UNIT 4—Motion

Teacher Materials

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TEKS Support
Teacher Notes
	"Simple Model"—Context Overview
	Mathematical Development
	Unit Project Suggestions
	Unit Project—Look Before You Leap
	Supplemental Activity 3 (optional project)—Evel Knievel Rides Again!
	Preparation Reading—It’s Showtime!
	Activity 1—Plan Ahead!
	Homework 1—Can You Say "Toy Boat" Three Times Fast?
	Activity 2—Watch Your Step
	Homework 2—Step By Step
	Activity 3—Staging a Near Hit
	Homework 3—Start Your Engines
	Activity 4—Falling For You
	Homework 4—Moving Along
	Activity 5—It Depends on Your Perspective
	Homework 5—Along These Lines
	Activity 6—Fall Fashions
	Activity 7—Close Call
	Supplemental Activity 1—Simulating a Near-Hit Stunt
	Supplemental Activity 2—Off Ramp
	Supplemental Activity 3—Evel Knievel Rides Again!
	Supplemental Activity 1—Simulating a Near-Hit Stunt
	Supplemental Activity 2—Off Ramp
	Supplemental Activity 3—Evel Knievel Rides Again!
	Handout 1—Motion Detector Set Up
	Handout 2—Walking the Walk
	Handout 3—BALLDROP Directions
	Handout 4—Falling Book Models
	Handout 5—Parametric Equations: Calculator Review Motion Detector Set Up
	Transparency 1—Book Drop Data
	Transparency 2—What Can Go Wrong?
	Transparency 3—Falling Book Models
Annotated Student Materials
	Preparation Reading—It’s Show Time!
	Activity 1—Plan Ahead!
	Homework 1—Can You Say "Toy Boat" Three Times Fast?
	Activity 2—Watch Your Step
	Homework 2—Step by Step
	Activity 3—Staging a Near Hit
	Homework 3—Start Your Engines
	Activity 4—Falling for You
	Homework 4—Moving Along
	Activity 5—It Depends on Your Perspective
	Homework 5—Along These Lines
	Activity 6—Fall Fashions
	Activity 7—Close Call
	Assessment—Hurry Up and Slow Down!
	Unit Project—Look Before You Leap
	Mathematical Summary
	Key Concepts
Solution to Short Modeling Practice
	Solution to Short Modeling Problem
Solutions to Practice and Review Problems

TEKS Support

• Linear equations
• Slope of linear equations
• Coordinate graphing of linear and quadratic equations

• Linear equations
• Quadratic equations
• Parametric equations
• Matrix representations
• Matrix additions

• 2 battery-operated vehicles
• vernier calipers
• carbon paper
• catch ramp or coffee can
• chalk line or masking tape
• 2 meter sticks
• acoustic motion detector
• CBL (if needed for your motion detector)
• photogate equipment
• shims
• string and eye screw
• stunt vehicle (toy car or ball)
• take-off ramp
• Programs BALLDROP, GATE, TIMER, EDITPART
• graphing calculator

Teacher Notes

"Simple Model"—Context Overview

Car and motorcycle stunts serve as the motivation behind the study of motion contained in this unit. Students model different stunts during the unit, including two near-collision stunts and a planned collision. An optional ramp-to-ramp jump is included as a supplemental activity.

In this unit students use motion detectors to collect distance-versus-time data on moving objects. Most of the work in the unit deals with the relationships between location and time for objects in motion along a straight line, in either the horizontal or vertical direction. The final supplemental activity looks at motion in a plane.

Mathematical Development

The opening activity introduces students to a variety of stunts. Students engage in the first two steps of modeling as they identify what they need to know in each situation and then think about how they will obtain the needed information. Next, informal explorations of student motion lead to a review of velocity as a rate of change and its representation as the slope of a line. Because real data are being collected and studied, the piecewise nature of real data becomes an issue early in the unit.

Linear distance-versus-time graphs resulting from student walks at a constant pace are followed by graphs that curve. The notion of "local linearity"—that a curve "looks like a line" if you zoom in far enough—is used as the basis for defining instantaneous velocity. First, students learn that a linear distance-versus-time graph means a constant velocity. Later, they discover that a quadratic distance-versus-time graph means a linear velocity-versus-time graph and constant acceleration.

In this unit, students use motion detectors to collect real data on the times and corresponding distances of moving objects. Because of variability in real data, perfectly linear data and perfectly quadratic data are rare. Therefore, linear and quadratic regression are used to fit models to data that appear linear and quadratic, respectively.

Unit Project Suggestions

If your students enjoy designing stunts, and if you have the time, you may decide to conclude the unit with two projects, "Look Before You Leap" and the optional project "Evel Knievel Rides Again!" The latter stunt requires background information on parametric equations covered in Animation.

Unit Project—Look Before You Leap

Supplemental Activity 3 (optional project)—Evel Knievel Rides Again!

In this activity students design and stage a small-scale version of a ramp-to-ramp jump. Students break the motion of the stunt vehicle into its horizontal and vertical components. Then they apply mathematical techniques covered in this unit to complete their stunt.

Students should review material from Animation before beginning this project. You may want to assign and then review the answers to Supplemental Activity 2 as part of this review. Handout 5 provides review of how parametric equations are entered and graphed on TI-82 or TI-83 calculators. Students should be encouraged to use both algebraic and graphical analysis in designing their stunt.

This activity consists of five parts: The Plan, Equipment Set-Up, Data Collection, Analysis–Model Formulation, and The Contest. The activity should take at least two days to complete.

For Part I, students work in groups to identify the information they will need in order to develop their models. Then they need to decide how they will obtain that information. After groups have finished their planning, pull the class together as a whole and discuss the answers to Items 1–3.

In Item 3(d) students may suggest placing a motion detector in front of the end of the ramp and making a velocity calculation from the distance-versus-time graph. This method may give fair results and, if suggested, it should be tried. However, the photogate method (described later in these notes) gives better results.

For Part II students set-up the equipment needed for their stunts. General guidelines for equipment set-up and experimental procedures are described in the student pages. Use this written description as support for directions that you give orally. Below are some suggestions on ramp construction and photogates.

Ramp Construction

There are two options for the ramp, a high-tech option and a low-tech option. Instructions for each follow.

High-tech Method

If you are not handy, this is a great opportunity to get together with a technology teacher to build a ramp similar to the one described below.

Here’s how to construct the ramp (see Figure 1).

Make a 1" ´ 8" ´ 3/8"-deep groove on one side of one end of each piece of wood. The grooves will form the "resting place" for the molding, which is the active part of the track. Attach the two pieces of wood with the 3" angle iron. Have the groove on the inside of the "L" at the ends of the wood away from the corner. Cut one end of one of the 1" angle irons in half. This should be just high enough to hold the ramp in place. Attach the 1" angle irons at the ends of the wooden "L." Carefully bend the molding so it fits in the grooves and is held between the two angle irons. Attach a string to the lower end of the apparatus. Attach a screw eye on the end of the string.

Low-tech Method:

Get a track for toy cars. Place a stack of books on the table. Attach one end of the Hot Wheels track to the top of the stack of books; attach the other end to the table. (Use masking tape to make the attachments.) If your track is fairly steep, you may want to brace it near the bottom (calculator manuals make a good brace). Make sure the take-off portion of the ramp is horizontal. For example, the bottom of the ramp in Figure 2 is supported by a cassette tape box that keeps the take-off section of the ramp horizontal and raises it so that the photogate’s beam hits the midsection of a golf ball (the stunt vehicle).

Whichever ramp you use, be sure that after it is set up no one touches it until this activity is completed. Stress DO NOT TOUCH!

The Photogate

The photogate method for determining the velocity of the stunt car as it leaves the track generally yields better results than a method that relies on motion detectors. The photogate method very accurately measures the time necessary for the car to travel its own length. It is extremely unlikely that students will suggest using a photogate to determine the stunt vehicle’s velocity as it leaves the ramp. Therefore, present and discuss this procedure, and consider a "sample calculation" so that students understand just what is being measured. (See Part III for a sample calculation.)

Regardless of which ramp method you select, you attach the photogate near the ramp take-off. If you do not have a photogate, you can make one by placing a penlight flashlight on one side of the ramp and the CBL light probe on the other. (See Figure 3.) Be sure the light source is shining directly into the probe. Adjust the height of the light (or the beam of the photogate) so that it will hit the longest part of the stunt vehicle as it rolls down the ramp.

A Vernier photogate is easier to set up than the penlight/light-probe photogate. If you use a Vernier photogate connected to a TI-CBL, you can run the program TIMER to collect the data on the time required for the stunt vehicle to pass through the beam of the photogate. If you make the penlight/light-probe photogate, you can run the program GATE to collect these data.

If, in Part III, a single homemade ramp and a golf ball are used, you may want to collect the data as a class. If commercial ramps and a variety of toy vehicles are used, have groups collect their own data.

Item 4 has the students find the combined effect of gravity and air resistance on the car. This is just like the book-drop experiment in Activity 4. If the stunt vehicle is fairly small, have students drop it away from the detector to avoid damage to the equipment. However, be sure they change the sign for a , the coefficient of t², before using it in their model.

Warning: If the stunt vehicle is fairly small, the motion detector may be unable to track its motion. Ask students to suggest a possible solution to this problem. For example, if the car is fairly heavy for its size or you are using a golf ball, students may get good results by ignoring air resistance and using the acceleration due to gravity listed in physics books (–32 ft/sec). Instead, students might decide to use the acceleration determined by their book-drop experiment.

Item 5 has students run some test runs with the car (or ball) in order to find the velocity of the car as it leaves the ramp. If students have trouble determining the off-ramp velocity, take time out to work through the following example:

Example: Suppose that the car is 4 cm long at the height at which the light beam is located. The photogate shows the light is blocked for 0.019987 seconds. How fast was the car going?
1.4 cm/0.019987 sec = 200.13 cm/sec

The trial test runs should also establish the line of flight for the car.

For Part IV, regardless of whether the data are collected by the class as a whole or by individual groups, allow each group time to get its own equations and interpret the data.

For Part V, conduct the contest with the full class. Each group should have a turn at locating the landing area and then staging their stunt. Two options for the contest challenge are described in the student material. You may give each group a separate challenge or let groups design challenges for each other. For the challenge, you should specify either the height of the can (landing ramp) or its distance from the jump ramp. Whoever designs the challenges should check in advance that the answers they produce will lead to locations somewhere in the "middle" of the car’s flight.

If you have plenty of time, add other challenges; this is a chance for students to be creative. For example, students could add a ring, such as the one shown in Figure 4, for the car to jump through.

On the other hand, if time is short, use the same challenge for all groups instead of specifying individual group challenges.

Allow time for groups to compute the locations (x,y) of their landing areas. Be sure that they get solutions analytically as well as by using the parametric graphs on their calculators. When it’s time to stage the stunts, be sure students place the landing area carefully. Move the ramp or can so that its center is over the x-value (horizontal distance from the release point) and its top is just below the computed y-value. Be certain it is placed along the marked line of flight. You might want to use a ring stand to support the landing area.

If a can is used for the catching ramp, suggest that it be tilted a bit toward the oncoming stunt vehicle, with the center of its opening located at the computed (x,y) coordinates. If a landing ramp is used, it should be somewhat wider than the take-off ramp since side-to-side motion is not being controlled carefully. Use a foot-wide piece of tri-wall cardboard or similar sturdy-but-light material.

To perform the stunt, have the designated roller(s) roll the stunt vehicle down the ramp and check whether their stunt plan worked. If the stunt vehicle doesn’t land safely but is close, roll again. If it isn’t close, have the group try to find the mistake in their calculations and reset the landing area. Insist that groups show you their corrected calculations before letting them stage the stunt a second time.

Preparation Reading—It’s Showtime!

Students should begin the unit by reading the preparation reading.

After students have completed the reading, use the following question to spark a brief discussion.

	Activity 1—Plan Ahead!

This activity describes three types of stunts: a two-vehicle, near collision stunt, a ramp-to-ramp jump, and a leap-over-an-oncoming-vehicle stunt. Students develop initial plans for modeling each of these stunts. Ideas generated during this activity may be used to guide explorations throughout the remainder of the unit, either as motivation for activities in the unit or as the bases for alternatives to unit activities.

Students should work in small groups, then share their ideas with the class.

In Items 1–3, students are given a stunt. They must decide what information might be relevant to the design of the stunt and how they might collect this information. Ask students to identify what steps in the modeling they have completed by answering these items. (In these items, students have identified the problem and determined which factors are most important to the problem.)

Conclude this activity with a discussion. Let groups share their plans and methods of gathering the necessary information. During your discussion of Item 4 remind students of the relationship between distance, D, rate, R, and time, T:

D = R ´ T

Use discussion of Item 4 as an opportunity to introduce the concepts of time-lapse graphs of motion, distance-versus-time graphs of motion, and average velocity. Some points to cover during this instruction as well as an example follow.

Example 1 shows time-lapse graphs for two walkers.

Walker #1 walks 12 ft in three seconds. Every second she covers the same distance.

• What is the walker’s rate?
  4 ft/sec



• How far does she walk each second?
  4 ft

Walker #2 walks 12 feet in 2 seconds. However, he walks twice as far during the second second as he did during the first.

• What is the walker’s average rate over the two-second walk?
  6 ft/sec



• What is his average rate from t = 0 to t = 1?
  4 ft/sec



• What about from t = 1 to t = 2?
  8 ft/sec



• Is the walker speeding up, slowing down, or walking at a constant rate?
  The walker is speeding up.



• Ask students what a distance-versus-time graph would look like for each of these walkers. (Students will produce their own distance-versus-time graphs in Activity 2.) Assume that the initial starting point is designated as distance = 0.
  Walker #1’s distance-versus-time graph consists of the points (0, 0), (1, 4), (2, 8), and (3, 12). Walker #2’s distance-versus-time graph consists of the points (0, 0), (1, 4), and (2, 12). The first set of points lie on a straight line, the second do not.

After reviewing the D = R ´ T formula and introducing distance-versus-time graphs, formally introduce the concept of average velocity.

Be sure to discuss the units of velocity. For example, if distance is measured in feet and time in seconds, the units for velocity are ft/sec. In addition, point out that the average velocity formula is the same formula as the one used to calculate the slope of the line joining the points (time 1, distance 1) and (time 2, distance 2). Provide several examples so that students can practice computing average velocity, including at least one where distance 2 is smaller than distance 1 (so that velocity is negative). Conclude this instruction by explaining the difference between velocity and speed—velocity has direction, which is indicated by its sign, while speed is always positive.

	Homework 1—Can You Say "Toy Boat" Three Times Fast?

This assignment reviews average velocity and asks students to interpret various representations of distance and time information.

	Activity 2—Watch Your Step

This activity is to introduces the use of a motion detector for collecting distance-versus-time data from moving objects.

Divide the class into as many groups as you have motion detectors, and distribute Handout 1 (or specific instructions for the equipment you will be using). Ask students to save this handout for future reference.

Each group should have one motion detector equipment set-up consisting of a motion detector and CBL or the equivalent. Demonstrate the motion detector and HIKER (a program for use with the motion detector) to record the motion of students walking along a line in front of the motion detector. (Note: In this unit, any program for the motion detector that can take readings every tenth of a second for at least 6 seconds and that displays the distance-versus-time graph as the motion occurs will be referred to as HIKER.)

Motion Detectors

An ultrasonic motion detector sends out a beam of ultrasonic sound. If an object is in the beam, this sound reflects off the object back to the sensor which detects the returning signal. The distance between the object and the motion detector can be determined from the time lapse between emitting the signal and detecting the return signal because the sound emitted by the detector travels at a known speed.

The motion detector must be connected to a calculator or a computer. You will need to use an intermediate device (such as a CBL) to link a calculator to a motion detector unless you are using a "smart" motion detector (such as the TI-CBR). Example 2 illustrates a typical calculator-motion detector set up.

The CBR (Calculator Based Ranger)

The Calculator Based Ranger (CBR) is a motion detector and CBL all in one. The programs needed to run the CBR can be downloaded directly from your CBR to your calculator. There is no need for additional programs for different experiments or to edit the data. All the capabilities that you will need can be accessed through the main menu of the RANGER program. Consult the manual for instructions.

Equivalent Motion Detector Set-ups

Both Casio and HP calculators have motion detector equipment that is similar to the CBL/Vernier motion detector. If you are not using TI-graphics calculators, you will need to adapt the TI-instructions to your brand of calculator.

For Item 1, select two students from your class to be Walker #1 and Walker #2. Use this as an opportunity to make sure that all groups understand how to operate the motion detector equipment. Also, try to get students to make the connection between walking at a steady pace and producing a linear distance-versus-time graph.

Before students begin Item 3, distribute five copies of Handout 2 to each group. Each group will need at least one calculator with HIKER. Remind the groups to use the worksheets to record a description of each walker’s motion as well as the graph that the motion generates on the calculator screen. (For example, students might record the following description: he walked at a steady pace for half of the walk and then stood still.). Encourage students to experiment with the equipment and generate a variety of graphs. Suggest that each student take a turn walking.

Be sure to leave enough time for Items 4 and 5. Groups will need at least 2 large sheets of paper to transfer their chosen graphs. After students have completed their graphs, have them post them around the room. Discuss the variety of graphs that students have produced. This discussion should be structured to help students analyze their results.

If, during the activity, a student puts his or her hand in a motion detector’s beam while another student is walking, don’t get upset. This will create a "bad point"—an outlier. The group can use TRACE to find the coordinates of that point and see exactly where (how far from the detector) the hand-waving student was. If the group chooses to post this graph, the class will have an opportunity to discuss what is meant by an outlier.

Use the class discussion to help students notice important features of a graph. The content of this discussion should include answers to the following questions:

1. Can you describe the motion of an individual walker when given a graph?
2. Can you find patterns among the graphs?

A good opening question for this discussion is to ask your students what information they can get from a distance-versus-time graph. Look for some of the following answers: a walker’s location (distance from the detector) at a given instant; whether a walker is going toward or away from the detector; whether the walker is moving or standing still; how quickly the walker is moving; whether the walker changed direction (e.g., first going toward the detector, then away from it or vice versa), etc. If students do not come up with these answers on their own, pose these answers as questions. For example, if the first answer is not forthcoming, ask "How can you determine a walker’s distance from the motion detector three seconds into their walk."

Have students explain their reasoning behind the chosen categories.

Examples of categories may include the following:

• graphs that resemble certain curves (such as parabolas) or lines
• graphs that have peak points
• graphs that have "flat" pieces
• graphs that are combinations of lines and curves
• graphs that show increase or decrease (moving toward or away from the detector)
• graphs that have piecewise components

Caution: Be sure to stress that a graph is not a "picture of the path" that was walked. Each student walked in a straight line. However, many of their distance-versus-time graphs were curved.

As part of this discussion ask general questions about what can and cannot be learned from distance-versus-time graphs of motion, move to the problem of classifying the posted graphs into categories.

If you are short on time, have students sketch a graph from another group before they leave class. Then they can complete Item 6 for homework.

	Homework 2—Step By Step

This assignment provides an opportunity for students to practice interpreting distance-versus-time graphs.

Discuss Item 1(g) and Item 2 before beginning the next activity. In particular, try to make the connection between the slope of a linear distance-versus-time graph and constant velocity. In addition, discuss the importance of fitting a linear regression only to the portion of data that appears linear.

	Activity 3—Staging a Near Hit

• Motion detector equipment
• Battery operated toy cars (at least two)
• Masking tape
• Supplemental Activity 1

The purpose of this activity is to provide students with an opportunity to complete their modeling of a "near-collision" stunt using battery operated toy cars. Students will then test their model by staging the stunt.

In selecting battery-operated vehicles for this stunt, you want to choose vehicles that are somewhat large and drive straight. One possibility is to ask students to bring in toy vehicles from home for use in their group. Here is a description of the car and truck used to generate the sample answers:

Two different style vehicles were used so that there was a good chance the vehicles would move at different velocities. The car is a red fire chief’s car that is 5 inches wide and 12 inches long. The truck is a monster truck that is 7 inches wide and 8 inches long. Both vehicles are battery operated. To start the car, you pressed a button. The message "We’re on our way" plays and then the fire chief’s car starts moving (complete with siren). (If noise is not a problem, the siren heightens spectator’s interest in this stunt.) The truck has an on/off switch and begins moving as soon as the switch is turned on. Both vehicles are relatively inexpensive and have proven to be durable.

There are two options for conducting this activity.

Option 1: You may collect data on two toy cars as a class and then let groups use the results to plan their stunt.

Option 2: Let each group conduct their own experiments on the toy cars and use their results to plan their stunt.

When you stage the stunt in Item 4, use masking tape for the roadway. Each group should have an opportunity to stage their stunt and test their plan. If a group’s stunt fails, give them an opportunity to rework their calculations and then re-stage their stunt. Insist that they furnish mathematical support for the revision to their plan before letting any group re-stage their stunt.

As an assessment for this activity, consider giving each student (or group) dimensions of two vehicles and their velocities. Students should plan the stunt based on these vehicles and then use Supplemental Activity 1 to test their plan.

	Homework 3—Start Your Engines

This assignment reinforces student understanding of graphical and symbolic representations of motion.

	Activity 4—Falling For You

In this activity, students begin their study of the motion of falling objects in order to model stunts involving falls.

This activity is divided into three parts: Imagining Falls, Recording Falls, and Analyzing the Data. In Part I, students think about how the distance-versus-time graphs produced by falling objects should look and why they should look that way. In Part II, students collect data on falling books (or, if you chose, other objects such as basketballs), and in Part III, they fit a model to their data.

Break students into as many groups as you have motion detector equipment. (Or use smaller groups and have two or more groups share equipment.) For the book-drop experiment, have groups use a variety of different objects—such as books of different weights—and drop them from different heights. This will ensure that patterns in the resulting models will be easy to detect when they are analyzed in Activity 6.

Before students begin Item 2, distribute Handout 3 or provide your own directions for using a BALLDROP program to collect data on falling objects. Technology Notes and Data Collection Notes follow.

Technology Notes

Equipment Set-up

You and your students will be dropping objects. Do not let any of these falling objects actually hit the detector. There are two methods for dropping objects and safely using a motion detector to record their fall.

Method 1: Drop the object on top of a frame around the detector. Details about protecting the motion detector are provided below in Item 2, "Construction of Protective Frames." (This is the recommended method.)

Method 2: Secure the detector above the floor, pointing down. Then drop the object away from the detector. For example, attach the detector to a ring stand on a table. (It is more difficult to obtain good results using this method.)

Construction of Protective Frames

High-tech frame: Get a 2 ´ 4 and cut it into four pieces, each at least 8" longer than the longest side of the motion detector. Aligning the frame before you actually make it will show you where the wood will go when it is fastened. Place two of the pieces on their 2" sides along the sides of a motion detector. Place the other two pieces at right angles to and across the top of the first two pieces. Be sure the top pieces do not cover any part of the motion detector but that all the wood is as close to the detector as possible (See Example 3.).

Use wood glue to glue the frame together. Once the glue dries, nail the wood together. This frame should be used to protect your motion detector when books or similar objects are dropped toward it.

Low-tech frame: Use three thick textbooks (old calculus books or dictionaries work well). The thickness of the books should be slightly higher than the motion detector when it is lying facing up. Construct your frame shown in Example 4.

Calculator Programs

HIKER programs do not collect data fast enough to capture the motion of falling objects. Instead, you will need to switch to BALLDROP.83P, or a program with settings similar to those in BALLDROP. Since BALLDROP takes 50 readings each second (once every 0.02 seconds), it does not display the data in "real time." Instead the data are displayed only after all data collection is complete. Any program using these settings—readings every 0.02 second for approximately 1.6 seconds—will be referred to as BALLDROP.

Remember, BALLDROP uses the TRIGGER key. You will hear the motion detector making readings as soon as BALLDROP is executed. However, the program does not record any of its readings until the TRIGGER key on the CBL is pushed.

Data Collection

Students should follow the directions that are given in Handout 3 to collect their book-drop data.

The student dropping the book should hold the book until after the TRIGGER has been pushed so that:

1. The first instant the book moves will be in the domain of the data.
2. The data will be piecewise (three different rules), reinforcing the notion that almost no motion activity involves just one equation.
3. Translation will be needed in order to interpret the equation of motion.

The actual book drop will go quite quickly. Once a group feels they have good data, students should move on to Item 3. Let groups who finish early work on Item 3 prior to the discussion of data editing. Then you may be able to use their results (from fitting a quadratic to all the data) in place of Transparencies 1 and 2. See Dealing with Data below.

Dealing with Data

By now you will probably want to let students use the program EDITPART with their data sets. EDITPART takes a data set apart, copying a designated piece to new lists. Thus students can use the calculators’ regression capabilities to do their curve fitting on large data sets more easily.

To use EDITPART, the total data set must be in L1 and L2. EDITPART will display the graph of the data and then ask for a lower and upper bound on the piece of the data you want to keep. To set these bounds, move the cursor to the ends of the piece you want and press ENTER.

When you have finished, the original data are still in L1 and L2. The part of the data that you selected will be in L3 and L4. A plot of L3 and L4 will be displayed. You can now safely use the regression capabilities of the calculator on L3, L4.

You may use Transparencies 1 and 2 to illustrate what happens if students don’t edit their data. Transparency 1 displays data collected in a book-drop experiment similar to the one that your students performed. (The detached horizontal piece resulted when the book fell off the frame and the motion detector began taking readings from the ceiling.) Transparency 2 shows the graph of the quadratic that resulted after applying quadratic regression to all the data. (According to the quadratic model, the book drops initially and then begins to fly!)

For Item 3(c), warn students to use all digits supplied by their data and not to round until the end of their calculations. Remind students to save their equations from Item 3(f). They will need them in later activities.

Item 4 is optional. Students walk in front of a motion detector and try to produce distance-versus-time graphs that look linear and parabolic, respectively.

	Homework 4—Moving Along

This assignment further develops the concept of instantaneous velocity.

Item 3 revisits the link between velocity and slope, and approximations to instantaneous values. In part (d), the idea of "zooming in until the curve is a line" (local linearity) is suggested for defining slope at a point on a curve. Discuss this item in class. In addition, check that students have made these two connections:

1. When velocity is constant, the instantaneous velocity at a particular time and the average velocity between any two times are the same.
2. When an object is moving at a constant velocity, its distance-versus-time graph is linear, and its slope is the velocity.

The idea here is to apply (1) and (2) to a situation in which velocity is not constant; thus, the distance-versus-time graph is curved. Students should think about this question: How do you define the "slope" of a curve at a particular point on the curve?

Students may need some assistance with the "zoom in" portion of 3(d). An outline of a possible demonstration follows:

Using a calculator with a viewscreen, graph y = –15.46x² + 12.78x +1.29. TRACE to one location on the curve. Use ZOOM BOX to draw a small square around that point. The image on your screen should resemble the one in Example 5. Ask students how the boxed piece of graph would look if it were magnified so that it filled the calculator screen. Most students will agree that it should look like a line. Press ENTER to execute ZBOX and then view the approximate "line." (You may have to repeat the process more than once for "sharper" curves.) Next, be sure students know how to find the slope of this "line." One method is to use TRACE to select two points at opposite ends of the "line" and then use those coordinates to approximate the desired slope.

Warning: Be sure students use all the decimal places that the calculator provides. Rounding should be done only at the very end of the slope calculations.

	Activity 5—It Depends on Your Perspective

In this activity students complete the modeling of a vertical near-hit stunt using translations of quadratic functions and, time permitting, test their models by staging the actual stunt. Translations serve to make the models developed for the book-drop experiment in Activity 4 more "portable" by using dependent and independent variables more appropriate for the context.

This activity may take two days for some classes. If time runs short, omit the test of the stunt.

In Item 5, students discover one of the limitations of their models for the motion of the falling book. The model doesn’t hold for values of t (the elapsed time since the motion detector began recording data) that correspond to times before the motion begins.

Item 7 asks students to translate their distance-versus-time graphs by introducing a new dependent variable h that measures the object’s height above the floor rather than its distance to the face of the motion detector. This is one of two translations that students will make and is a good opportunity to review material from Testing 1, 2, 3. Later, in Item 9 a new independent variable, t_fall is introduced. Remind students to save their models from Item 9(e) for Activity 6.

Item 11 asks students to carry out their stunt. Be sure to use sturdy toy car’s for this stunt and drop light-weight books (such as calculator manuals rather than hard-cover textbooks) so that you don’t break the car. This is particularly important since student reaction times in releasing the book at the proper moment can have a major effect on the success of the stunt. (Investigations of the sensitivity of the stunt’s success to the release time makes a nice project.)

	Homework 5—Along These Lines

Items 1 and 2 of this activity ask students to interpret linear and parabolic distance-versus-time graphs. Be sure that students complete Items 1 and 2 and save their answers for use in the assessment.

In Item 3, students practice translating a model by switching to independent and dependent variables that are more appropriate to the context.

	Activity 6—Fall Fashions

This activity introduces the concept of acceleration in the context of models for falling objects. At the end of this activity, students should realize that for falling objects distance-versus-time is quadratic and velocity-versus-time is linear. They should also realize that the slope of the velocity-versus-time graph gives the acceleration and that acceleration is essentially the same for the different objects dropped in Activity 4. In addition, students should discover that the coefficient a in the model d = at² + c is one-half the acceleration and that this equation provides a general model for the motion of any falling object.

Item 2 reviews the calculation of instantaneous velocity and concludes with the definition of acceleration. Acceleration is a difficult concept for students to grasp. Explain that acceleration occurs only when a force acts on an object. In the case of the book, gravity and air resistance supply the force causing the book to accelerate during its fall. Gravity pulls the book down. Air resistance pushes up on the book (because air resistance always acts in the direction opposite to the motion). Because of air resistance, student results for acceleration will be slightly less in magnitude than the acceleration due to gravity found in physics texts (–32 ft/sec.²)

After students complete Item 3, distribute Handout 4. You will need Transparency 3 for recording class results. Ask each group to report their results for Items 1–3 and record them on the transparency. Have students copy the class results to Handout 4. They should continue to work in the same groups that they were in for Activity 5.

After completing Items 4–6, students will have models describing how things fall. They will have general equations describing distance versus time, velocity versus time, and acceleration versus time (which, in this case, is constant). Beware of the mix of letters used as constants. It is customary to use a to represent acceleration. It is also customary to write general quadratic models as

The two a-values are not the same. Encourage students to use g for acceleration due to gravity to avoid confusion. Then

	Activity 7—Close Call

In this activity, students plan a more complicated near-collision stunt. Students discover that the force on a toy vehicle that moves down an incline is related to the steepness of the plan. In turn, the force on the vehicle acting parallel to the plane causes the vehicle to accelerate.

Have students set up one or more inclined planes. They can make the planes by bracing up one end of a long piece of plywood (or sturdy carboard) on a stak of books or boxes or large storage cans. They should experiment with the height of the plane to find the lowest height for which their toy vehicle will easily roll down the plane without needing a tap. During experiments with inclined planes at various heights, students should always keep their planes more steeply slanted than this lowest-height inclined plane.

In this activity, students will be working with Newton's second law which is usually stated as:

Net force = Mass × Acceleration

F_net = ma

Net force means the sum of all forces acting on an object, or the unbalanced force acting on a body.

Solve Newton's second law for the acceleration.

Discuss that this equation shows the acceleration is proportional to the net force and inversely proportional to mass.

The units for acceleration, force, and mas can be very confusing due to their misuse in common speech. The following three statements may help.

• Force is a push or a pull that can cause something to accelerate. Force is expressed in units of pounds or Newtons.



• Mass is a measure of the quantity of matter present. Mass is expressed in units of slugs or kilograms.



• Acceleration is the rate of change of velocity. Acceleration is expressed in units of ft/sec² or m/sec².

It is best if students measure distances in meters and use a balance scale to find the mass of their toy vehicles in grams and convert the mass to kilograms. That way force will be in Newton's: 1 Newton = (1kg)(1m/sec²). However, some CBL prgrams such as BALLDROP and HIKER record distances in feet. (Whereas, the CBR gives students a choice of units, meters or feet.) In this case, students will have to convert acceleration in feet/second² to meters/second². You will probably need to give students the conversion that:

1 ft/sec² = 0.3048 meters/sec²

If you choose to use the English system of units, then measure force in pounds, mass in slugs, and acceleration in feet/second². Students will be unfamiliar with mass in slugs. To convert a weight, in pounds, to mass, in slugs, divide the weight by the acceleration of gravity. This is an application of Newton's second law.

Students will need to attach the motion detector to the inclined plane so that it points in a direction parallel to the plane. One easy way to do this is to set the motion detector at the end of the plane, attach masking tape (or packaging tape) along the back of the motion detector and then fold the tape under the plane. You may want to use additonal tape on the sides of the motion detector to make it more secure.

Time permitting, end this activity by allowing students to stage their stunts. If students want more of a challenge, use two battery-operated cars with different velocities and challenge students to design a stunt similar to the one described in the opening to this activity.

	Supplemental Activity 1—Simulating a Near-Hit Stunt

This activity can be used as an assessment for Activity 3. Give each group, or each student a different set of velocities and dimensions for the car and truck.

	Supplemental Activity 2—Off Ramp

This activity is designed to support Supplemental Activity 3.

	Supplemental Activity 3—Evel Knievel Rides Again!

This activity can serve as a unit project for students who have completed Animation. See Unit Projects for details.

Supplemental Activity 1—Simulating a Near-Hit Stunt

THE STUNT DESIGN

The idea of this stunt is to simulate a near-hit stunt between a car and a truck as they cross an intersection. The truck travels east and the car north along the roadway shown in Figure 1. Remember, the key to this stunt is to cause some anxiety for those watching. The car and truck should cross the intersection as closely as possible without colliding.

Your teacher will give you information about your car and truck. Enter this information into a table (See Figure 2.)

1. First, you must plan your stunt.



1. Determine where to position the car and truck for the start of the stunt.

Car’s distance from intersection E = ________________ft.

Truck’s distance from intersection F = _______________ft.

Sample answer:

Car’s distance from intersection E = 5 ft.

Truck’s distance from intersection F = 3.5 ft.

2. How did you chose the starting positions in (a). Explain mathematically why you expect your stunt to work.

Position for car: We decided to place the car five feet from the intersection. So, E = 5. Based on this decision, we then worked to determine an appropriate value for F, the truck’s distance from the intersection.

In order to avoid a crash, we need to have the front of the car (12.5 + 3.5)in. = 16 in. or 4/3 ft beyond the center of the intersection. Here’s how long it will take the car to travel (5 + 4/3) ft or approximately 6.33 ft:

t = 6.33ft/(1.9 ft/sec) » 3.3 seconds.



The truck travels more slowly. So, we’’ have to place it closer to the intersection than the car. Here are the calculations for its starting position F. Here’s how far the truck will travel in 3.3 seconds:

(3.3 sec)(0.9ft/sec) » 3.0 ft

We want the truck to reach a location 2.5 inches (or about 0.2 ft) before the intersection in 3.3 seconds. (See diagram above.) So, F » 3.5 ft.

THE SIMULATION

You will use the program DRIVE to test whether or not your stunt is successful. Before running the program, check that your mode settings match the default settings (see Figure 3).

2. Set a window appropriate for this stunt. Remember, the intersection is the point (0,0). So select a window that will allow you to see all the action. For example, you might set Xmin = –E and Ymin = –F. Then set Xmax and Ymax so that the intersection visible.

3. Here’s the information that you’ll be asked to enter when you execute the program DRIVE. Decide on the values that you will enter before you run the simulation. Warning: Check to see that you have used the correct units.



1. You will need to enter the dimensions, in feet, for each vehicle. Determine the values for each of the following:

CAR WIDTH ______

CAR LENGTH _____

TRUCK WIDTH _____

TRUCK LENGTH _____

2. You will need to enter the positions of the vehicles (distance in feet from the intersection).

POS. CAR: value of E _____

POS. TRUCK: value of F _____

3. You will need to enter the velocity of the vehicles (ft./sec).

VEL. CAR _____

VEL. TRUCK _____

4. You’re almost ready. The program needs additional information about time.

STOP: How many seconds you want the action to run before stopping.

STOP _____

TIME INCREMENT: Time increment between views of car/truck positions.

TIME INCREMENT ________

START VIEW: Enter 0 to view the action from the start. However, you may specify a larger number (less than the number that you entered for STOP) if you want to pick up the action closer to the intersection.

START VALUE ______________

5. Execute the program DRIVE and enter the values from (a) – (d). After you have entered this information, you should see two rectangles on your screen. These represent the two vehicles. Press ENTER repeatedly to advance the motion by one TIME INCREMENT. Watch as the vehicles near the intersection. Do your vehicles avoid crashing? Do they pass very close as they cross the intersection? If the car and truck crash or if they cross the intersection far apart, re-work your design and try again.

Sample answer: The car and truck just missed each other as they crossed the intersection.

Note: If you had trouble telling whether or not the car and truck collided, change to a smaller viewing window—perhaps [–2, 2] × [–2, 2]—select a smaller value for TIME INCREMENT, and increase the value you enter for START VIEW. Then run the stunt again.

Supplemental Activity 2—Off Ramp

Imagine that Evel Knievel drives his motorcycle off a 40-ft. tall ramp at 60 mph. The ramp is parallel to the ground. Assume that the cycle and rider will accelerate vertically at –32 ft/sec.² (Ignore any air resistance.)

1. Write an equation to model the height, h (ft.), of the motorcycle t seconds after it leaves the ramp.

h = –16t² + 40

2. What equation describes the downward velocity of the cycle?

v = –32t

2. Write an equation that represents the distance, d (ft.), the cycle has traveled forward t seconds after it leaves the ramp.

d = 88t

3. Use you equations from Items 1 and 2 to answer the following items.



1. How long does it take for the motorcycle to hit the ground? Explain how you got your answer.

h = –16t² + 40; h = 0 at t » 1.6.

2. How far has the motorcycle travel horizontally at the time it hits the ground?

d » 88(1.6) = 140.8 ft

4. Draw three graphs describing the motorcycle’s motion:



1. (horizontal) distance versus time



2. height versus time



3. height versus distance (the path of the motion)

Supplemental Activity 3—Evel Knievel Rides Again!

This is your chance to plan and then execute a scaled-down version of Evel Knievel’s ramp-to-ramp jump. You’ll use a toy car (or a ball) for the vehicle instead of a motorcycle.

Part I: THE PLAN

In this activity you will create a mathematical model of the position of a toy car (or ball) as it sails across your classroom. Once you have your model, you will use it to locate the best position for a landing ramp or a can that will catch the car safely. Figure 1 shows roughly how the set up will look.

The jump ramp should be set up so that your car sails off exactly horizontally. Another ramp (or a can on a ring stand) will serve as the landing region for your car. Your decision of where to place the ramp involves determining:

• the correct direction,
• the correct distance from the jump ramp, and
• the correct height for the landing area in order to catch the car in flight.

As a start, discuss Items 1–3 in your group. (Remember that forces produce acceleration.)

1. What force or forces do you think affect the car’s motion in the vertical direction? What about in the horizontal direction?

Vertical direction: Gravity pulls the car down. Air resistance pushes up. Air resistance should have only a small effect if the car (or ball) are fairly heavy for their size.

Horizontal direction: Air resistance acts in a direction opposing the car’s forward motion. However, if the car is fairly heavy relative to its size, air resistance should have little affect.

2. Acceleration is always the result of force. How will you measure the acceleration that results from the force(s) you identified in Item 1?

Drop the car over (or under) the motion detector to get a model in the same way that was done with the falling objects in Activity 4. Air resistance in the horizontal direction will be ignored. In other words, we’ll assume that acceleration in the horizontal direction is 0 ft/sec.²

1. If no force acts on the car in one particular direction, what can you say about the acceleration in that direction? What about the velocity in that direction?

The acceleration is 0. Therefore the velocity is constant.

2. What measurement will you need to make in order to find a good model for motion in such a direction?

You need to know the velocity at any time during this motion. You might, for example, measure the horizontal velocity the instant the car leaves the ramp.

3. Is there a direction for which you can argue that there is 0 (or almost 0) force? If so, which one; if not, why not?

Horizontal motion is affected only by air resistance, and for small, but relatively heavy, cars or balls its effect is negligible.

4. How will you obtain the measurement you just described in (c)?

Students may suggest placing the motion detector in front of the end of the jump ramp and making a velocity calculation from the distance-versus-time graph of the car’s motion as it leaves the ramp. What may work better, is to move the jump ramp so that the ball travels a short distance on the table before falling off its edge. Friction will cause the rolling ball to slow slightly, but this method may give fair results. (Note, the motion detector can be secured on a ring stand at the proper height to detect the motion.)

The "photogate" method, very carefully measuring the amount of time necessary for the car to travel its own length, is much more accurate. (It is also very unlikely that students will suggest this!)

Part II: Equipment Set-Up

Set up the equipment that you will need for your stunt. Here are some general guidelines for the equipment set-up. Your teacher will provide more specific directions.

• Attach the jump ramp to a table so that the jump-point of the ramp is horizontal. (Check it with a level. Use shims as needed.) The higher the jump point, the better. Secure the ramp in place with C-clamps (or masking tape).
• You need to be able to release the car exactly the same way each time. One way to ensure this is to attach a small block of wood to the top of the ramp. Then to start a run, move the car up to touch this block and then let it roll down the ramp. (You may find another equally suitable method.)
• You will need room to work around the jump ramp and the landing area. Be sure both areas are clear.
• Using a plumb bob, find and mark the point on the floor directly under the jump point of the ramp.
• Set up a photogate at the jump point of the ramp. (If you do not have a photogate, make one by placing a penlight flashlight on one side of the ramp and a light probe on the other. Be sure the light source is shining directly into the probe.) Adjust the height of the photogate or light so that it will hit the longest part of the car (or middle of the ball) as it rolls down the ramp.

Be sure the jump-ramp set-up is not touched until the entire activity is completed!

Part III: Data Collection

4. What is the acceleration due to gravity on the car (or ball) that you will be using for this activity? How did you get your answer?

Sample answer:

We rolled a golf ball down the ramp. The golf ball is quite small. The motion detector was not able to track it when it fell. We decided to ignore air resistance and use the value of acceleration due to gravity found in physics books: –32 ft/sec² or –9.8 m/sec² or –980 cm/sec.² (In this situation, students might decide to use their experimental results from their book-drop experiments in Activity 4.)

5. Make three trial runs down the ramp and find the average velocity of the car off the ramp for those trials.

Sample answer: 207 cm/sec. (Realistic velocities could be anywhere around 200 cm/sec.)

Warning: On these three test runs, use the carbon paper to find exactly where the car hits the floor. It should hit approximately in the same place each time. If not, you should refine your release methods. Once you have done so and landings are more consistent, then repeat Item 5.

6. Find the middle of the three landing points for your test runs in Item 5. Place a mark there. Mark a line connecting the point on the floor directly below the jump point to your landing mark. (A chalk line or line of masking tape line will work well.) This line will be the line on which you will place your landing ramp or can.

Part IV: ANALYSIS—MODEL FORMULATION

7. Using the information you found in Items 4 and 5, develop a parametric model for the flight of the car. Your parametric model should include two equations: one describing the vertical motion and the other the horizontal motion. (Be sure all of your distance measurements are in the same units. Using centimeters and seconds for units works well.)

Parametric models should be in the form: x = v₀t, y = 0.5at² + h₀; a should be consistent with answers to Item 4 converted to the proper units; v₀ should be consistent with answers to Item 5.

Sample answer: x = 207t; y = –490t² + 77.5 (The motion detector was unable to get readings on the falling golf ball, so we used the value of acceleration due to gravity found in physics books.)

8. Use your model from Item 7 to predict how long the stunt car (or ball) will be in the air? How far from the plumb bob mark will it land? Does your answer make sense given the results of your test runs in Item 5?

Sample answer:

The vehicle will hit the ground in 0.40 sec and land 82.8 cm from the plumb bob mark. The center mark for the test run data was 81.5 cm. So, the answer based on the model is consistent with the results of our test runs.

Part V: THE CONTEST

The challenge here is to place the landing area so that it catches your stunt jumper safely.

Your teacher, or a student from another group, will give you a challenge by telling you either:

• the horizontal distance in front of the jump ramp for the landing area. (Sort of like saying, "Jump over 3 crates of rattle snakes.") After you get your distance, decide how high you need to set the catch ramp or ring stand for a safe landing at this distance.
• the height of the landing area. After you get your height, decide the distance from the ramp to place the landing area.

9. Describe the challenge you are given. Then use your model to compute your solution. Explain your reasoning.

Sample answer:

The Challenge:

Place the landing ramp (the can) 40 cm from the bottom of the take off ramp. How high should the ramp be placed?

Our Solution:

Find the time when the ball reaches a distance of 40 cm; solve 40 = 207t.

t = 0.193 seconds.

Now find the height of the ball at t = 0.193 seconds: –490(.193) + 77.5 » 59.2 cm.

Now get ready for the big jump! But first, here are some hints before you continue:

• Check that no one has altered the ramp, table, or other equipment since your three test runs.
• Be sure to release the car exactly the same way on the actual roll as you did on the three trials runs.
• Insist that more than one person do all the calculations and check their results. Have more than one person do the measurements for locating the landing area.
• Remember, the idea is to use your mathematical model for the flight of the car to position the landing area in this flight path so the car will land safely.

Once all the measurements have been made and the ramp or can is set, clear the area and roll the car (or ball) down the ramp.

Handout 1—Motion Detector Set Up

This handout is for use with a TI-82 or TI-83 and a TI-CBL.

You are about to collect data from a moving object using a motion detector.

1. Clear a walkway in the classroom in front of the motion detector. Since the beam detects objects between 1.5 and 24 feet away, a rectangular region about 10´ 25 feet will do nicely. (See Figure 1.)



Figure 1. Cleared area

2. Place the motion detector on a table at one end of the cleared area with its front face (see Figure 2) aimed toward the open region.



Figure 2. Motion detector front

3. Attach the motion detector to the SONIC port on the side of the CBL.

4. Use a link cable to attach the CBL to a TI-82 or TI-83. See Figure 3 for final configuration.



Figure 3. Motion detector set-up

5. Run the HIKER program according to your teacher’s directions. When the person is positioned in front of the motion detector, execute the program. The person should begin moving as soon as a point is displayed on the graph. Remember, you only have six seconds.

Note: In this unit, any program that takes readings for 6 seconds (in some cases, it may be longer), records readings every tenth (0.1) second, and displays the distance-versus-time graph as the motion occurs will be referred to as HIKER.

Handout 2—Walking the Walk

Descriptions of Walks

Handout 3—BALLDROP Directions

You will need the following equipment:

• Calulator with motion detector BALLDROP program.

Note: In this unit, any program that takes readings for approximately 1.6 seconds, records readings about every 0.02 seconds, and collects the all the data before displaying it will be referred to as BALLDROP. BALLDROP programs begin recording data after the TRIGGER button on the CBL is pressed.

• Motion detector equipment
• Protective frame
• Book

1. Attach a motion detector to the sonic port on the CBL.



2. Attach the CBL to a TI-82 or TI-83.



3. Place the protective frame on the floor around the motion detector (see Figure 1).

4. Execute the BALLDROP program. You may hear the motion detector clicking, but it will not begin recording data until you have pressed the TRIGGER button on the CBL.
5. Hold the book directly over the motion detector. Have someone push the TRIGGER button on the CBL and then say "Go." A moment after you hear the word "Go," drop the book so that it falls directly over the motion detector. Redo the activity until you have what you feel is a good graph. Check your calculator lists to see where your program has saved the times and distances.
6. Link calculators and make sure that everyone in your group has the data.

Handout 4—Falling Book Models

In column 2, write your answer to Item 1 (Activity 6), your model for the book’s distance given the elapsed time since the book was released. In column 3, record your equation for the book’s velocity from Item 2(b)(Activity 6), and in column 4 the book’s acceleration (Item 3, Activity 6).

The first row of Figure 7.1 contains this information from Sonia’s. Add the information from your group and then from the other groups in your class.

Handout 5—Parametric Equations: Calculator Review Motion Detector Set Up

The following calculator displays show possible settings for using parametric equations and your calculator to graph the motion of car’s sailing off ramps.

Select paramtric mode.

Enter the equations of the x and y motion.

Select a window that will show the x and y motion for a reasonable time.

Either graph and TRACE . . .

. . . or use tables to answer the question.

You might need to change the window to be more precise with your graph.

Now you can be precise to the 0.01 second.

Or you could change the table set-up to be more precise.

Transparency 1—Book Drop Data

Transparency 2—What Can Go Wrong?

Transparency 3—Falling Book Models

Annotated Student Materials

Preparation Reading—It’s Show Time!

You’ve probably watched scenes like these in the movies:

The hero, with no other route of escape, jumps off a roof top. Miraculously, he lands in the back of a passing pickup truck.

During a chase, the hero speeds down the street on his motorcycle. He crosses an intersection narrowly missing a truck.

Trial-and-error alone would be a poor method for planning such stunts. A mistake could cost stunt drivers their lives. A successful stunt requires careful planning. Mathematical calculations and an understanding of the laws of physics are often an important part of this planning.

Early in his career, Evel Knievel often relied on "gut-level instincts" to help him create stunts. The results were sometimes more spectacular than intended. In one show, for example, Evel placed a row of open crates containing rattlesnakes between two ramps. A hush fell over the crowd when he signaled for the start. Evel revved up his motorcycle’s engine. Then he sped up the first ramp, sailed over the rattlesnakes . . . and fell short of the landing ramp. When he landed he took down the side of one of the crates. This freed the snakes. Unharmed, Knievel sped off into the sunset. The crowd quickly left the stands.

Jeff Lattimore’s specialty in Chittwood’s Thrill Show is "The Leap for Life." Jeff has performed it successfully for years. In his stunt, Jeff climbs a ladder and stands on an eight-foot stool. Then the ladder is removed, leaving Jeff stranded just as a car comes speeding toward him. Jeff jumps a moment before impact. The car hits the stool and snaps it out from under him. Jeff lands safely on the ground.

Near crashes, ramp-to-ramp jumps, and leaps over oncoming vehicles are standard stunts at motorcycle and car thrill shows. In this unit, you’ll plan similar stunts. You’ll test small-scale versions of these stunts using toy cars or trucks or you’ll simulate your stunts using a calculator or computer.

	Activity 1—Plan Ahead!

 = 1, 2, 3

1. Suppose that it is your job to design a motorcycle-truck, near-collision stunt. What information would you specify in your design? What kind of instructions would you give to the stunt drivers?

Sample answer: The stunt design should include a layout of the intersection and a description of the two vehicles that will be used. The stunt drivers should be told (1) where to position their vehicles at the start of the stunt, (2) how fast their vehicles should be moving when they enter the intersection, and (3) how quickly they should get their vehicles up to this speed. For safety, drivers also might be told to wear helmets and fasten seat belts.

2. Next, suppose that you have been hired to design a ramp-to-ramp motorcycle jump.



1. Draw a sketch of the setup for the take-off and landing ramps.

Sample answer: Sketch of the ramp set-up:



Note: Students may decide to sketch a take-off ramp where the motorcyclist drives up the ramp rather than down the ramp as pictured above.

2. What information would be helpful in planning your stunt?

Sample answer: You should know how fast the motorcycle is going when it leaves the take-off ramp and how far it will fall by the time it reaches the landing ramp (so you’ll know how high to make the landing ramp).

3. How might you obtain this information?

Sample answer: In order to determine how fast the motorcycle is going when it leaves the first ramp, you could build a horizontal extension as shown in the diagram below.



Then have a rider drive down the ramp onto the extension and make a video of his ride. You can approximate the motorcycle’s speed at the bottom of the take-off ramp by determining the distance the motorcycle travels over a small time interval near the end of the ramp. Careful examination of the video can provide this information.

The question of how far the motorcycle will fall as it travels between ramps is more difficult to answer. If you know how fast the motorcycle is going when it leaves the take-off ramp, you can calculate how long it will take for the cycle to cover the distance between ramps. All that’s left is to determine how far a motorcycle with a rider will fall during this period of time. Maybe you could drop large objects (you wouldn’t want to drop actual motorcycles with riders) from various heights and videotape the falls to determine how far objects fall in a given amount of time.

3. Jeff Lattimore’s "Leap for Life" is a successful leap-over-an-oncoming-vehicle stunt. (Remember, he jumped and the car passed directly under him.) What important elements would you need to consider if you were designing such a stunt?

Sample answer: How fast the car should approach the stool, the height of the stool, how long it will take Jeff to reach the ground, the height of the car, the timing for Jeff’s jump.

4. In this unit, the first stunt that you’ll design is a two-vehicle, near-collision stunt using battery-operated, toy cars or trucks. Imagine that a battery-operated, toy car is moving along a straight line across the floor of your classroom. How could you determine how fast the car is traveling? How could you tell whether or not the car is moving at a constant speed?

Sample answers: Place two markers along the path that the car will travel. Turn the car on, start a stop watch when the car reaches the first marker, and then stop the watch when the car reaches the second marker. To approximate the toy vehicle’s speed, measure the distance between the two markers and divide by the time on the stopwatch. If you use this method to estimate the car’s speed along two different sections of its trip and find that the speeds are roughly the same, then it may be reasonable to assume that the toy car is moving at a constant speed. If the two speeds differ substantially, then you know that the car is not traveling at a constant speed.

5. Suppose that you are driving down a highway at 65 mph. You pass a highway sign that tells you you’ll have to drive 16 more miles to get to your exit. If you continue driving at the same speed, how many minutes will it take to get to the exit?

Driving 65 mph, means that you’ll cover 65 miles in 1 hour. It should only take 16/65 hour to drive 16 miles. That’s (16/65)hr ´ 60 min/hr » 14.8 min (or just under 15 minutes).

	Homework 1—Can You Say "Toy Boat" Three Times Fast?

The average velocity, V_1,2, of an object between time 1 and time 2 is determined by the following ratio:

V_1,2 =
V_1,2 = , where d₂ is distance at time₂ and t₂ = time₂.

Note that velocity will have units of such as , , or

1. Suppose that a toy boat is moving toward you. At time t = 0, the boat is 20 feet away from you. Three seconds later, at t = 3, the boat is 2 feet away.



1. What is the boat’s average velocity. (Be sure to include the units for velocity.)

The boat’s average velocity is (2 – 20) ft/(3 – 0) sec = –6 ft/sec

2. How does the boat’s velocity differ from its speed?

The speed is 6 ft/sec. Speed is always positive. Velocity changes sign depending on whether the boat is moving toward you or away from you.

2. Next, imagine that your toy boat is moving away from you along a straight line. Suppose that Figure 1 displays the distance between you and your boat each second during an 8-second trip.




Elapsed time (sec)	Distance between you and your boat (ft)
0	1.0
1	1.2
2	1.8
3	4.0
4	6.5
5	9.0
6	11.5
7	14.0
8	15.0

Figure 1. Time and distance data for a toy boat

1. Draw a graph that shows the relationship between the boat’s distance and the elapsed time. (Distance should be on the vertical axis and time on the horizontal axis. Be sure to label the scaling on your graph.)



2. What is the boat’s average velocity over the one-second interval from t = 0 to t = 1? (Be sure to include the units for velocity.)

The average velocity is (1.2 – 1.0) ft/(1 – 0) sec = 0.2 ft/sec.

3. What is the boat’s average velocity from t = 1 to t = 2? Is the boat speeding up or slowing down as it continues its trip?

The average velocity is (1.8 – 1.2) ft/(2 – 1) sec = 0.6 ft/sec. The boat is speeding up.

4. Find a three-second interval over which the boat appears to be traveling at a constant velocity. What is this velocity?

There are two three-second intervals: from t = 3 to t = 6 or from t = 4 to t = 7. The velocity is 2.5 ft/sec.

3. Next, imagine that your toy boat is moving away from you along a straight line. The distance-versus-time graph in Figure 2 shows the distance between you and your boat each half-second.



Figure 2. Distance-versus-time graph for a toy boat.

1. What is the average velocity of your boat from t = 0 to t = 9? (Be sure to include the units for the velocity.)

Average velocity = (18 ft — 0 ft)/(9 sec – 0 sec) = 2 ft/sec

2. What is the average velocity over the two-second time interval from t = 0 to t = 2? What about from t = 2 to t = 4? Is the boat moving faster during the first two seconds of its trip or the second two seconds? How could you tell from the graph in Figure 2?

The average velocity from t = 0 to t = 2 is 2 ft/sec and from t = 2 to t = 4 is 4 ft/sec. The boat is traveling faster during the second two-second interval than during the first two-second interval. You can observe from the graph that the change in the heights of the dots corresponding to t = 2 and t = 4 is greater than the change in the heights of the dots corresponding to t = 0 and t = 2. This means that the boat is traveling a greater distance during the second two-second interval than during the first two-second interval.

3. Find a two-second interval over which the boat appears to be traveling at a constant velocity. What is this velocity? Can you find more than one such two-second interval?

The boat appears to be traveling at a constant velocity over the intervals from t = 2 to t = 4 and from t = 7 to t = 9. Over the interval from t = 2 to t = 4, the boat is traveling 4 ft/sec. Over the interval from t = 7 to t = 9 the boat is traveling at 0 ft/sec

4. Describe in words what is happening to the toy boat’s velocity during its 9-second trip.

Sample answer: During the first two seconds of the trip, the boat starts out slowly and picks up speed. Then, it travels at a constant velocity for the next two seconds after which it begins to slow down. The boat comes to a stop 7 seconds into the trip.

4. Suppose that your toy boat is moving toward you instead of away from you. Draw a distance-versus-time graph that might represent its 9-second trip.

Because distance is decreasing over time, student graphs should be decreasing. In other words, their graphs should drop as you trace along them from left to right. For example, students could draw a linear graph with a negative slope.

	Activity 2—Watch Your Step

 = 4, 5, 6

DEMONSTRATION

Your teacher should select two students, Walker #1 and Walker #2: Instructions for their walks follow:

Walker #1: Stand about 2 feet from the motion detector. As soon as the motion detector begins clicking, walk away from the detector at a steady pace.

Walker #2: Stand about 10 feet from the motion detector. As soon as the motion detector begins clicking, walk toward the motion detector at a steady pace. Stop walking when you reach the motion detector.

1. The motion detector records the walker’s distance from the detector and the corresponding time. It plots each of the points, (time, distance), in real-time.



1. Before Walker #1 begins his walk, predict what the graph of distance versus time will look like. Why do you think it will look this way?

Sample answer: The graph should look like a straight line that moves up as you trace along the line from left to right. Because the person is walking at a steady pace, equal increments in time should produce the same changes in distance.

2. After the detector has recorded Walker #1’s motion, make a sketch of his distance-versus-time graph. Do you think the walker moved at a steady pace? How can you tell from the graph?

Sample answer: The graph had an initial horizontal piece to the left and then appeared to be a line with positive slope. When the walker began moving, he walked at a fairly constant pace. The pattern of the scatter plot of the distance-time data appears to closely follow a straight line.

2. Answer the following Items after Walker #2 has completed her walk.



1. Make a quick sketch of Walker #2’s graph. How is Walker #2’s graph similar to Walker #1’s graph? How is it different?

Walker #2’s graph has a brief horizontal section at the left, followed by a line segment with a negative slope, then a horizontal piece at about d = 1.5. Both Walker #1’s graph and Walker #2’s graph contained an initial horizontal segment (at the left end of the graph) followed by a straight line segment. For Walker #1, the slope of the line was positive; for Walker #2, the slope was negative.

2. According to Walker #2’s distance-versus-time graph, how close was she to the detector when she stopped walking? How close was Walker #2 to the detector when she actually stopped walking? What accounts for this difference?

According the graph, Walker #2 stopped about 1.5 feet in front of the detector. However, she actually stopped much closer to the detector. The detector is not able to track objects closer than about 1.5 ft from the detector.

WALKING

Your group should set up the motion detector as described in Handout 1 or as described by your teacher.

3. Select a member of your group to be the first walker. Watch carefully as the first walker completes his walk. On Handout 2, describe, in words, what you see the walker do. Then sketch the graph that is produced on the calculator screen. If possible, repeat this process so that each group member has a chance to be the walker.

Sample answers:

Walker #1

Description of walker’s motion:

The student stood about 2 feet from the detector. After the program began, the student paused for a moment and then walked steadily away from the motion detector.

Graph produced by walker’s motion:



Walker #2

Description of walker’s motion:

The student stood about 12 feet from the motion detector. When the program began, the student stood still for a moment before slowly walking toward the detector.

Graph of produced by walker’s motion:



Walker #3

Description of walker’s motion:

The student started walking away from the motion detector just before the program began. Then he stopped, turned around, and walked toward the detector.

Graph produced by walker’s motion:



Walker #4

Description of walker’s motion:

The student stood about 15 feet from the detector. After the program began, she hopped toward the motion detector.

Graph of walker’s motion:



Note that for approximately the first third of the hop-walk, there are several stray points approximately the same distance from the motion detector. These points indicate that sometimes the detector is picking up an object located behind the student.

TALKING

4. With your group, select two graphs from Handout 2 to present to the class. Once you have agreed on those two graphs, transfer your sketches and descriptions of the walks onto large sheets of paper.