. Aim the sensor directly at a
wall. Hints: The maximum distance of any graph is 4 meters (12 feet) from the CBR. The minimum range is 0.5 meters (1.5 feet). Make sure that there is nothing in the clear zone (see page 7).
Run the RANGER program (see page 5 for keystrokes for each calculator). From the MAIN MENU choose APPLICATIONS. Choose METERS. From the APPLICATIONS menu choose DISTANCE MATCH. General instructions are
displayed. DISTANCE MATCH automatically takes care of the settings.
Press to display the graph to match. Take a moment to study the graph. Answer
questions 1 and 2 on the activity sheet.
Position yourself where you think the graph begins. Press to begin data collection.
You can hear a clicking sound and see the green light as the data is collected.
Walk backward and forward, and try to match the graph. Your position is plotted on

the screen.

When the sample is finished, examine how well your walk matched the graph, and

then answer question 3.

Press to display the OPTIONS menu and choose SAME MATCH. Try to improve your
walking technique, and then answer questions 4, 5, and 6.
14 GETTING STARTED WITH CBR

Explorations

In DISTANCE MATCH, all graphs are comprised of three straight-line segments.
Press to display the OPTIONS menu and choose NEW MATCH. Study the first
segment and answer questions 7 and 8.
Study the entire graph and answer questions 9 and 10. Position yourself where you think the graph begins, press to begin data collection,
and try to match the graph.
When the sampling stops, answer questions 11 and 12. Press to display the OPTIONS menu and choose NEW MATCH. Study the graph and answer questions 13, 14, and 15. Press to display the OPTIONS menu. Repeat the activity if desired, or return to the
MAIN MENU, and then choose QUIT to exit the RANGER program.

GETTING STARTED WITH CBR 17
Position the car at least 0.5 meters (1.5 feet) from the CBR, facing away from the CBR in
a straight line. Hints: Aim the sensor directly at the car and make sure that there is nothing in the clear zone (see page 7).
Before starting data collection, answer question 1 on the activity sheet. Run the RANGER program (see page 5 for keystrokes for each calculator). From the MAIN MENU choose SETUPSAMPLE. For this activity, the settings should be:
REALTIME: TIME (S): DISPLAY: BEGIN ON: SMOOTHING: UNITS: NO 5 SECONDS DISTANCE [ENTER] LIGHT METERS
Instructions for changing a setting are on page 38.
Choose START NOW. Press when you are ready to begin. Start the car and quickly move out of the clear
zone. You can hear a clicking sound as the data is collected and the message TRANSFERRING. is displayed on the calculator.
When the data collection is concluded, the calculator automatically displays a DistanceTime plot of the collected data points.
Compare the plot of the data results to your prediction in answer 1 for similarities and

differences.

18 GETTING STARTED WITH CBR
Activity 2Toy car (cont.)
The values for x (time) in half-second intervals are in the first column in question 2.
Trace the plot and enter the corresponding y (distance) values in the second column. Note: Include results only from the linear part of the plot. You may need to disregard inconsistent data at the beginning of the data collection. Also, you may need to approximate the distance (the calculator may give you distance for 0.957 and 1.01 seconds instead of exactly 1 second). Pick the closest one or take your best guess.
Answer questions 3 and 4. Calculate the changes in distance and time between each data point to complete the
third and fourth columns. For example, to calculate @Distance (meters) for 1.5 seconds, subtract Distance at 1 second from Distance at 1.5 seconds.
The function illustrated by this activity is y = mx + b. m is the slope of a line. It is

1. varies (in meters) 2. varies (in meters) 3. varies (in seconds); T (one period) = total time of 10 periods10; averaging over a larger sample tends to minimize inherent measurement errors 4. the total arc length, which should be approximately 4 times the answer to question 2; because an arc is longer than a straight line 5. sinusoidal, repetitive, periodic; distance from the xaxis to the equilibrium position 6. each cycle is spread out horizontally; a plot spanning 10 seconds must fit more cycles in same amount of screen space, therefore cycles appear closer together 7. (total # of cycles)(5 seconds) = cyclessecond; easier to view full cycles, and fewer measurement errors 8. f = 1T, where T is time for 1 period 9. decreased period; increased period (Pendulum length is directly related to period time; the longer the string, the longer the period. Students can explore this relationship using the calculators list editor, where they can calculate the period for various values of L.) 10. A (amplitude) = total distance that the pendulum travels in 1 period 11. both sinusoidal; differences are in amplitude and phase 12. equilibrium position 13. when position = maximum or minimum value (when the weight is at greatest distance from equilibrium). 14. It doesnt. T depends only on L and g, not mass.

Ideas for weights:

See pages 612 for hints on effective data collection.

Physical connections

An object that undergoes periodic motion resulting from a restoring force proportional to its displacement from its equilibrium (rest) position is said to exhibit simple harmonic motion (SHM). SHM can be described by two quantities.
The period T is the time for one complete cycle. The amplitude A is the maximum displacement of the object from its equilibrium position (the position of the weight when at rest).
For a simple pendulum, the period T is given by:

T = 2p

where L is the string length and g is the magnitude of the acceleration due to gravity. T does not depend on the mass of the object or the amplitude of its motion (for small angles). The frequency f (number of complete cycles per second) can be found from: 1 f = T, where f is in hertz (Hz) when T is in seconds. The derivatives of a sinusoidal plot are also sinusoidal. Note particularly the phase relationship between the weights position and velocity.

1. time (from start of sample); seconds; height distance of the ball above the floor; meters or feet 2. initial height of the ball above the floor (the peaks represent the maximum height of each bounce); the floor is represented by y = 0.

Extensions

Integrate under Velocity-Time plot, giving the displacement (net distance traveled) for any chosen time interval. Note the displacement is zero for any full bounce (ball starts and finishes on floor).
GETTING STARTED WITH CBR 25

parabolic

Begin with a test bounce. Drop the ball (do not throw it).
Hints: Position the CBR at least 0.5 meters (1.5 feet) above the height of the highest bounce. Hold the sensor directly over the ball and make sure that there is nothing in the clear zone (see page 7).
Run the RANGER program (see page 5 for keystrokes for each calculator). From the MAIN MENU choose APPLICATIONS. Choose METERS or FEET. From the APPLICATIONS menu choose BALL BOUNCE. General instructions are displayed.
BALL BOUNCE automatically takes care of the settings.
Hold the ball with arms extended. Press. The RANGER program is now in Trigger
mode. At this point, you may detach CBR from the calculator.
Press. When the green light begins flashing, release the ball, and then step
back. (If the ball bounces to the side, move to keep the CBR directly above the ball, but be careful not to change the height of the CBR.) You can hear a clicking sound as the data is collected. Data is collected for time and distance, and calculated for velocity and acceleration. If you have detached the CBR, reattach it when data collection is finished.
Press. (If the plot doesnt look good, repeat the sample.) Study the plot. Answer
Observe that BALL BOUNCE automatically flipped the distance data. Answer

questions 3 and 4.

26 GETTING STARTED WITH CBR
The Distance-Time plot of the bounce forms a parabola.
Press. From the PLOT MENU, choose PLOT TOOLS, and then SELECT DOMAIN. We
want to select the first full bounce. Move the cursor to the base of the beginning of the bounce, and press. Move the cursor to the base at the end of that bounce, and then press. The plot is redrawn, focusing on a single bounce.
The plot is in TRACE mode. Determine the vertex of the bounce. Answer question 5 on

the activity sheet.

Press to return to the PLOT MENU. Choose MAIN MENU. Choose QUIT. The vertex form of the quadratic equation, Y = A(X H) 2 + K, is appropriate for this
analysis. Press. In the Y= editor, turn off any functions that are selected. Enter the vertex form of the quadratic equation: Yn=A(XH)^2+K.
On the Home screen, store the value you recorded in question 5 for the height in
variable K; store the corresponding time in variable H; store 1 in variable A.
Press to display the graph. Answer questions 6 and 7. Try A = 2, 0, 1. Complete the first part of the chart in question 8 and answer

question 9.

Choose values of your own for A until you have a good match for the plot. Record
your choices for A in the chart in question 8.
Repeat the activity, but this time choose the last (right-most) full bounce. Answer
questions 10, 11, and 12.
Repeat the data collection, but do not choose a single parabola. Record the time and height for each successive bounce. Determine the ratio between the heights for each successive bounce. Explain the significance, if any, of this ratio.
GETTING STARTED WITH CBR 27
1. What physical property is represented along the x-axis? _____________________________________ What are the units? ___________________________________________________________________ What physical property is represented along the y-axis? _____________________________________ What are the units? ___________________________________________________________________ 2. What does the highest point on the plot represent? ________________________________________ The lowest point? _____________________________________________________________________ 3. Why did the BALL BOUNCE program flip the plot? __________________________________________ 4. Why does the plot look like the ball bounced across the floor? _______________________________
5. Record the maximum height and corresponding time for the first full bounce. __________________ 6. Did the graph for A = 1 match your plot? _________________________________________________ 7. Why or why not? _____________________________________________________________________ 8. Complete the chart below. A -How do the data plot and the Yn graph compare?

9. What does a positive value for A imply? __________________________________________________ What does a negative value for A imply? _________________________________________________ What does a zero value for A imply? _____________________________________________________ 10. Record the maximum height and corresponding time for the last full bounce. __________________ 11. Do you think A will be bigger or smaller for the last bounce? ________________________________ 12. How did A compare? __________________________________________________________________ What do you think A might represent? ___________________________________________________
28 GETTING STARTED WITH CBR

Activity 5Rolling ball

Function explored: parabolic. Plotting a ball rolling down a ramp of varying inclines creates a family of curves, which can be modeled by a series of quadratic equations. This activity investigates the values of the coefficients in the quadratic equation, y = ax 2 + bx + c.
3. varies (should be half of a parabola, concave up) 4. a parabola (quadratic) 5. varies 6. varies (should be parabolic with increasing curvature) 7. 0 is flat (ball cant roll); 90 is the same as a free-falling (dropping) ball
calculator CBR calculator-to-CBR cable mounting clamp large (9 inch) playground ball long ramp (at least 2 meters or 6 feeta lightweight board works well) protractor to measure angles books to prop up ramp TI ViewScreen (optional)
The motion of a body acted upon only by gravity is a popular topic in a study of physical sciences. Such motion is typically expressed by a particular form of the quadratic equation, s = at2 + vit + si where
Discuss how to measure the angle of the ramp. Let students get creative here. They might use a trigonometric calculation, folded paper, or a protractor. See pages 612 for hints on effective data collection.
s is the position of an object at time t a is its acceleration vi is its initial velocity si is its initial position
In the quadratic equation y = ax2 + bx + c, y represents the distance from the CBR to the ball at time x if the balls initial position was c, initial velocity was b, and acceleration is 2a.

Advanced explorations:

Since the ball is at rest when released, b should approach zero for each trial. c should approach the initial distance, 0.5 meters (1.5 feet). a increases as the angle of inclination increases. If students model the equation y = ax2 + bx + c manually, you may need to provide hints for the values of b and c. You may also direct them to perform a quadratic regression on lists L1, L2 using their calculators. The balls acceleration is due to the earths gravity. So the more the ramp points down (the greater the angle of inclination), the greater the value of a. Maximum a occurs for q = 90, minimum for q = 0. In fact, a is proportional to the sine of q.

Other changes occur once the data from real-world events is collected. CBR lets your students explore underlying relationships both numerically and graphically.

Explore data graphically

Use automatically generated plots of distance, velocity, and acceleration with respect to time for explorations such as:
What is the physical significance of the y-intercept? the x-intercept? the slope? the maximum? the minimum? the derivatives? the integrals? How do we recognize the function (linear, parabolic, etc.) represented by the plot? How would we model the data with a representative function? What is the significance of the various coefficients in the function (e.g., AX2 + BX + C)?

Explore data numerically

Your students can employ statistical methods (mean, median, mode, standard deviation, etc.) appropriate for their level to explore the numeric data. When you exit the RANGER program, a prompt reminds you of the lists in which REALTIME=NO data for time, distance, velocity, and acceleration is stored.
GETTING STARTED WITH CBR 33
CBR plotsconnecting the physical world and mathematics
The plots created from the data collected by RANGER are a visual representation of the relationships between the physical and mathematical descriptions of motion. Students should be encouraged to recognize, analyze, and discuss the shape of the plot in both physical and mathematical terms. Additional dialog and discoveries are possible when functions are entered in the Y= editor and displayed with the data plots.
A Distance-Time plot represents the approximate position of an object (distance from the CBR) at each instant in time when a sample is collected. y-axis units are meters or feet; x-axis units are seconds. A Velocity-Time plot represents the approximate speed of an object (relative to, and in the direction of, the CBR) at each sample time. y-axis units are meterssecond or feetsecond; x-axis units are seconds. An Acceleration-Time plot represents the approximate rate of change in speed of an object (relative to, and in the direction of, the CBR) at each sample time. y-axis units are meterssecond 2 or feetsecond 2; x-axis units are seconds. The first derivative (instantaneous slope) at any point on the Distance-Time plot is the speed at that instant. The first derivative (instantaneous slope) at any point on the Velocity-Time plot is the acceleration at that instant. This is also the second derivative at any point on the DistanceTime plot. A definite integral (area between the plot and the x-axis between any two points) on the Velocity-Time plot equals the displacement (net distance traveled) by the object during that time interval. Speed and velocity are often used interchangeably. They are different, though related, properties. Speed is a scalar quantity; it has magnitude but no specified direction, as in 6 feet per second. Velocity is a vector quantity; it has a specified direction as well as magnitude, as in 6 feet per second due North. A typical CBR Velocity-Time plot actually represents speed, not velocity. Only the magnitude (which can be positive, negative, or zero) is given. Direction is only implied. A positive velocity value indicates movement away from the CBR; a negative value indicates movement toward the CBR.

TI83_Calc.book Page 77 Wednesday, February 25, 2004 9:50 AM

Objectives

Explore the relationship between position and velocity Explore the relationship between functions and their derivatives Connect mathematical relationships to real-world phenomena

Activity 3

Materials
TI-84 Plus / TI-83 Plus Calculator-Based Ranger (CBR)
Move My WayA CBR Analysis of Rates of Change

Introduction

Speed and velocity are concepts that you have observed since early childhood. Because velocity is the derivative of displacement, taking a closer look at position and velocity can develop a deeper understanding of how derivatives and their functions are related. In this activity, you will use a Calculator-Based Ranger (CBR) data collection device with your graphing handheld to collect motion data with certain characteristics and use that data to draw conclusions about the relationship between position and velocity.

Exploration

A CBR emits an ultrasonic beam that reflects off the closest object and returns to the CBR. Depending on the speed of sound and the time it takes for the sound wave to return, the distance between the object and the CBR is calculated.
1. Begin by selecting :CBL/CBR from the APPS Menu. Press any key. Then select 3:RANGER and press . Select 2:SET DEFAULTS. The
CBR will collect distance data for 15 seconds. You will see the graph as it is created and be able to analyze what type of movement causes the graph to increase or decrease.
2004 TEXAS INSTRUMENTS INCORPORATED
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Calculus Activities

2. Move back and forth in front of the CBR, or
point the CBR at the wall and move back and forth from the wall, changing directions at least two times. The CBR collects data representing the distance from the CBR as a function of time. This data represents the distance between either you and the CBR or the CBR and the wall, depending on your experimental setup.
3. Make a sketch of your data on the grid
shown. Write a description of the motion.
4. The velocity is considered positive when the distance from the CBR is increasing
and negative when the distance from the CBR is decreasing. Identify regions on your graph where the velocity is positive and where it is negative. Clearly indicate those regions on the graph.
5. The velocity is zero when you stop, even if it is momentary. Remember that in
order to change directions you had to first stop, even if you did not think you came to a complete stop. You cannot move forward and then move backward without first stopping. Label all points on your graph where the velocity is zero. What do these correspond to on the position versus time graph?
6. From your data, make a sketch of what the
velocity versus time graph will look like for your walk. It is easiest to begin by plotting the zero points and then graphing the velocities as positive or negative. Do not worry about estimating the velocity values. The idea is just to sketch what the graph will look like.
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Activity 3: Move My WayA CBR Analysis of Rates of Change
7. Instantaneous velocity is the average velocity as the change in time
approaches zero: d ----- t In mathematical notation: d v = lim -----t 0 t where d is the change in distance from the CBR and t is the elapsed time. To approximate the instantaneous velocity, you can divide the change in distance by the change in time for each set of points taken. This is the average velocity over each small time interval. Because the time intervals are small, these values will approximate the velocity function. Select 1:Edit in the STAT Menu. Move the cursor to the top of L3 so that the L3 is highlighted, and press and to clear the list. Do the same for L4. When the lists are clear, calculate the average velocity for each time interval, and place it in L4 by moving the cursor to the top of L4 and selecting 7:List from the LIST OPS Menu. Calculate List(L2)/List(L1). The List operates by taking the difference between successive items in a list.
8. If the CBR collected 94 data points, how many values will be in L4? In other
words, how many differences will be calculated? Explain your answer.
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9. Because there will only be 93 values in L4,

you will need to have a list of times with only 93 data points to plot. The actual velocity that you calculated will occur at the midpoints of all the time intervals. But because the time intervals are short, you can use the time values from L1 and delete the first one to get a good idea of what the velocity graph looks like. Select 1:Edit from the STAT Menu, and move the cursor to the top of L3 so that the L3 is highlighted. Input L1 and press to place the values of L1 into L3. Highlight the first value, and press { to delete it.
10. Turn on a stat plot for velocity as a function of time (L3 and L4) as shown in the screen to
the right. Sketch the actual velocity graph on the screenshot below. How does it compare with your prediction?
11. What does positive velocity look like on a distance versus time graph?
12. What does positive velocity look like on a velocity versus time graph?
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13. Make a sketch of the velocity versus time
graph for the position versus time graph shown. Again, do not worry about plotting each velocity value exactly. This is just a sketch to show that you understand when the velocity is positive, negative, or zero.
14. Use the RANGER application to collect
velocity data so that your graph matches the one shown to the right. Your graph will not match exactly, but you should be able to approximately match each section. After executing prgmRANGER, select 1:SETUP/SAMPLE and set up the collection options as shown. Be sure to set the SMOOTHING to HEAVY. Try to match the graph. Describe your motion.
15. Did your starting point affect the graph of velocity versus time? Explain.
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16. Make a sketch to show what the position
versus time graph would look like for the velocity graph that you matched.
17. After completing this activity, complete the chart shown:
When the function graph is. Increasing Decreasing Changing from increasing to decreasing Changing from decreasing to increasing A constant value The derivative graph is.
18. Summarize at least three concepts that were reinforced during this activity.