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. Label them with your group’s name. Hang your presentations on a wall in your classroom.

Sample answers:

See Item 1 for sample graphs.

5. After all the graphs have been posted, take a few minutes to look at them. Sort these graphs into groups. Record how you decided which graphs belong in the same group. Explain your reasoning.

Sample answers:

Linear graphs, piecewise linear graphs, and curved graphs.

Graphs that are increasing over the six-second trip, decreasing over the six-second trip, sometimes increasing and sometimes decreasing over the six-second trip.

6. Select one graph from another group. Describe how the walker moved in order to produce this graph.

Sample answer: See descriptions from sample answers to Item 3.

	Homework 2—Step by Step

1. Suppose that Julie was Walker #1 when her class did Item 1, Activity 2. Her distance-versus-time graph appears as the first graph in Figure 4. The program used to collect the data recorded time in seconds and distance in feet. TRACE was used to find the points that are shown in the other graphs.



Figure 4. Graph of Julie’s walk

1. How far was Julie from the motion detector when the program began running?

1.98 ft

2. How long did Julie stand still before moving? Did she move toward or away from the detector?

0.9 sec She moved away from the detector.

3. How much time did Julie spend walking?

4.9 – 0.9 = 4 sec

4. How far did she walk?

15.98 – 1.98 = 14 ft

5. How fast did she walk?

14/4 = 3.5 ft/sec

6. Write an equation that models the relationship between distance, d, and time, t, during the period that Julie was moving. State the values for t that make sense in your model.

Equation: d = 3.5t – 1.17. This model makes sense for values of t between 0.9 and 4.9.

7. Does the d-intercept in the equation you found in (f) have meaning in the context of this situation? (The d-intercept is the value of d when t = 0.) What about the slope? Explain.

The d-intercept would give you the distance from the motion detector at t = 0. However, t = 0 is not one of the values that makes sense for this model. So, the d-intercept has no meaning in this context. The slope of the linear equation gives Julie’s velocity as she moves away from the motion detector.

2. Suppose the distance-time data in Figure 5 were recorded when George walked in front of a motion detector. A distance-versus-time graph for his walk appears next to the data table.




Time (sec)	Distance (ft)
0	13
1	13
2	13
3	11.5
4	9.9
5	8.6
6	6.8
7	5.6
8	3.9



Figure 5. Table and graph of distance-time data

1. How long did George wait until he began walking? Did he walk toward or away from the motion detector?

George waited for two seconds before he began walking. He walked toward the motion detector.

2. Enter the data from the table into calculator lists. Fit a least-squares line to these data. What is your equation?

d = –1.23t + 14.40

3. Graph the data and your equation from part (b). Does this line do a good job describing the portion of the graph that represents George’s motion?



This line does not do a good job describing the part of the graph representing George’s motion. The line should slant downward more steeply.

4. Remove any data that was not recorded during the time that George was moving. Fit a least-squares line to your edited data. What is your equation?

Students may decide to edit the first two or three data points. In either case, their regression equation will be (approximately) d = –1.51 t + 16.0.

5. Graph the data and your equation from part (d). How well does this line fit the portion of the graph that represents George’s motion?

Students may decide to graph either the edited data or the original data. Notice that the equation from (d) fits the portion of the graph representing George’s motion very well. Below is a graph of the original data and the least-squares equation from (d).



6. Approximately what is George’s velocity when he is walking? How can you read this information from your linear equation in part (d)?

George walks at a constant velocity of approximately –1.51 ft/sec His velocity is the same as the slope of the line describing his motion.

Students may decide to calculate his velocity using two points; for example, students might use (3, 11.5) and (8, 3.9) to calculate George’s velocity. In this case, the average velocity is –1.52 ft/sec This is very close to the slope of the equation from part (d).

3. Suppose that Jessie is walking at a steady rate of 4 miles per hour. At 5:00 she is three miles from home and continues walking at the same steady rate. At what time will she reach her house? How did you determine your answer?

Using the formula d = rt, solve 4 = 5t for t. This gives T = 4/5 = 0.8 hours. So, it will take Jessie (0.8 hours)(60 minutes/hour) = 48 minutes. Jessie will arrive home at 5:48.

4. The graph in Figure 6 is a time-lapse graph of Mich’s motion as she walked in front of a motion detector. This graph is similar to ones that you drew for the unit Animation. The solid line indicates Mich’s path. Sample times have been added to the graph to show when she passed particular locations.



Figure 6. Time-lapse graph of a walker’s motion

1. Is Mich traveling at a constant velocity? If not, is she speeding up or slowing down? How can you tell?

The walker is not moving at a constant velocity. The walker moves 1 foot during the first second, 2 feet during the second, 3 feet the third, 4 feet the fourth, 5 feet the fifth, and 6 feet the sixth. The walker is speeding up as she moves along her path from t = 0 to t = 6. Therefore, her velocity is increasing and not constant. You can also tell that the velocity is not constant because the dots on the graph are not equally spaced.

2. What is her average velocity for the entire 6-second walk? Over what one-second interval do you think her average velocity was closest to her average velocity for the entire trip?

Sample answer: The average velocity during the trip is 3.5 feet per second. From t = 2 to t = 3 Mich was moving an average of 3 ft/sec From t = 3 to t = 4, she was moving an average of 4 ft/sec So, Mich’s average velocity from t = 2.5 to t = 3.5 is probably fairly close to 3.5 ft/sec.

3. Draw a distance-versus-time graph for this walk. Describe, in words, the shape of the graph. (For example, does the graph look like a straight line? Does it curve up or down?)

Sample answer: The graph curves upward.



4. Suppose the location of the motion detector had been at d = 24 instead of at d = 0. (In other words, the motion detector was placed on the opposite side of the room.) Repeat part (c) for this situation.

Answer:

5. Mizan lives 3 miles from his school. Suppose that he missed his bus and walked home. It took him 3/4 hour.



1. What was Mizan’s average speed?

His average speed was 3 mi/0.75 hr = 4 mph.

2. Suppose that after walking a half hour, Mizan was half-way home. Realizing that he was going to be late, he jogged the rest of the way. What was his average speed during the first half-hour of his walk? What was his average speed during the last 15 minutes?

Mizan’s average speed for the first half-hour of his walk was 1.5 mi/0.5 hr = 3 mph. His average speed for the last quarter hour was 1.5 mi/0.25 hr = 6 mph.

	Activity 3—Staging a Near-Hit

 = 7, 8, 9, 10

In this activity, you’ll first plan, and then stage, a two-vehicle near-collision stunt.

1. Assume that both roads are straight and that they intersect at right angles. Make a rough sketch of the intersecting roads.

Sample answer:

2. Next, you will need two battery-operated vehicles (cars, trucks, robots).



1. Describe the toy vehicles that you will be using for your stunt. Be sure to record any dimensions that you think will be important to the success of the stunt.

Sample answer: For our stunt we used a big red fire chief’s car and a blue monster truck. The car was 12.5 inches long inches long and 5.25 inches wide. The truck was 8 inches long and 7 inches wide. Both were battery operated. You started the car by pressing a button which caused a voice to respond "We’re on our way" after which the car began moving. The truck had an on/off switch and began moving as soon as the switch was turned on.

2. Use the motion detector to gather information about how your battery-operated cars (or trucks) move. Describe the shapes of your graphs. Based on these graphs, were your toy vehicles traveling approximately at a constant velocity? If not, were they speeding up or slowing down? Explain.

Sample answer: The distance-versus-time graphs both look linear during the times when the vehicles were moving. That means that both vehicles were traveling at constant velocities.

3. Describe with an equation the distance-versus-time data collected by the motion detector while each car (or truck) was moving. Explain how you got your equation. Include sketches of the graphs produced by the motion detector readings as part of your explanation.

Sample answer

Car:

We looked at the car’s distance-versus-time data. Here’s the graph:



Since the portion of the graph that corresponds to the car’s motion is linear, you can fit a least-squares line to the portion of the data collected while the car was in motion. In order to do this, the data must be edited. A scatter plot of the edited data appears below.



Least-squares equation: d = 1.53t – 1.23.

Truck:

The graph appears below.



Least-squares equation: d = 0.67t + 2.22.

Note: Instead of fitting a least-squares line to part of the data, students may decide to select two data points and use them to determine the equations.

4. What is your best estimate for how fast each vehicle moves? How do you get your estimate?

Sample answer:

Car:

Based on the slope of the least-squares equation, the car moves at approximately 1.5 ft/sec

Truck:

Based on the slope of the least-squares equation, the truck is traveling at approximately 0.67 ft/sec

3. Remember, the key to this stunt is to cause some anxiety for the students who are watching. The vehicles should pass at the intersection as closely as possible without crashing. With this in mind, describe how you will stage this stunt:



• Where on the intersecting roadway will you place the vehicles?
• Will you start both vehicles at the same time? If not, how do you plan to stagger the starts?

Include in your description the mathematics supporting your stunt design.

Sample answer:

Position for car: The front of the car should be placed five feet from the middle of the intersection.

In order to avoid collision, 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 when the truck arrives. Here’s how long it will take the car to travel (5 + 4/3) feet:

t = 6.33ft/(1.5 ft/sec) » 4.2 sec



Next we calculate the starting position for the truck. The truck travels more slowly than the car. Here’s how far the truck will travel in 4.2 seconds:

(4.2 sec)(0.7 ft/sec) » 2.9 ft

We want the truck to reach a location 2.5 inches (or about 0.2 ft) before the intersection in 4.2 seconds. (See diagram above.) So, place the truck at 3.1 ft or about 3' 1" to the left of the intersection.

Get ready for the show!

4. Perform the stunt. Did the stunt go according to your plans? If not, go back and re-work the details of your stunt design. (Check your calculations. Be sure that you accounted for the vehicles’ sizes.) Revise your stunt plans and try your stunt again.

Sample answers:

Sample answer #1: The stunt worked perfectly!

Sample answer #2: Our vehicles crashed. We forgot to take into account the dimensions of the vehicles.

Sample answer #3: Our stunt didn’t work very well. We think that the batteries in the cars wore down and that changed the vehicles’ velocities. We could check this assumption by taking new motion detector readings.

	Homework 3—Start Your Engines

1. Suppose a battery-operated toy car is allowed to travel down a straight highway until its batteries run out.



1. Draw a sketch of how you think its distance-versus-time graph would look. Assume that distance is measured from the car’s starting position. Be sure to put labels and scales on your axes.

Sample answer:



2. Describe in words what your graph tells you about the car’s velocity during its trip.

Sample answer based on (a): When the car is turned on it runs at approximately a constant velocity for a period of time. At first the graph looks fairly straight. As the car continues to move, its batteries weaken and the car begins to slow down. Eventually the car stops. The toy car in (a) runs continuously for 20 minutes before it comes to a stop.

3. Based on your graph, what is this toy car’s average velocity? Show your calculations.

Sample answer: According to the graph in (a), the car’s average velocity over its 20 minute trip is around 1250 ft/20 min. = 62.5 ft/min or a little over 1 ft/sec

2. Suppose that a battery-operated toy car moving in front of a motion detector produced the distance-versus-time graph in Figure 7.



Figure 7. Distance-versus-time graph for a toy car

1. Approximately how far was the car from the motion detector when it began recording data? How far was the car from the motion detector when it stopped recording data?

It was about 2 feet from the motion detector when the detector began recording data and about 22 feet from the detector when it stopped recording data.

2. After the motion detector began recording data, how long did it take before the car began moving?

About 1.5 seconds.

3. During the time that the car was in motion, how fast was it moving? Show how you got your answer.

It was moving (22 – 2) ft/(6 – 1.5) sec » 4.4 ft/sec.

4. Suppose that Cora wanted to determine a model for the car’s motion. She fit a least-squares line to the data collected by the motion detector. A graph of her model appears in Figure 8. According to Cora’s model, what was the approximate velocity of the car? How did you get your answer?



Figure 8. Least-squares line superimposed on scatter plot

According to Cora’s model, the car was traveling approximately 3.9 ft/sec Because Cora’s model is a linear equation, the slope gives the car’s velocity.

5. Is your approximation of the car’s velocity based on Cora’s model (part (c)) too high, too low, or about right? What went wrong?

It is too low. Cora should have removed the data that corresponded to the horizontal section of the graph before she fit a least-squares line. Her model does not do a good job describing the pattern of graph that corresponds to the moving vehicle.

3. The screens in Figure 9 show a distance-versus-time graph for a wind-up toy car powered by a spring. The times, x, in seconds and corresponding distances, y, in feet are indicated for three different times.

 

Figure 9. Distance-versus-time graph for wind-up car

1. Is the car traveling at a constant velocity? How can you tell from the first graph in Figure 9?

No, the car is not traveling at a constant velocity. Because the graph bends upward, the car is speeding up.

2. What is the car’s average velocity from t = 0.0 to t = 0.5? What is it from t = 0.5 to t = 1.0? What about from t = 1.0 to t = 1.5? Based on these velocities is the car speeding up, slowing down, or traveling at a constant rate?

0 ft/sec; (3.838 – 2) ft /0.5 sec » 0.9 ft/sec; (6.7085 – 3.838) ft/0.5 sec » 5.7 ft/sec

The car is speeding up; its velocity is increasing.

4. Suppose that the motion from Jason’s car produced a horizontal line for its distance-versus-time graph. Describe how Jason’s car moved relative to the motion detector to produce this graph.

Sample answers:

Jason’s car drove in a circular arc with the motion detector at the center. This kept its distance from the motion detector constant.

Jason’s car never moved.

	Activity 4—Falling for You

 = 11, 12, 13

You’ve probably seen a stunt similar to this in the movies:

The hero, trapped on a rooftop by villains, jumps for his life. (The audience gasps.) Fortunately, the hero lands in the back of a passing pickup truck.

Part I: IMAGINING FALLS

1. Imagine that the hero steps backward off a rooftop 25 feet above the ground. He falls straight to the ground (and lands in an air bed).



1. Suppose that a motion detector, placed on the ground, is able to track the hero’s fall. Sketch a distance-versus-time graph that might result from the detector’s readings. Add appropriate scales and labels to your graph’s axes. Explain why you drew the graph the way that you did.

Sample answer:



Because the hero gets closer to the motion detector as he falls, the height of our group’s graph decreased as time elapsed. We decided that the hero would fall slowly at first and then pick up speed during his fall. So, we made the graph curved rather than straight.

2. Do you think your graph does a good job describing the hero’s fall? Suggest ways that you might collect data that would help you decide.

Sample answer #1: You could video tape stunt men or women falling from buildings. Then you could slow the video down and pause at equal intervals of time to measure the stunt person’s height above the ground each time the video was paused.

Sample answer #2: Drop various objects (dummies, large dolls, balls, books, etc.) and use a motion detector to gather distance-versus-time data on their falls.

Part II: RECORDING FALLS

To answer Item 1(b), you really need data on falling heroes. But you can’t drop a person to see how he falls. Instead you’ll drop a book and record data from its fall. (Or you’ll drop some other non-living object.)

1. Drop a book, and use a motion detector to collect distance-versus-time data on its fall. Your teacher will supply the directions. When you are happy with your data, link calculators and make sure that everyone in your group has the data. (Store your data in a location where it won’t be erased.)

Data will vary from group to group. See (b) for graphs of sample data.

2. Make a sketch of the distance-versus-time graph produced by your data. Be sure to add appropriate scales and labels to the axes.

Sample Answer #1:

For the following graph the motion detector was placed on the floor. The window settings were [–0.14, 1.76] × [–1.31, 9.02].



Sample Answer #2:

For the following graph the motion detector was placed above the falling object. The window settings were [–0.15, 1.74] × [–1.16, 7.99].



3. Explain what each piece of your graph in part (b) means. If you followed directions exactly, you should have three distinct pieces. (Don’t worry if you have an extra small piece here or there.)

Sample Answer #1:

The object wasn’t moving for a short time after the program began. The decreasing portion of the graph corresponds to the book’s fall. The book hit the motion detector’s protective frame and then fell off. Then the motion detector began taking readings of the ceiling which produced the horizontal line segment not connected to the rest of the graph.

Sample Answer #2:

The object wasn’t moving for a short time after the program began. The increasing portion of the graph corresponds to the object’s fall. The detector lost the object just before it hit the floor. Then the motion detector began taking readings of the floor which produced the horizontal line segment not connected to the rest of the graph.

4. Look at the portion of your graph that represents the motion of the object while it is falling. Was the object falling at a constant velocity? If not, did the object speed up or slow down as it fell? How can you tell from the graph?

Sample answer: (Based on Sample Answer #1 in (b).)

The portion of the graph corresponding to the book’s fall is curved rather than straight. This means that the object did not fall at a constant velocity but that its velocity decreased as it fell. (Because the velocity is negative, a decrease in velocity means it gets more negative. So, the object travels more quickly as it nears the motion detector.) If you trace from left to right along this portion of the graph, it initially drops gradually and then more steeply. The change from gradual decline to steep decline indicates the object is falling faster at the end of its fall than at the beginning.

Part III: ANALYZING THE DATA

3. Next, you will determine an equation that describes the motion of the book. You’ll need the results from part (f) in Activity 5. So, remember to save your work!



1. Edit your data so that you have only the portion of data that represents the object’s fall. (You may want to use a program such as EDITPART to help you edit your data.) Continue to remove any stray points so that the graph of the edited data is fairly smooth.

See sample answer part (b).

2. Make a quick sketch of this graph. Be sure to state the window settings and label your axes in your sketch.

Sample answer:



3. Select a time near the beginning of the book’s fall. Recall that you have calculated a moving object’s average velocity from time 1 to time 2 using the ratio:



How could you use this ratio to approximate the book’s velocity at your selected time? What is its approximate velocity?

Sample answer:

Selected time: t = 0.58 second. Use the average velocity over a very small interval containing t = 0.58 to approximate the book’s velocity at this time. When t = 0.56001, the book is 3.5975 feet above the ground and when t = 0.6001, the book is 3.3886 feet above the ground. Using these points, the velocity of the book when t = 0.58 is approximately –5.2 ft/sec.

4. Select a time near the end of the book’s fall. Repeat part (c) for this time.

Sample answer:

Selected time: t = 0.74 second. When t = 0.72, the book is 2.4739 feet above the ground and when t = 0.76, the book is 2.067 feet above the ground. Using these points, the book’s velocity when t = 0.74 is approximately –10.2 ft/sec

5. Based on your calculations for (c) and (d) is the book speeding up, dropping at a constant rate, or slowing down as it falls?

Sample answer: The velocity is getting more and more negative as the book falls. This means the book is speeding up during the fall. Since speed is the absolute value of velocity, the book’s speed increases from around 5.2 ft/sec to 10.2 ft/sec during the drop.

6. What type of equation do you think might describe the data shown in part (b)? Fit an equation to your graph in (b). Then make a residual plot. Based on the residual plot, does your equation appear to do a good job in describing these data? (If not, select another type of function and try again.)

Sample answer: d = –15.46t² + 12.78 t + 1.29, where d is the distance between the object and the motion detector and t is the elapsed time since the calculator began recording readings. Note that distance is recorded in feet and time in seconds. Save this equation for use in Activity 5.

The points in the residual plot appear to be randomly scattered. However, there are two sizable residuals, one negative and one positive. The window settings for this plot were [0.45, 0.81] × [–0.006, 0.008]. So, relative to the size of the actual distances, these residuals are very small.

4. Jackie and Darryl drew the graphs in Figures 10 and 11 as part of their answers to Item 1.



Figure 10. Jackie’s graph



Figure 11. Darryl’s graph

Select one of your group members to be the walker. Your walkers will try to create distance-versus-time graphs that resemble the shapes in Figures 10 and 11. (Don’t worry about matching the scaling on the axes.)

1. Write instructions telling each walker how to walk in order to produce the desired distance-versus-time graphs.

Sample answer:

Walker #1: Assume that Walker #1 is trying to produce a graph similar in shape to Figure 10. She should stand on the other side of the room from the motion detector. She should begin walking toward the motion a moment before the detector begins recording data. She should walk at a constant pace.

Walker #2: Assume that Walker #2 is trying to produce a graph similar in shape to Figure 11. He should stand on the other side of the room from the motion detector. He should begin walking toward the detector at about the same moment the motion detector begins recording data. He should begin his walk slowly and gradually pick up speed.

2. After the walkers have completed both walks, identify any differences between the shapes in Figures 10 and 11 and the distance-versus-time graphs produced by the walkers. Discuss possible explanations for those differences.

Sample answer:

Walker #1’s graph looked like a straight line with a y intercept of 7 and a negative slope. However, there was a short horizontal section at the left end of the graph. The walker had moved too close to the detector and it was no longer reporting correct distance readings.

The mid-section of Walker #2’s graph looked about right. However, there were short horizontal segments to the right and left Apparently, the walker did not begin walking the moment the motion detector began recording readings and at the end of his walk was too close to the motion detector to get accurate distance readings.

	Homework 4—Moving Along

 = 7, 11

Ms. Keating’s math class had a contest to see who could walk the best parabola. Each group planned how its walker would have to walk so that the graph produced by the motion detector’s readings would look like a parabola. Items 1 and 2 refer to this experiment.

1. Suppose that Anita was the walker for her group. Anita’s distance-versus-time graph is shown in Figure 12. The coordinates corresponding to points A, B, and C are (1, 8), (2, 16.25), and (3, 20), respectively.



Figure 12. Graph of Anita’s walk

1. What was Anita’s average velocity from t = 1 to t = 3? What about from t = 1 to t = 2? From t = 2 to t = 3?

5.5 ft/sec; 8.25 ft/sec; 2.75 ft/sec

2. Describe how Anita walked from t = 1 to t = 3. Use your velocities from part (a) in your description.

Anita walked more quickly initially traveling an average of 8.25 ft/sec from t = 1 to t = 2. As she continued walking her paced slowed. From t = 2 to t = 3, her average velocity was only 2.75 ft/sec For the two seconds from t = 1 to
t = 3 she averaged 5.5 ft/sec

3. Imagine that there is a device called a "velometer" that measures a person’s velocity at each instant during a walk. (This is much like a car’s speedometer that measures your speed each instant that you are driving. When velocity is positive, velometer readings are the same as speedometer readings.) Does the average velocity you compute for the interval from t = 1 to t = 2 best represent the velometer reading for t = 1, t = 2, or some other instant? Explain.

Sometime between t = 1 and t = 2. Because Anita is slowing down during the interval from t = 1 to t = 2, her velocity at t = 1 is greater than 8.25 ft/sec and her velocity at t = 2 is less than 8.25 ft/sec.

4. What can you say about Anita’s velometer readings between t = 3 and t = 5?

Sample answer: They should all be negative because Anita is moving toward the motion detector during the second half of her walk. Because of the symmetry of the parabola, the velometer reading at t = 4 should be the negative of the velometer reading at t = 2 and the velometer reading at t = 5 should be the negative of the velometer reading at t = 1.

2. Anita’s group used quadratic regression to fit an equation to the data corresponding to times that Anita was actually walking (between t = 1 and t = 5). The group decided that the model:

d = –2.75t² + 16.5t – 5.75

did a good job describing these data.

1. Anita’s average velocity between any two given times, say time 1 and time 2, can be calculated from the ratio:



What would happen if you tried to use this same ratio to find her instantaneous velocity (velometer reading) at t = 2 seconds?

You would get a ratio of 0/0.

2. One way to approximate Anita’s instantaneous velocity at t = 1.5 is to find her average velocity over a very small time containing t = 1.5. Select an interval centered at t = 1.5. What is your interval? Use this interval and the group’s model to estimate Anita’s instantaneous velocity at t = 1.5.

The chosen interval is from 1.4 to 1.6. The average velocity over this interval is (13.61 – 11.96)/(1.6 – 1.4) = 8.25 ft/sec.

3. Use the group’ quadratic equation to estimate Anita’s instantaneous velocity at t = 3.

Sample answer: For t = 3: If you use the interval from 2.9 to 3.1, the average velocity is 0 ft/sec

4. Repeat part (c) for t = 4.5. How is your answer related to the one in part (b)? How do you explain this relationship?

For t = 4.5: If you use the interval from 4.4 to 4.5, the average velocity is –8.25 ft/sec

The portion of the parabola that lies to the left of t = 3 is symmetric to the portion that lies to the right of t = 3. (From t = 1 to t = 3, Anita walks away from the detector. Then she reverses her direction and retraces her steps.) Because t = 1.5 is 1.5 units to the left of t = 3 and t = 4.5 is 1.5 units to the right of t = 3, her speeds at t = 1.5 and t = 4.5 are identical. The difference in direction accounts for the opposite signs on the velocities.

5. If a distance-versus-time graph is linear, you can determine the instantaneous velocity at any time from the slope of the line. However, the graph d = –2.75t² + 16.5t – 5.75 is curved. How might you define the "slope" of this graph at t = 0.5?

Sample answer: Determine a small interval centered at t = 0.5. For example, t = 0.45 to t = 0.55. Use the equation for d to find the distance corresponding to these times. Then use these two points to calculate slope by the ratio: (change in distance)/(change in time).

3. Figures 13 and 14 display distance-versus-time graphs when Anita’s class performed the falling-book experiment. Figure 13 shows the graph of all the data recorded by the motion detector. Figure 14 displays only those data relevant to the book’s motion while falling. Figure 14 is completed by overlaying the graph of a quadratic model fit to these data.



Figure 13. Falling book’s distance-versus-time graph




Figure 14. Portion of graph representing the fall,
with quadratic model overlay

1. Return to your data from Activity 4. If you have not already done so, complete graphs like Figures 13 and 14 for your data. What is your model?

Sample answer: d = –15.46t² + 12.78t + 1.29.

2. After your book was released, did it fall at a constant velocity? How can you tell from your graphs?

Since the book’s distance-versus-time graph is curved and not a straight line, the book is not falling at a constant velocity.

3. Explain how you might use your quadratic model to estimate the book’s velocity at the instant t = 0.5 second? (If this instant is not during the "fall" part of your data, choose another time that is.) What is your estimate?

Sample answer: This answer is based on the model d = –15.46t² + 12.78t + 1.29. To estimate the velocity at the instant t = 0.5, you could use the average velocity over a very short time interval centered at t = 0.5 seconds. If, for example, you used the interval from t = 0.45 to t = 0.55, you could use your model to determine the corresponding distances of around 3.91 and 3.64, respectively. This gives an average velocity of –2.68 ft/sec

4. Use a calculator to graph your model in a window appropriate for the book-drop data. Then trace to the point on your graph that is closest to the time t = 0.5 (or, if that time is not during the "fall," to some other time when the book was still in the air). Magnify a small section of graph containing your point by zooming in on this point several times. Sketch and describe this magnified section of your graph.

The graph should appear to be a straight line.

5. Use your answer to part (d) to reanswer Item 2(e); that is, define the "slope" of a curve at a particular point on the curve. Explain your method and the idea behind it.

To find the slope of a curve at a particular point, zoom in on that point until the curve resembles a line. Then select two points on this "approximate" line, and determine slope by the usual formula.

6. Explain what measuring the slope of your graph at time t = 0.5 tells you about the book’s fall during your experiment in Activity 4.

The slope tells you the instantaneous velocity of the falling object at t = 0.5 second.

4. For stunts, it’s often important to know when the falling object reaches a certain height.



1. Use your model to find how long it took from the time the book was released until it reached a height 1 ft above the motion detector. Explain how you got your answer.

Sample answer based on the model d = 15.46t² + 12.78t + 1.29. When t = 0.85, the book was 1 ft above the detector. The book was released at t = 0.41. This means that it took the book 0.44 seconds to reach a height 1 ft above the motion detector. Here’s how these times were found. I graphed the model and the line d = 1. Then I used the CALC/Intersect feature on a TI-83 to find the time when the model intersected with the line d = 1. To find the time the book was released, I used the CALC/Maximum feature on a TI-83.

2. Explain why you won’t find the answer to this question in your data.

The motion detector doesn’t record distances for objects closer than about 1.5 ft. So you won’t find a distance of 1 in the data collected by the motion detector. In addition, you the motion detector may not have taken a reading exactly the moment the book was dropped.

	Activity 5—It Depends on Your Perspective

 = 14, 15

Recall Jeff Lattimore’s stunt where he jumps off a stool into a fall. (See the preparation reading.) Imagine performing the following scaled-down, simplified version of Jeff’s stunt.

Hold your book above the floor. Aim a battery-operated toy car on a straight-line path toward the drop point for the center of the book. Next, mark a point along the car’s path as shown.

To begin the stunt, turn the car on. When the front of the car reaches the mark, drop the book. The "trick" to this stunt is to place the mark in such a way that the car has enough time to pass beneath the book, thus narrowly avoiding getting hit.

1. In Item 3(f), Activity 4, you determined a quadratic model that described the data in a distance-versus-time graph. In your model, the dependent variable, d, was the book’s distance from the motion detector; the independent variable, t, was the elapsed time since the detector began recording data. Sketch a graph of your equation (not the data) for t between 0 and 2 seconds. Then shade the portion of graph that actually makes sense for the falling-book context.

Note: The sample answers to this activity are based on the model d = –15.46t² + 12.78t + 1.29.

Sample answer: A graph of d = –15.46t² + 12.78t + 1.29 in the window [–0.5, 1] × [–1,5] appears below.

2. Approximate the coordinates of the vertex of the graph that you drew for Item 1. Interpret the meaning of the vertex in the context of this problem.

The coordinates of the vertex are approximately (0.41, 3.93). The first coordinate gives the time the book was dropped, and the second the distance between the book and the motion detector at the instant the book was dropped.

3. How long was your book in the air? Could you answer this question from your data? Why or why not?

Sample answer: Because the motion detector won’t detect an object that is closer than about 1.5 feet from the detector, you could not answer this question using your data.

d = 0 when t » 0.92. That means that the book hit the detector approximately 0.92 seconds after the motion detector began recording data. Thus, the book was in the air from t = 0.41 to t = 0.92 seconds or for about one-half of a second. (This answer ignores the fact that the book hits the protective frame surrounding the motion detector. However, if the frame is not much higher than the detector, this shouldn’t add much error to the answer.)

4. Use your equation for d to determine the average velocity of the book from t = 0 sec to t = 0.2 sec Does your answer make sense in the context of the book-drop experiment? Explain.

Sample answer: (3.2276 – 1.29) ft/(0.2 – 0.0) sec » 9.7 ft/sec

This answer does not make sense. The book is not dropped until t = 0.41 seconds. If t < 0.41, it does not make sense to use this model.

5. Select a time during the book’s fall.



1. Without calculating the book’s velocity at this time, is the velocity positive or negative? How do you know?

Sample answer:

I selected t = 0.8 seconds. The book’s velocity is negative. As the book falls, its distance from the motion detector (assuming the motion detector is on the floor) is decreasing. The book is falling faster as time elapses.

2. Approximate the instantaneous velocity (the velometer reading) of the book at your selected time. Describe the method that you use.

Sample answer:

To determine the velocity or "slope" of the graph at the point (0.8, 1.6), trace as close to this point as possible. Then zoom in on this point. Repeat this process a second time. Now the graph in the calculator screen will look like a line. Select two points on either side of (0.8, 1.6), say (0.78058511, 1.845897) and (0.8025266, 1.5892933). Use these points to calculate the slope and don’t round until you have completed your calculations. The book’s velocity at t = 0.8 seconds is approximately –11.7 ft/sec.

For now, assume that you’ll be performing the stunt described in Figure 15 holding the book at the same height as you did when you collected your group’s book-drop data in Activity 4. In fact, you can use the equation that you graphed in Item 1.

1. The dependent variable, d, for the model you graphed in Item 1 is not the same as the book’s height above the floor. The motion detector has thickness! Let h represent the book’s height above the floor. What is the relationship between h and d?

Sample answer: The face of the motion detector is approximately 11/8 in. or about 0.11 ft above the floor. So, h = d + 0.11.

2. Write a model that describes the relationship between h, the height of the book from the floor, and t, the elapsed time since the motion detector began recording data.

Sample answer:

Assume that the motion detector is on the floor and the book falls toward the motion detector. Add the constant 0.11 to your model for d to obtain the model for h: h = –15.46t² + 12.78t + 1.40.

3. Graph your equation for h versus t and d versus t in the same window. How is the graph for h related to the graph for d?

Sample answer:

The graph of h versus t looks exactly like the graph of d versus t except that it is 0.11 units higher.

4. What are the coordinates of the vertex of your graph for h? Use 4-digit accuracy for your answer. (You will need this degree of accuracy for later calculations.) What method did you use to determine the vertex?

Save your answer for use in Activity 6.

Sample answer: The coordinates of the vertex are (0.4133, 3.9311). We used the 2nd, CALC, Maximum feature on a TI-83 to determine the vertex. The display produced by this routine was rounded to four decimal places.

Testing 1, 2, 3 introduced the quadratic formula for solving quadratic equations. Recall, if you want to solve an equation of the form
ax² + bx + c = 0, you can use the formula



5. Use the quadratic formula and your model from part (b) to determine the "hit-the-floor" time for your book. You should get two answers. One of them makes sense in the book-drop context and the other does not. Which of your two answers makes sense?

Sample answer: 0.92; –0.10. Only the first answer makes sense in terms of the book drop experiment.

6. How long would it take from the time the book was released for it to actually hit the floor?

Drop time was approximately (0.92 – 0.41) seconds = 0.51 seconds.

7. Use the information you developed in Item 7 to design a book-drop, toy-car, near-hit stunt. Do not carry out your design yet; just plan it. Do you foresee any problems in your plan?

Answers will depend on the height of the book when dropped, the book’s dimensions, the car’s dimensions, and the speed of the car.

Sample answer:

Assume the following: The book is dropped from 3.93 feet; the book was 8.5 inches in width (this is the length that the car must pass under); the car moves 1.53 ft/sec; the car is 12 inches long.

In 0.51 seconds, the car will travel 0.78 feet or approximately 9.36 inches. The car must be 4.25 inches beyond the drop point when the book hits the ground. This means that the mark should be placed 5.11 inches in front of the drop point.

This plan does not take into account the height of the car. This omission may cause problems for the success of the stunt. The book may reach the height of the back-end of the car before the car has passed under the book.

1. What do you get when you substitute t = 0 into your equation from Item 7? What does this tell you about the motion of the book during the stunt? Based on your answer to this question, why might it be more appropriate to start timing the stunt after the book is released?

Sample answer: When t = 0, h = 1.40. However, this doesn’t give you any information relevant to the stunt. The book has not yet been released. We only care about descriptions for times after the fall has begun.

Recall that the independent variable, t, for your model in Item 7(b), is the elapsed time since the motion detector began recording data. (You might want to rename this variable t_record.) This is not the most natural independent variable to use in this context. What you really need to know is how long it would take from the time the book was released for it to hit the ground. Or for the stunt in Figure 15, how long it would take from the time the book was released for it to reach the height of the car.

2. Let t_fall represent the elapsed time since the book was released. How is t_fall related to the original independent variable t (or t_record).

Sample answer:


t_fall = t – 0.4133 or t_fall = t – (first coordinate of vertex)

3. Select values for t ( separated by 0.05 second) during the book’s fall and record them in a table similar to Figure 16. Then use your equation from Item 7(b) to approximate the height of the book at these times. Use 4 decimal places when recording the heights. (Note, your times and corresponding heights will probably vary from those in Figure 16.)




t, elapsed time (sec) since the detector began recording data	tfall, elapsed time (sec) since the book was released	h, height above the floor (ft)
0.45		4.0204
0.50		3.9250
0.55		3.7524
0.60		3.5024
0.65		3.1752

Figure 16. An example: sample times and corresponding heights.

Sample answer: See Figure 16.

4. The time data in column 1 of your table from part (c) refer to the elapsed time since the motion detector began recording data. You used these times to calculate the corresponding values for h. Now, use your relationship in (b) to complete column 2.

Sample answer: Subtract approximately 0.4133 seconds (the first coordinate of the vertex) from each entry in column 1.

5. Make a scatter plot of the data in column 3 versus the data in column 1. (The points on your scatter plot will lie on the graph of your equation for h versus t from Item 7(b)). In the same window, make a scatter plot of the data in column 3 versus the data in column 2. Compare your two plots. How is the second plot related to the first?

The second scatter plot would shift left about 0.41 units.

6. The old independent variable, t, that measured the elapsed time since the motion detector began recording data is not the most natural variable to use in the context of this stunt. A better independent variable would be t_fall, the time elapsed since the book was released. Adjust your equation for h so that the book’s height can be determined from the values for t_fall. (In determining this adjustment, use all four decimal places from your answer to Item 7(d).)

Sample answer:

t_fall = t – 0.4133 or t = t_fall + 0.4133. So, substitute this expression in your model for h: h = –15.46(t_fall + 0.4133)² + 12.78(t_fall + 0.4133) + 1.40

Some students may apply quadratic regression to columns 2 and 3. This would result in the simplified model h = –15.46 t_fall ² + 4.04.

Save your model from Item 9(f) for use in Activity 6.

9. Use your model from Item 9(f) to verify your stunt plan in Item 7. Which model is easier to use?

Setting the equation h = –15.46 t_fall ² + 4.04 = 0 and solving for t_fall tells you how long it takes for the book to hit the floor. So, the final plans should be the same, but the computations should be easier using the model from Item 9(f). Using the model from Item 9(f) would make it easier to take the height of the car into consideration.

For example, if the height of the car is 2 in., the book would reach the height of 2 in. above the floor in only 0.36 sec In this time, the car travels only 0.55 ft or about 6.6 in. This means that the mark must be placed 2.35 in. in front of the drop point. (So, the car needs to be partially under the book before you release the book.)

10. If time permits, execute your planned stunt. Comment on your success.

Sample answer: It didn’t work! The book hit the car. Perhaps we didn’t release the book at exactly the right time.





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	Homework 5—Along These Lines

1. Suppose that Jane walks in a straight line and that her motion is recorded by a motion detector. Figure 17 shows her distance from the sensor every second.




Time (sec)	Distance from Sensor (ft)	Velocity (ft/sec)
0	3
1	6
2	9
3	12
4	15
5	18
6	21
7	24
8	27

Figure 17. Distance-time data for Jane’s walk

1. Use the data in Figure 17 to plot a distance-versus-time graph of Jane’s motion.



2. Describe the shape of your graph from part (a).

The points appear to lie on a line.

3. Based on your description in part (b), what should be true of Jane’s velocity?

It should be constant and positive in value.

4. To estimate Jane’s instantaneous velocity at a particular time, use the average velocity for the smallest time interval centered at that time. For example, use the average velocity from t = 0 to t = 2 to approximate Jane’s instantaneous velocity at t = 1. What is this velocity?

3 ft/sec

5. Complete column 3 of Figure 17. (Note: You have already calculated the entry for t = 1 in part (d). This should be your first entry in column 3.) Describe the pattern of your entries in column 3. What does this say about Jane’s motion?




Time (seconds)	Distance from Sensor (feet)	Velocity (feet/second)
0	3	xxxx
1	6	3
2	9	3
3	12	3
4	15	3
5	18	3
6	21	3
7	24	3
8	27	xxxx

Jane’s velocity never changed. It was always 3 ft/sec

6. Find an equation that describes the relationship between Jane’s distance from the motion detector and time.

d = 3t + 3

2. Jane’s best friend, Rhonda, wanted to help. She also walked in a straight line in front of the motion detector. Some of her data, for 1-second intervals, are shown in Figure 18.




Time (sec)	Distance from Sensor (ft)	Velocity (ft/sec)
0	2
1	3.15
2	6.8
3	12.95
4	21.6
5	32.75
6	46.4
7	62.55
8	81.2

Figure 18. Distance-time data for Rhonda’s walk

1. Plot a distance-versus-time graph for Rhonda walk. Be sure to including labels and scales on your graph.






2. Describe the shape of Rhonda’s graph. What does it tell you about her velocity as she walked?

It is a curve, opening upwards and increasing. It looks like it might be part of a parabola. The upward curve means that her velocity is increasing as she walks.

3. Copy Figure 18 and complete the velocity column. Use the same method that you used in Item 1(e).




Time (seconds)	Distance from Sensor (feet)	Velocity (feet/second)
0	2	xxxx
1	3.15	2.4
2	6.8	4.9
3	12.95	7.4
4	21.6	9.9
5	32.75	12.4
6	46.4	14.9
7	62.55	17.4
8	81.2	xxxx




4. Study the entries in column 3. What do the entries tell you about Rhonda’s velocity? Can you detect any pattern in the entries?

Rhonda’s velocity increased as she walked. As you move down the column from one entry to the next, the entries increase by 2.5 units.

5. If you had trouble finding a pattern in your entries in column 3, it might be helpful to examine a graph of velocity versus time for Rhonda’s walk. Make such a graph and comment on it’s shape. What equation describes this graph? How did you determine your equation?



The graph of velocity versus time is linear. The points fall along a straight line.

6. What equation describes your graph in part (e)? How did you determine your equation?

The equation: v = 2.5 t – 0.1. Linear regression was used to determine the equation. (Students could also determine the equation using two points from their completed table.)

7. Find an equation for Rhonda’s distance from the motion detector in terms of time. How did you determine your equation?

d = 1.25t² – 0.1t + 2. Quadratic regression was used to determine this equation.

Save your tables and equations from Items 1 and 2 for use later in the assessment.

3. Keith dropped the book for his group in Activity 4. To model the book’s motion, his group used the equation

h = –15.32t² + 5.53t + 3.13

where h is the height above the floor and t is the elapsed time since the motion detector began recording data.

1. Sketch a graph of the group’s model.



2. Based on this model, how long after the motion detector began recording data did Keith release the book? How did you determine your answer?

The book was released approximately t = 0.18 seconds after the motion detector began recording. This is the t-coordinate of the vertex. You can use the 2nd/CALC/Maximum feature on a TI-83 to get this answer. You can also use the formula t = –b/2a.

3. How high was the book the instant Keith released it?

Keith held at the book 3.63 feet above the floor.

4. Suppose that the motion detector had not been in the way. How long would it have taken from the time the book was released until it hit the floor?

The book hit the floor at approximately t = 0.67 sec The book was in the air from t = 0.18 sec to t = 0.67 sec, a total of 0.49 sec, or about one-half second.

5. How fast was the book moving when it hit the floor?

It was falling at about 15 ft/sec (Note: Its velocity was about –15 ft/sec)

6. The independent time variable, t, in Keith’s model refers to the elapsed time since the motion detector began recording data. Suppose Keith’s group wants to describe the relationship between h and t_fall, where t_fall is the elapsed time since Keith released the book. How do they do this? What is their equation?

Sample answer:

t = t_fall + 0.18; replace t in the equation for (f) with this expression. The result is y = –15.32 t_fall² + 3.63.

Students could also generate a table of values similar to the one done in Item 9(c), Activity 5. Then they could use quadratic regression to fit a quadratic to the h versus t_fall data.





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	Activity 6—Fall Fashions

 = 14, 15

1. Refer to your model from Item 9(f), Activity 5. Your dependent variable, h, is the height of the book above the floor; your independent variable, t_fall, is the elapsed time since the book was released. If you have not already done so, rewrite your equation so that it contains no parentheses, and combine terms. Then round the constants to two decimal places. Write your simplified equation.

Sample answer:

h = –15.46(t_fall + 0.4133)² + 12.78(t_fall + 0.4133) + 1.40

= –15.46t_fall² + 0.000764 t_fall + 4.0411.

Rounding the constants to two decimal places gives h = –15.46 t_fall² + 4.04.

1. Use your model (equation) from Item 1 to complete a copy of the table in Figure 19. You may want to divide the calculations among the members of your group. (Round your final results to two decimal places.)




Time, tfall (sec)	Instantaneous velocity, v (ft/sec)
0
0.1
0.2
0.3
0.4
0.5

Figure 19. Table of velocity-versus-time values
(using translated times)

Sample answer:





Time, tfall (sec)	Instantaneous velocity, v (ft/sec)
0	0
0.1	–3.09
0.2	–6.18
0.3	–9.28
0.4	–12.37
0.5	–15.46




2. Make a scatter plot for the velocity-versus-time data in your completed table. What equation describes the relationship between velocity and time? How did you determine your equation?

Sample answer:



v = –30.92t (rounded least-squares equation).

Acceleration is the rate of change of velocity with respect to time. (This concept is not new to you. If you step on your car’s accelerator (the gas pedal), your car will speed up.) You calculate average acceleration, a_1,2, from time 1 to time 2 by the ratio:



If distance is measured in feet and time in seconds, the units for acceleration are feet per second per second or feet per second² or ft/sec².

3. Use your table from Item 2 to find the following average accelerations.



1. What is the average acceleration from t = 0 to t = 0.1? Be sure to include the units as part of your answer.

Sample answer: –30.9 ft/sec²

2. What is the average acceleration from t = 0.1 to t = 0.2?

Sample answer: –30.9 ft/sec²

3. What is the average acceleration from t = 0.2 to t = 0.3?

Sample answer: –30.9 ft/sec²

4. Your velocity equation in Item 2 is linear. What is its slope? How does the slope of the velocity equation compare to the average accelerations that you computed in parts (a) – (c)? Does this make sense? Explain.

The acceleration appears to be constant. (The answer to Item (b) differed very slightly from the answers to Items (a) and (c).) The value of that constant is the slope of the linear equation for velocity. Since slope is the rate of change of the graphed quantities, the slope in v vs. t is the rate of change of velocity with respect to time, which is acceleration.

5. What does it mean when the acceleration is negative? What about when acceleration is 0?

When acceleration is negative, velocity gets smaller as time elapses. Since velocity is negative in this situation, negative acceleration causes the velocity to become more and more negative as time passes. (That means that the book is dropping faster as time elapses.) When acceleration is 0, velocity is constant.

Your results so far apply only to stunts involving dropping your object from the height that you used. In order to plan actual stunts, you’ll need to generalize your work. To begin this process, you’ll pool your results with those from other groups. Copy the class information onto Handout 4.

4. What patterns, if any, do you see in the information on Handout 4?

The answers below assume that the motion detectors were on the floor and that the books fell toward the motion detector.

Acceleration:

Students should find that the acceleration is constant. The value of the constant may vary from group to group, but most likely will be somewhere between –29 to –32 ft/sec². (The acceleration due to gravity is –32 ft/sec². Air resistance opposes the book’s downward movement and should cause the magnitude of the book’s acceleration to be a little less than 32.)

Students may decide to use the average of the group acceleration values for their approximation of a falling book’s acceleration. If some outliers are present, students may decide to remove them before taking the average.
Velocity Equations:

The velocity equations were of the form v = g t, where g is acceleration.

Distance-versus-time equations:

The height-versus-time equations were of the form h = at² + c, where a is (1/2)g and c is the drop-height. See Items 5 and 6, below.

5. All the height-versus-time models in Handout 4 should be of the form h = at² + c. Interpret the values of a and c in the context of the falling book. (Save your results for use in Item 7(d) of Activity 6.)

c is the height of the book the instant it was released; a is half the acceleration.

6. What’s the connection between acceleration and your model (equation) for velocity? (Save your results for later use.)

v = (acceleration) ´ t; acceleration is its slope.

7. Suppose that you drop a book from a height of 20 feet. Use ideas from Items 4–6 to determine a model for this situation.



1. Write a model describing the height of the book in terms of the elapsed time since the book was released. For what time values does your model make sense? Did you make any assumptions? If so, what were they

Sample answer: Assume that the acceleration of the book is –31.0 ft/sec² The model is h = –15.5t² + 20. This model makes sense from t = 0 to t = 1.1 seconds.

2. How long will it take for the book to hit the ground? Explain how you got your answer.

Sample answer based on answer to (a):

Algebraic solution:

–15.5t² + 20 = 0

–15.5t² = –20

t² = 20/15.5

t » 1.1 or –1.1; only the first solution makes sense in the context of the falling book.

Graphical solution:

The positive t-intercept of the parabola is approximately located at t = 1.1 sec

3. Write a model describing the relationship between velocity and time. According to your model, what is the book’s velocity the instant it hits the ground?

Sample answer based on answer to (a): v = –31.0t.

The book is traveling (–31.0)(1.1) = –34.1 ft/sec when it hits the ground.

	Activity 7—Close Call

Here's a new stunt.

The hero, who is being chased by villains, is driving down the road as fast as his car will go. The villains are gaining on him! Moments before the hero reaches an intersection, a truck parked on the hill loses its breaks and begins rolling down the hill. The truck arrives at the intersection narrowly missing the hero's car. The villains, however, are not as fortunate. They crash into the side of the truck.

In this activity, you'll plan, and time permitting, stage a scaled-down version of this stunt. You'll need at least one non-motorized car or truck and two battery-operated vehicles.

Earlier in this unit, you discovered that a battery-operated toy car moved approximately at a constant velocity. This meant that, once the car got up to speed, the car's acceleration was approximately zero. Later, you dropped the book and recorded its distance from the motion detector versus time. You discovered that a quadratic equation did a good job in describing your data. The book did not fall at a constant velocity, instead the force of gravity caused the book to accelerate during its fall.

1. Imagine a truck parked on a hill. The breaks fail and the truck starts to roll. Make a sketch of how you think the truck's distance-versus-time graph should look. What does your graph imply about the truck's acceleration? What does your graph imply about the truck's velocity during its roll down the hill?



2. Unbalanced forces cause acceleration and acceleration produces change in velocity. Figure 20 shows time-lapse graphs of two cars driving down a straight highway in the direction of the arrows.






Figure 20. Time lapse graphs of two cars

1. Which car's acceleration is positive? Which car's is negative?

The acceleration for Car #1 is positive and for Car #2 is negative.

2. Which car is speeding up? Which car is slowing down?

Car #1 is speeding up while Car #2 is slowing down.

3. Acceleration is caused by an unbalanced force. On which car is an unbalanced force acting in the direction opposite the car's motion? On which car is an unbalanced force acting in the direction of the car's motion?

The unbalanced force is acting in the direction of Car #1's motion and acting in the direction opposite to Car #2's motion.



1. Set up an inclined plane using a piece of plywood braced up by books or boxes on one end. Experiment with the height of the brace to find the lowest height for which a toy vehicle will roll down the plane when the car is released from rest and the plane is "thumped" lightly. What is the height of the inclined plane? What is its length? How is the horizontal distance directly under the plane related to the plane's height and length?

Student answers will vary.

Sample answer (using a relatively short inclined plane):

3.5 inches ( 0.29 ft. The plane was 48 inches or 4 feet long. The horizontal distance under the plane could be measured or determined using the Phytrhagorean Theorem:

(height)² + (horizontal distance)² = (plane length)². So, in this case, the horizontal distance was approximately 3.99 ft.

2. Set up the plane so that it is considerably higher than in part (a). What is the height of your plane? Next, attach a motion detector at one end of the plane so that it points in a direction parallel to the plane. Record the motion of a car as it rolls down the plane.

Sample answer: 0.92 ft.

3. Use a program such as EDITPART to select only the portion of data that relates to your vehicle's motion down the inclined plane. (Omit any data corresponding to horizontal portions of the distance-versus-time graph.) Describe the shape of your data. Then use the regression capabilities on your calculator to fit an equation that is adequate to describe the pattern of your data. (Make sure to check that the dots in a plot of the residuals versus the times appears to be randomly scattered.) What is your equation?

Sample answer: The data almost looks like a line. However, there is a slight curve to its shape. The residual plot based on a linear model appeared strongly curived. So, a linear model is not adequate to describe the shape of our data. However, the quadratic equation y = 2.56t² + 1.817t + 4.9 appeared to fit our data reasonably well. The dots in a residual plot appeared to be randomely scattered.

4. Based on your data, or your equation in c), what is your car's acceleration as it rolls down the inclined plane?

The accleration is twice the coefficient of t². We confirmed this by using our data to estimate the instantaneous velocity at various times and then used those velocities to estimate the instantaneous acceleration. Using this method, we found that the instantaneous acceleration was almost constant and around 5.12 ft/sec².

Newton's second law of motion gives the relationship between the net force acting on an object in the direction of the acceleration, the object's mass, and its acceleration. This relationship can be expressed in equation form as:

F + ma,

where F is the force in the same direction as the acceleration,
      m is the mass, and
      a is the acceleration.

When your toy vehicle is placed on the inclined plane, gravity pulls on the vehicle in the direction perpendicular to the ground. This force can be decomposed into component forces acting in two directions, one perpendicular to the inclined plane (which is balanced by the plane pushing on the car) and the other parallel to the inclined plane. (See Figure 21.)



Figure 21. Decomposition of forces acting on toy vehicle

The force that causes the car to move down the plane is F_m. (Assume that there is no friction as the car rolls down the ramp.) In the metric system force is measured in Newtons.

1. In Item 3, what is the force, in Newton's acting on your vehicle in the direction parallel to the inclined plane? You will need to use a balance scale to find your vehicle's mass. (You may also need to convert English units of ft/sec² into metric units, m/sec²: use the conversion 1 foot/sec² = 0.3048 meter/sec².

Sample answer: The toy vehicle's mass was 285 gm or 0.385 Kg. The vehicle's acceleration was approxiamtely 5.12 ft/sec² or about 1.56 m/sec². That means that the force that caused the vehicle to accelerate down the plane was (0.385 Kg)(1.56 m/sec²) » 0.60 Newton's.

2. If you lowered the inclined plane, would the force on the car acting in the direction parallel to the incline plane be larger or smaller than your answer to part(a)? Explain.

If your lower the inclined plane, F₈ would be the same. However, when you draw a diagram similar to the one in Figure 2, F_p will be longer and F_m will be shorter. This means that the force that causes the car to accelerate down the ramp will be smaller. Since the mass of the car has not changed, the smaller force will result in a smaller acceleration.

3. Lower the inclined plane. What is its height? Again roll your toy vehicle down the plane and record its motion with a motion detector. What equation models its distance from the detector versus time? Based on your model or your data approximately what is the acceleration of your vehicle?

This time the ramp was 0.656 ft high. The equation modeling the vehicle's distance over time was y = 1.57t² + 2.64t + 0.75. The vehicle's acceleration was approximately 3.14 ft/sec².

4. Use the acceleration determined in part c) to estimate the force acting on the vehicle in the direction parallel to the inclined plane. Is this force greater or smaller than the force that you determined in part b)?

Sample answer: F lk = (0.385 Kg)0.957 m/sec²) » 0.37 newtons. The force is smaller.

5. Experiment with several more heights until you can find a relationship between the height of the plane and your vehicle's acceleration. What is your relationship? Then test your relationship using a different ramp height.

Sample answer:
(Answers will vary depending on the mass of the vehicle and the amount of friction between the vehicle's tires and the ramp.)

We found that the linear relationship a = 8.16h – 2.35 (height in feet, and acceleration in ft/sec²) appeared to describe the relationship between ramp height and acceleration reasonably well. We tested this relationship by setting the ramp to a height of 0.84 ft and determined that the car's acceleration would be around 4.5 ft/sec². In the trial run, the car's acceleration turned out to be 4.36 ft/sec².

6. Plan a simplified version of the stunt described above. Mark with masking tape a road that passes by the inclined plane. The hero's car will drive along this road. Decide on the height of your hill (the inclined plane). Make a mark on the masking-tape roadway that will signal the person holding the truck on the ramp that it is time to release the truck. Your job is to work out the mathematics for this stunt so that the car passes the trucks path just before the truck reaches the masking tape roadway.

Student answers will vary depending on the battery-operated car, the toy truck used on the inclined plane, the length and height of the plane, etc. First, students should determine the truck's acceleration. Then they can use the formula d = 0.5 at² to determine the time it will take for the truck to reach the bottom of the ramp. Students may need to check the car's velocity. Then they can determine the position of the mark signaling when to release the truck.

7. Time permitting, test your stunt design from Item 6.

	Assessment—Hurry Up and Slow Down!

1. Look back at your data tables from Figures 17 and 18 in Homework 5. Use those data to complete the following analysis.



1. Add a fourth column, for acceleration to each table. Use the average velocity for the smallest interval centered at a particular time to approximate the acceleration at that time. For example, use the average acceleration from t = 0 to t = 2 to approximate the instantaneous acceleration at t = 1.

Notice that you must have distance data before and after the time for which you are calculating velocity. We cannot calculate instantaneous velocity for t = 0 or t = 8. The first instantaneous velocity is known at t =1, and we cannot calculate instantaneous acceleration until t = 2.

For Jane:

Time (sec)	Distance from Sensor (ft)	Velocity (ft/sec)	Acceleration (ft/sec2)
0	3	xxxx	xxxx
1	6	3	xxxx
2	9	3	0
3	12	3	0
4	15	3	0
5	18	3	0
6	21	3	0
7	24	3	xxxx
8	27	xxxx	xxxx





For Rhonda:

Time (sec)	Distance from Sensor (ft)	Velocity (ft/sec)	Acceleration (ft/sec2)
0	2	xxxx	xxxx
1	3.15	2.4	xxxx
2	6.8	4.9	2.5
3	12.95	7.4	2.5
4	21.6	9.9	2.5
5	32.75	12.4	2.5
6	46.4	14.9	2.5
7	62.55	17.4	xxxx
8	81.2	xxxx	xxxx




2. Based on Jane’s velocity, was she walking toward or away from the motion detector. How can you tell?

She was walking away from the detector. You can tell because her velocity was positive.

3. After Jane started walking, what was her acceleration? What does her acceleration tell you about her velocity as she walked?

Her acceleration was 0 ft/sec². Her velocity never changed. She walked at a constant rate.

4. What was Rhonda’s acceleration? What does her acceleration tell you about how she walked? What does the sign of her acceleration tell you about her velocity?

Her acceleration was 2.5 ft/sec². Her velocity changed during her walk. Her velocity increased becoming more and more positive as she walked.

2. Imagine that you are driving down the highway at 65 mph.



1. Is your acceleration positive, negative, or zero? Explain.

Your velocity is constant so your acceleration is 0 miles/hour².

2. A slow truck is ahead of you and you decide to pass. So you step on the gas pedal. Is your acceleration positive, negative, or zero? Explain.

Your acceleration is positive. Your velocity increases.

3. After you pass the truck, you let your velocity return to the 65 mph speed limit. What can you say about your acceleration as you return to 65 mph? What effect does your acceleration have on your velocity?

Your acceleration is negative. Your velocity decreases.

3. Roland is driving on the Interstate at 55 mph. He comes up behind a slow bus and decides to pass. He steps on the gas pedal and goes around the bus. Once around the bus, he slows back to 55 mph. Figure 1 shows a graph of his velocity versus time.



Figure 1. Graph of velocity versus time

1. Describe Roland’s acceleration during this trip. When is it positive? negative? zero?

His acceleration is zero when he is driving at a steady 55 mph. When he stepped on the gas pedal to pass the bus, he increased his velocity at a steady rate. That means that his acceleration was positive. When he slowed down, he decreased his velocity, and his acceleration was negative.

2. Sketch a graph of his acceleration versus time.

4. Irene walked in front of a motion detector. Her distance-versus-time graph looked like a parabola. Her group found that the equation

d = –0.555t² + 7t + 2

described the pattern of her data fairly well. (Distances were measured in feet and time was measured in seconds.)

1. Based this equation, decide whether Irene’s acceleration was a constant. If you think it was, predict its value (just from the equation). If you think it was not, explain why.

Sample answer: It was a constant, with value of –1.11 ft/sec².

2. Complete a copy of the table in Figure 2 using the equation that Irene’s group found. What method did you use to approximate the velocity and acceleration entries?




Time (sec)	Distance from sensor (ft)	Velocity at time t (ft/sec)	Acceleration (ft/sec2)
0		xxxx	xxxx
1			xxxx
2
3
4
5			xxxx
6		xxxx	xxxx

Figure 2. Data from Irene’s walk.





Time (sec)	Distance from sensor (ft)	Velocity at time t (ft/sec)	Acceleration (ft/sec2)
0	2	xxxx	xxxx
1	8.445	5.89	xxxx
2	13.78	4.78	–1.11
3	18.005	3.67	–1.11
4	21.12	2.56	–1.11
5	23.125	1.45	xxxx
6	24.02	xxxx	xxxx





Students can use a small-interval method or a zoom in approach to calculate the velocity and acceleration entries.

3. What does the sign of Irene’s velocity tell you about how she walked?

It is positive, so she walked away from the detector.

4. What was Irene’s acceleration? What does the sign of her acceleration tell you about how she walked?

–1.11 ft/sec². Because her acceleration was not zero, her velocity changed as she walked. Because the acceleration is negative, her velocity became less and less positive. That means she was walking away less rapidly. In other words, she was slowing down as she neared the six-second mark.

5. Extend the table to 10 seconds. Then, if necessary, revise your answer to part (e) based on this additional information.




Time (sec)	Distance from sensor (ft)	Velocity at time t (ft/sec)	Acceleration (ft/sec2)
5	23.125	1.45	–1.11
6	24.02	0.34	–1.11
7	23.805	–0.77	–1.11
8	22.48	–1.88	–1.11
9	20.045	–2.99	xxxx
10	16.5	xxxx	xxxx

After 6 seconds, the velocity becomes negative and increases in magnitude. This means that she is now moving toward the motion detector. Because of her negative acceleration, her velocity continues to become less and less positive, but she does not continue to "slow down." Instead she picks up her pace as she walks toward the motion detector.

5. After collecting data from the book-drop experiment in Activity 4, Keith’s group found that

d = –15.46t² + 3.93

did a good job in describing its motion. (Recall that d was the distance (ft) from the motion detector and t was the elapsed time (sec) since the book was released.)

1. Sketch a graph of this model in a window that makes sense in the context of the falling book.



2. Your graph in (a) records the relationship between the distance of the book from the motion detector and time. It does not record the path of the book. Sketch a time-lapse graph that describes the motion of the book along its path. Identify points on the path at 0.10-second intervals. Assume that Keith is standing at the zero x-location and that he holds the book one-foot away from his body before he drops it.



3. In Animation, you used parametric equations to describe the motion of objects. In order to write a parametric equation for your graph in (b), you would need an equation for x that specifies the x-coordinate of the book’s location at time t. In addition, you would need an equation for y that specifies the y-coordinate of the book’s location at time t. Write a set of parametric equations that describes your graph in (b).

x = 1, y = –15.46t² + 3.93

4. Change the mode settings on your calculator from function to parametric and change from connected to dot format. (For example, if you are using a TI-83, you would press MODE and change the settings from Func to Par and from Connected to Dot.) Graph your parametric equations from (c). Does your calculator-produced graph resemble the one that you drew for (b)?

Yes, the calculator graph resembles the hand-drawn graph in (b).



5. How can you tell from the time-lapse graph that the book’s velocity is changing?

The distance between dots for equally spaced time increments get larger as the book drops.

Unit Project—Look Before You Leap

You’ve probably seen movies in which the hero, trapped on a rooftop, jumps off and lands in the back of a pickup truck driving parallel to the building. (The truck is usually carrying something soft such as mattresses or squishy garbage.)

In this project you will design this stunt. In your plans you’ll need to decide the building’s height, the location of the mark used to signal the hero that it’s time to jump, the pickup’s speed when it reaches this mark, and the relevant dimensions of the pickup.

1. As part of your plan, answer the following questions. (Be as realistic as possible.)



1. How tall is the building?

Sample answer: 30 ft

2. What are the length and height of the truck’s cab and bed:

Sample answer: The back of the pickup is 10 ft long and 4.5 ft high (with the mattresses). The cab of the pickup is 5 ft long and 7 ft high.

3. How fast should the truck be going when it reaches the signal mark? Convert your answer to feet per second.

Sample answer: When the pickup reaches the signal mark it should be traveling a constant 25 mph or approximately 36.7 ft/sec.

2. Develop a model that describes the motion of the hero during his fall from the building. State any assumptions that you make.

Sample answer: Assume that the hero’s acceleration is approximately –31 ft/sec (This allows for a slight effect due to air resistance.) The model describing the hero’s motion is h = –15.5t² + 30.

3. How long will it take the hero to reach the height of the cab? How long will it take him to reach the height of the back of the truck? How long will it take him to hit the ground?

Sample answer:

The hero will reach a height of 7 ft in about 1.2 seconds. It will take the hero approximately 1.3 seconds to reach a height of 4.5 ft He will hit the ground at around 1.4 seconds.

4. Next, you’ll need to determine where to place a mark on the road. The hero should begin his fall the instant the front of the truck reaches this mark.



1. Where should you place this mark so that the hero will land safely in the back of the truck? Remember that you don’t want him hitting the cab of the truck. (Specify the distance between the mark and the hero’s drop point.) Explain how you determined your answer.

At first we decided to place the mark 39.7 ft in front of the drop point. If the hero jumps the instant the truck reaches this mark, then the front of the truck will have moved (1.3 sec)(36.7 ft/sec) » 47.7 feet beyond the mark. That means that the hero should land 8 feet back from the front of the truck.

However, we didn’t check to see if the hero would clear the top of the cab. The hero will reach a height of 7 ft at about 1.2 seconds. After 1.2 seconds, the front of the truck will be approximately 44.0 ft beyond the mark which means the hero will hit the top of the truck (he’ll hit approximately 4.3 ft from the front of the truck which is still the cab section). So, we had to change the location of the mark in the road.

Instead, place the mark one foot closer to the drop point—at 38.7 ft. Then at 1.2 seconds the front of the truck will be 5.3 feet beyond the drop point. So, the hero will not hit the cab’s roof. At 1.3 seconds the front of the truck will be approximately 8 feet beyond the drop point. (Note, however, that the one-foot change in "mark" translates into a change of only 0.027 seconds in the hero’s jump time. That’s much smaller than his reaction time. He may have a problem "hitting the mark" this precisely. The hero has about a 5 ft "safety margin," which means the jump must take place within a 0.13-second interval to be successful. If he misses that window of opportunity, the stunt will fail!)

2. Make a sketch indicating where the hero will land in the back of the pickup.

5. How fast will the hero be falling the instant he lands in the back of the truck? Explain how you determined your answer.

Sample answer:

When t = 1.3, the hero lands in the back of the truck. If you magnify a small section of graph surrounding the point (1.3, 4.5), the resulting graph will look like a line. Select two points on this line and calculate the slope. Here are the approximate coordinates for two points: (1.29, 4.273014) and (1.31, 3.469094). Using these points, the approximate velocity is –40.2 ft/sec.

6. Convert your answer to Item 5 to miles per hour. Do you think the hero is likely to "feel" the impact?

Convert to miles per hour by multiplying by (3600 sec/hr.)/(5280 ft/mi.). The hero’s velocity is approximately –27.4 mph. He will definitely "feel" the impact of his landing. He could get hurt unless there is some sort of padding in the bottom of the truck bed. Remember, too, that the truck is moving at 25 mph relative to him (horizontally), so that he will likely slide into the rear gate at close to 25 mph.

7. How sensitive is the success of this stunt to the hero’s timing? Investigate how close to "on time" he must jump in order safely land in the back of the pickup.

Sample answer: There is almost no room for error. As noted above, he will clear the cab section by only 0.3 ft (about 3.5 inches), and he has only 6 feet to spare at the rear of the bed. That sounds like a lot of space, but at 36.7 ft/sec, that 6 feet represents a margin of error of only 0.16 seconds! At this speed our hero is possibly doomed! A slower truck would help.

Mathematical Summary

In this unit you have determined equations describing various motions. In several situations, you developed equations based on time-distance data collected by motion detectors. In this process, you discovered that linear distance-versus-time graphs are produced by moving at a constant rate, and curved graphs are produced by moving at a non-constant rate.

The average velocity from time 1 to time 2 may be calculated using the ratio (distance 2 – distance 1)/(time 2 – time 1).

Similarly, after determining velocities at time 1 and time 2, you can determine the average acceleration by calculating the ratio (velocity 2 = velocity 1)/(time 2 – time 1).

The simplest motion studied was that of a walker (or car) moving in a straight line at a constant rate. This type of motion can be modeled by distance-versus-time equations of the form

d = vt + d₀

where d is the distance
t is the elapsed time.
The slope, v, is the object’s velocity and d0 gives the object’s initial distance (its distance when t = 0).

The motion of a dropped object is somewhat more complex to analyze. In this case, the object’s velocity is not constant. Such motion is modeled by quadratic equations. In general, motion described by quadratic equations is characterized by parabolic distance-versus-time graphs, linear velocity-versus-time relationships, and constant acceleration.

The instantaneous velocity at any time may be thought of as the "slope" of the distance versus time graph. You can approximate this slope by "zooming in" on the point of interest until the graph looks like a line. Then select two points on this "line" and determine slope in the usual way.

The story of a falling object can be told by its quadratic model:

h = at² + h₀

where h is the height and t is the elapsed time since the object was released.

The value of a is half the acceleration and h₀ is the initial height.

Key Concepts

Acceleration: The rate of change of velocity with respect to time. Note that force is the cause of acceleration. Typical forces are due to gravity, friction, or the push of a hand.

Average acceleration: Average acceleration from time 1 to time 2:

Average velocity: Average velocity from time 1 to time 2:

Instantaneous velocity at time t₀: The rate of change of location with respect to time at the instant t = t₀. You can approximate the instantaneous velocity by tracing to the point on the distance-versus-time graph corresponding to t₀, zooming in on this point until the graph resembles a line, selecting two points on this "line," and using these points to compute the slope. This "slope" is your approximation of the instantaneous velocity at t₀.

Newton's second law: Net force = mass × accleration.

Time-lapse graph: A graph of the path of a moving object that includes sample times displayed on the graph to show when the object reaches a particular location on the graph.

Solution to Short Modeling Practice

Solution to Short Modeling Problem

Part 1: Finding the Scale Factor

The scale factor is found by physically measuring, in mm, the height of the fountain wall in the photo, and forming the ration of the real height, if feet, as written on the photo.

We measured the height to be 9.0mm. This may change due to printing or copying scale changes, but the whole photo will change as well. Using our data the scale factor is:

Part 2: Collecting Data from the Photograph

Time

The slug just leaving the nozzle is at zero height at zero time. Now consider the next slug up. The time of flight is 0.2 seconds and the one above it has been moving for 0.4 seconds.

Height

Our measurement on the photo shows the distance from the nozzle to the second slug to be 40.0mm. The real world height is found by multiplying by the scale factor:

Similar scaling must be done for each slug.

The students should get approximately these values. There will be differences caused by measurement technique and rounding.

Part 3: Test the Quadratic Model

The students should graph the time and distance data, either on paper or by graphing calculator, and determine the coordinates of the vertex to be very close to

(1.6 sec,40.8 ft).

The coefficient of t², a, will always be × of the acceleration of the object. This is discussed in the unit model. In this case the acceleration of the water slug is –32 ft/sec², and a = –16 ft/sec². Using the known values for a, h, and k the vertex form of the equations is

y = –16(t – 1.6)² + 40.8

The graph of this equation does match the data very well. This is evidence that Edward is on the right track.

Part 4: Converting to a Second Model

Simplifying the equation above gives:

y = –16t² + 51.2t – 0.16

Note: The data on the fountain was derived from the equation

y = –16t² + 50.0t

The student’s results should match the coefficient of t², between 48 and 52 for the coefficient of t, and the constant term should be between –0.2 and +0.2. The process for finding the velocity Edward wants is to use the model developed above and change the value of the coefficient of t. This quantity is the initial velocity of the water slug when it leaves the nozzle. A velocity of –51ft makes the water rise to 40 feet before it starts down. We want it to rise to 25 feet.

Part 5: Solving Edward’s Problem

Students should choose a smaller value for velocity, and graph the result manually or with technology. If the highest point is not 25 feet they should make another educated guess and graph the result. They should find that an initial velocity of 39.8 and 40.2 ft/sec will have the water reaching 25 feet.

Solutions to Practice and Review Problems

Exercise 1

1. The weight of 2 gallons of water is about 17 lb.
   The weight of 6 gallons of water is about 50 lb.



2. The slope of the graphed line is = 8.3 lb/gal.

The intercept of the graphed line is 0.

3. The equation of the graphed line is Weight = 8.3 lb/gal (Volume) + 0.

Exercise 2

The students’ answers may vary, but should be close to those suggested here. You can use your own judgment as to the validity of the responses to this question.

1. 100 mi per inch
2. 100,000 persons per inch
3. 5 sec per cm, or 10 sec per 2 cm
4. 0.5 m per cm, or 1 m per 2 cm
5. 4 C° per cm, or 5 C° per 1/2 in. (tricky second choice, here)

Exercise 3

= $5.25 per hour

The units of the slope are dollars per hour.

The slope represents the hourly wage of the laborer.

Exercise 4

1. The lines have nearly the same slope–that is, it appears that the increase in consumer spending for durables kept pace with the spending for nondurables.
2. The two lines have nearly the same y-intercept, that is, at the start of the period, consumer spending for services and for nondurables was nearly the same.

Exercise 5

1. The slope of the graphed line is rise¤run.

= 0.002 lb per week/Cal per day

2. The equation of the graphed line is

Weekly weight loss = 0.002 lb per week/Cal per day ´ Daily exercise calories

3. On the graph, the weekly weight loss with 350-calorie per day exercise is 0.7 lb¤ wk.
4. If the woman decreases her dietary calories from 2400 calories to 2150 calories the reduction in calories is 250 calories per day or 1750 calories per week. Body fat is 3500 calories per pound so the calorie reduction is equivalent to 1750 calories per week ¸ 3500 calories per pound = 0.5 lb per week. The reduction in calories gives the graphed line a nonzero y-intercept of 0.5 lb per week. The new equation for the graphed line is

Weekly weight loss = 0.002 lb per week/Cal per day ´
Daily exercise calories + 0.5 lb/wk.

Exercise 6

1. 42.5 ÷ 60 = 0.708 gal/sec

V = 0.708T

2. For T = 20

V = (0.708)(20) = 14.2 gal

3. T =

For V = 5

T = or about 7.06 sec

4. V = (0.708)(40) = 23.32

The radiator is not clogged.

Exercise 7

1. The curve definitely appears nonlinear.
2. The students must estimate values from the graph. Therefore, answers between 118 and 119 lb¤ cu ft for the M.D.D. and between 16 and 17% for the O.M. should be considered acceptable.

Exercise 8

1. The graph, since the line is curved, illustrates a nonlinear trend.
2. The shape of the graph most resembles a y = x² relationship.
3. The students’ answers may vary slightly, depending on the reading of the graph. The slope of the small section from 1962 to 1969 will serve to provide an average slope for the 1960s.

Slope =

Slope = $0.39 per year (rounded)

For the 1970s, the span of 1972 through 1978 will suffice.

Slope =

Slope = $0.79 per year (rounded)

No, the values of the slope for the two periods are NOT the same. In fact, the slope for the ‘70s is almost twice that for the ‘60s. This means that the automotive workers’ pay during the ‘70s was increasing almost twice as fast as it was during the ‘60s. (If the graph were linear, you would expect to see the same slope during both periods.)

Exercise 9

2. Yes, y = x²
3. A little more than 45 mi/hr

Exercise 10

1. The graph of the dc motor performance is nonlinear and the graph of the ac motor is linear.
2. As the torque demanded of the dc motor grows smaller, the speed of the motor is able to increase.
3. For the ac motor, the speed remains relatively constant, increasing only slightly as the torque diminishes.
4. Only the dc-motor graph appears nonlinear, and it most closely resembles the y = 1¤ x relationship—the inverse relationship, choice 3.

Exercise 11

1. The graph is very close to being linear in the range from 0 K to 20 K, and again in the range from 30 K to 80 K.
2. The slope is the ratio of the change in voltage to the change in temperature. The students’ answers may vary slightly, depending on how they read the values from the graph. For the first range, from 0 K to 20 K,

Slope =

Slope = –25 mV/K

For the second range,

Slope =

Slope = –2 mV/K

3. Since the slope is so much greater (over 10 times as great) in the low-temperature region, the device is very sensitive to temperature changes in this region. A temperature change of only 1 K  would result in a 25-mV change in output—a voltage change easily detected.

Exercise 12

1. In standard form, with h = 10,000 feet, we have the following.

D 2 + 520D = h

D 2 + 520D = 10,000

D 2 + 520D - 10,000 = 0

We can solve this equation using the quadratic formula, where a = 1, b = 520, and c = –10,000.

D =

D =

D = - 260 ±  1/2

D = 20°F or 500°C

It may take some sense of reality here for the students to be able to distinguish the meaningful value. It is of course 20°F. The temperature of boiling water will be 20°F less than 212°F at an altitude of 10,000 feet.

2. Answers will vary depending on your altitude. You should be ready to supply the students with your altitude (try your local weather station), or have a reference book available for them to look it up. You might want to consider assigning a different city to each student. You could then have them report their results to the rest of the class.

Exercise 13

1. The area of the rectangles is the length times the width. For the horizontal rectangle, the area A_H is 10 cm ´ x. The vertical rectangle is a bit trickier. Its length is 16 cm - x, and its width is x. Thus the area of the vertical rectangle A_V is (16 cm - x) ´ x. Thus,

A_H = 10x and A_V = (16 - x)^x

2. The total area A = A_H + A_V.

A = 10x + (16 - x)^x

A = 10x + 16x - x²

And if A = 60 cm², then substitute, simplify, and put into standard form.

60 = 26x - x²

x² - 26x + 60 = 0

3. We’ll solve this quadratic using the quadratic formula, where a = 1, b = –26, and c = 60.

h =

h =

h = 13 ±  1/2

h = 2.56 cm or 23.44 cm

The first solution here is the correct one. The angle-iron stock should be about 2.5 cm thick for the proper cross-sectional area. Obviously a thickness of 23 cm should be discarded, since it would be greater than the whole angle iron itself!

Exercise 14

1. Cross multiply to obtain:

(Top left) ´ (Bottom right) = (Bottom left) ´ (Top right)

(x) ´ (x) = (50) ´ (x + 50)

x² = 50x + 2500

x² - 50x - 2500 = 0

2. The values for x that satisfy the above equation can be found using the quadratic formula, where a = 1, b = –50, and c = –2500.

x =

x =

x = 25 ±  1/2

x = 80.9 watts or - 30.9 watts

Since a negative wattage makes no sense, the correct size for the second filament would be 80.9 watts.

3. Thus, the first filament is 50 watts, the second is 80.9 watts, and the "third" is 50 + 80.9, or 130.9 watts. The ratio of second to first is 80.9/50, or about 1.62. The ratio of the third to the second is 130.9/80.9, or about 1.62. Yes, they are equal, as we desired.

By analogy with the equations and solution from above, we can see that the solution for x would be as follows.

x =

x = 20 ± 1/2

x = 64.7 watts or –24.7 watts

Again, since a negative wattage makes no sense, the correct size for the second filament would be 64.7 watts.

Exercise 15

1. 

Substituting the slope and the coordinates of point A into the Point-Slope Formula.





See the graph for part (a).

2. 

x = 18,916 feet

See the illustration for (a).

3. 
4.