unit 1: motion - sample

INQUIRY PHYSICSA Modified Learning Cycle Curriculum

by Granger Meador

SAMPLE of Unit 1: Motion

This online sample includes the Teacher’s Guide,Student Papers, and Sample Notes for Unit 1 of thecurriculum.

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INQUIRY PHYSICS inquiryphysics.orgA Modified Learning Cycle Curriculumby Granger Meador, ©2010

Unit 1: Motion

Teacher’s Guide

these TEACHER’S GUIDES are copyrighted and all rights are reservedso you may NOT distribute them or modified versions of them to others

However, the STUDENT PAPERS, SAMPLE NOTES, and any PRESENTATIONS for each unithave a creative commons attribution non-commercial share-alike license; you mayfreely duplicate, modify, and distribute them for non-commercial purposes if you giveattribution to Granger Meador and reference http://inquiryphysics.org

http://www.design-simulation.com/

1 Motion Teacher's GuideInquiry Physics

Key Concepts

Speed is the measurable rate of change in the position of an object. Acceleration is the measureable rate

of change in speed. Both graphical and numerical representations of position, speed, and acceleration

can be utilized to describe and predict movement.

Student Papers

Lab: Galilean Ramp (analyze motion of ball/cart down a track)

W orksheet A: Calculating Motion (initial motion problems and a graph)

W orksheet B: Interpreting Motion Graphs

W orksheet C: Combining the Variables of Motion (formulating additional equations)

W orksheet D: 1-Dimensional Motion Problems

W orksheet E: Quiz Review

Introduction

Students are aware that such ideas as speed, velocity, and acceleration exist, but they are often unaware

of how to distinguish one of those ideas from another. Since the students are not proficient with the

concept of vectors, this investigation only uses speed. Do NOT feel compelled to introduce the velocity

concept here; that will come later in unit two. Tell the students that the symbol v will be used for speed to

avoid confusion later on.

Three key equations arise from this investigation:

Average speed is the change of distance with respect to time, or

Acceleration is the change of speed with respect to time, or

If acceleration is constant, average speed is also given by

INQUIRY PHYSICS TEACHER 'S GUIDE FOR UNIT 1: MOTION PAGE 2 OF 18

UNIT 1: MOTION Lab: Galilean Ramp, pages 1-3

Pre-Lab

Preface this lab with definitions, examples, and discussion of accuracy, precision (or tolerance, depending

on your text or personal usage), and parallax. Also instruct them on the proper use of significant figures,

and refresh them on the basic metric system (SI) prefixes. Also refresh your students on the basic

equations and graphical forms of linear, parabolic, and hyperbolic functions. This will help them interpret

their graphed data and develop the concepts regarding d vs. t, v vs. t, and a vs. t graphs.

Exploration

Equipment for each group (of 3 to 4 students):

grooved wooden track OR air track OR other dynamics track with ball or cart, about 1.5 to 2 m long

if using a wooden track, a 7 mm groove should run lengthwise along the track for a steel ball

(e.g. large ball bearing) to roll along; a wooden block or other stop will be needed at the bottom of

the track

ring stand OR blocks to incline the track

ball for wooden track (diameter $ 2 cm and mass $ 60 g) OR air track glider OR wheeled toy/cart

meter stick - show the students how to read the meter stick to as many decimal places

(precision/tolerance) as possible; typically they'll be reading 3 to 4 significant figures

stopwatch

masking tape (unless track is ruled)

PART ONE:

Only pass out the first two pages of the lab to begin with; the third page is designed to be pre-loaded into a

printer for printing of a distance vs. time graph (or students can put a hand-plotted graph in that space).

The fourth page is similarly designed for a speed vs. time graph, and will not be used until a day or two

after the lab begins. Don’t worry if your students can’t use computers or calculators - later in this guide

you’ll be shown how to handle a parabolic relationship without such equipment.

For the data collection, each group should set up the equipment in exactly the same way so that the class

data can be compiled, compared, and used to invent the concepts. Each student in a group should have

an assigned role (e.g. ball/cart release, timing, recorder, and ball/cart stop and return). For this first lab,

stress that students should check each other on their measurements.

Before the students collect data, discuss with them the various experimental errors to be avoided or

minimized, such as: which part of the ball or cart is to be held at the mark on the tape (the front, not the

middle or back), how should the release and timing be coordinated (verbal cue from timer), how the

ball/cart should be released (by swiping a pencil held in front of it straight down the track away from the

ball/cart, not pulling it upward or sideways to avoid backroll and spin), and why the longest run is

measured first (because it is the one least affected by timing errors; the shorter runs where timing errors

are more significant are saved for the end when the experimenters are more practiced).

PART TW O:

The labs are written so students could, if desired, use the Graphical Analysis program from Vernier

Software (www.vernier.com) to input and graph their data. Space has been left blank on lab pages 3 and

4 so that they can be pre-loaded into a printer. Another option is for students to analyze the motion using

Microsoft Excel or another spreadsheet, or to graph the data by hand. I do NOT advocate using

probeware to analyze the motion, as this can easily short-circuit the development of the concepts. Delay

using probeware until after unit two, when the vector concepts are in place; one can then have students

predict and analyze motion graphs collected as they move in front of a motion detector.

If using a calculator or computer to graph, first show the students how to use the machinery. Have them

make their graphs, get them approved, print them, and answer the questions on lab pages 2 and 3. In the

approval process, check that they have plotted (0,0); if they haven't, engage them in a discussion to

illustrate its validity.



What if my students do not have access to a computer or calculator for graphing?

Many of the graphs in this curriculum are linear, so students can simply plot the points by hand and

eyeball a best-fit line. Careful plotting and drawing of the best-fit line will yield equations with useful slope

values. However, in this first lab they face a squared relationship, and hand-drawn parabolas are seldom

accurate.

If your students need to construct graphs by hand (or if the graphing software cannot handle quadratic

fits), that need not prevent them from decisively determining the numerical relationship between distance

and time. They’ll just need to draw two different graphs - one to determine the type of relationship and the

second to find the numerical values in it. You will probably need to lead them through the process with

sample data or a class average.

1. Plot the data and note the shape of the best-fit curve and its general meaning.

First have the students graph distance vs. time (time always goes on the x-axis, even when it is

the dependent variable). Have them eyeball a best-fit curve to the data points, which should yield

a reasonable half-parabola. You will then need to ask them to identify the mathematical meaning

of that shape, or lead them to realize that it implies that distance is directly proportional to the

square of time.

2. Re-compute for the discovered relationship and re-plot accordingly.

The next step is to test for that squared relationship. The students will now compute the square of

the time for each part of the experiment. Then have them graph distance on the y-axis and time

squared on the x-axis. If their data is good, the points should plot out in a roughly straight

diagonal line.

3. A squared relationship will yield a straight line on the second graph.

Ask the students what the line means. It means that distance is directly proportional to the square

of time. Have them eyeball a best-fit line for the second graph. (Here it is useful to remind them

that a best-fit line need not hit even a single plotted data point nor go through the origin. A best-fit

line runs through the “middle” of the data point distribution.)

4. Using the slope of the best-fit line to obtain the relationship.

Once the students have drawn the best-fit straight line, have them compute its slope. The

resulting equation will be d = k t + b where d is the distance, k is the slope, t is the square of the2 2

time, and b is the graph’s y-intercept. Voila! They now have about the same equation a fancy

computer or calculator would have calculated when forming a best-fit parabola to the original

distance vs. time graph (the only difference is that the t term in the quadratic automatically has a

coefficient of zero: d = k t + 0 t + b).2

5. Use the results.

Now the students can fill in the equation for the graph and move on. If you have them make any

predictions about distance or time, have them use the second graph (the linear one of d vs. t )2

since the best-fit line will likely be more true to the data than the best-fit parabola.

Do NOT assume students will easily follow all of this! A common error is thinking that the second graph

indicates distance and time (instead of time squared) are directly proportional. Some students will simply

go through the drill of making the graphs without thinking about what they are doing, unless you question

them verbally or in written form.

Sample answers for those two pages are shown next.


Distance (cm) Time (s) Average Time (s)

175.005.20 HAVE THE

STUDENTSLEAVE THISCOLUMN BLANKUNTIL LATER,WHEN THEYWILLCALCULATE THESPEED ANDRECORD ITHERE

150.004.38

125.004.08

100.003.53

75.002.74

50.002.55

25.001.93

The Idea answer all questions in complete sentences

1. Identify the independent and dependent variables in this experiment.

The distance is the independent variable, and the time is the dependentvariable.

Create a graph of distance traveled along the incline versus average time. Graphs involving time always plot time horizontally on thex-axis and the other variable vertically on the y-axis. This can violate the usual practice of placing the independent variable on the x-axisand the dependent on the y-axis.

You need to decide if (0,0) is a valid point to include. Make sure each member of the group has a graph.

2. What is the shape of the line on your graph? (Is it straight or is it curved? If it is curved, state whether it looks parabolic orhyperbolic, etc.)

It is curved, like a parabola.

3. The shape of a graph illustrates the mathematical relationship between the independent and dependent variables. What doesyour graph specifically show you about the relationship between distance and time?

The distance is directly proportional to the square of the time.

4. Express the relationship you described in question 3 as a proportionality: d % t2

5. According to the graph, what was the ball/cart doing as it went down the track?

It was speeding up/accelerating/moving faster.



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6. The graph has a best-fit curve. What type of fit did you perform, or instruct the computer or calculator to perform (linear,quadratic, inverse, etc.)?

We used a quadratic fit.

7. In question 4, you expressed the basic proportionality between the independent and dependent variables. Your best-fit curveallows you to now express the precise equation for your group's data. Use your graph to fill in the missing values in thisequation. Round off the values to the appropriate number of significant figures.

-3.74 12.4 4.54d = ________ + ________ t + ________ t2

Soon we will examine how the speed of the ball/cart was changing.



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UNIT 1: MOTION Lab: Galilean Ramp

Using pages 1-3 to develop concepts:

Conceptual Invention

Combine the student graphs either using transparencies or, more effectively, by inserting each group's

data into a single Graphical Analysis or other program file and having the computer perform a best-fit

curve for the entire data set. Discuss the graphs and the answers to questions 1 through 7.

You will need to emphasize on question 3 how the graph shows that distance is proportional to the square

of time, and NOT time is proportional to the square of distance.

Students often read too much or too little into answering question 5. Help them realize that the graph

indicates distance is rising at a faster rate than time: that the ball/cart is covering more and more distance

during each time interval as it rolls down the track. The use of the term acceleration should be neither

discouraged nor encouraged; do not attempt to define acceleration yet.

Ask the students what units distance over time would have. (They should respond m/s.) Now you can tell

them that distance divided by time is called speed and is symbolized by a v to form their first

equation:

Note that the bar over the v has been omitted for now, because the concept of average speed has not

been developed.

Do not accept the term velocity for now; tell them they will deal with that idea later. Emphasize how speed

is always expressed in units of distance over time (e.g. m/s, mi/h, furlongs/fortnight).

Expansion of the Idea

Instruct the students on the proper approach to problem-solving and showing all of their work.

For example:

1. State givens

2. Show original equations being used

3. Show all work, including units (show numbers plugged into the equation, and always show the

units on any measurement)

4. Round the answer to the proper significant figures and box it

W ork an example out with them in their notes, and then assign W orksheet A, their first problem set.

Sample answers for W orksheet A are shown next.


208 days

55.0 m

20.0 s

Worksheet A: Calculating Motion

1. The Spirit and Opportunity robot rovers landed on Mars in 2004 and explored it surface for years. The rovers’ spacecraft and the rovers themselves travelled at wildly different speeds.

a. The Spirit rover could move across the Martian landscape at a maximum of 2.68 m/min. How many minutes would it take for it to travel 10.4 m, the length of a typical classroom?

v = 2.68 m/min t = d / v d = 10.4 m = (10.4 m)/(2.68 m/min) =t = ? = 3.8806 min

b. Spirit journeyed to Mars in a spacecraft that traveled about 487 gigameters (487×10 m or9

303 million miles) from Earth to Mars, averaging about 27,100 m/s (60,600 mi/h). Use theSI units to calculate how many Earth days it took for the spacecraft to complete its journey.

d = 487× 10 m ( or 4.87 × 10 m) 9 11

v = 27,100 m/st = ? days

t = d / v = (487 × 10 m) / (27,100 m/s) = 17,970,480 s = 17,970,000 s 9

= =

2. A runner in a 1.00×10 meter race passes the 40.0 meter mark with a speed of 5.00 m/s.2

a. If she maintains that speed, how far from the starting line will she be 3.00 seconds later?

runt = 3.00 s d = v t = (5 m/s)(3 s) = 15 mv = 5.00 m/s

start start rund = ? d = 40 m + d = 40 m + 15 m =

b. If 5.00 m/s was her top speed, what is the shortest possible time for her entire 1.00×10 m run?2

d = 100 mv = 5.00 m/s t = d / v = (100 m) / (5 m/s) = t = ?

3.88 min

UNIT 1: MOTION Worksheet A: Calculating Motion


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3. The graph above describes the motion of a golf ball. Note that it graphs distance from a position, not distance traveled. Theball is placed on the green at 5 meters from the cup at t=0 seconds.a. How far from the cup was the ball at t=1 second?

5 metersb. What was the speed of the ball at t=1 second?

0 meters/second; it is not moving yetc. How far from the cup was the ball at t=5 seconds?

2 metersd. What was the speed of the ball as it moved towards the cup?

d = 5 m; t = 5 s v = d/t = 5 m / 5 s = 1 m/se. What happened at t=7 seconds?

the ball reached the cup

4. Two bicyclists are riding toward each other, andeach has an average speed of 10.0 km/h. Whentheir bikes are 20.0 km apart, a pesky fly beginsflying from one wheel to the other at a steady speedof 30.0 km/h. When the fly gets to the wheel, itabruptly turns around and flies back to touch thefirst wheel, then turns around and keeps repeatingthe back-and-forth trip until the bikes meet, and thefly meets an unfortunate end.

How many kilometers did the fly travel in its totalback-and-forth trips?

fly fly fly flyd = ? d = v t = (30 km/h)(1 h) = 30.0 km

flyv = 30.0 km/h

bicycled = 10.0 km

bicyclev = 10.0 km/h

fly bicycle bicycle bicyclet = t = d /v = 10 km / 10 km/h = 1 h

There are several variants to solving this problem; consider havingstudents show their differing solutions on the board.

UNIT 1: MOTION Worksheet A: Calculating Motion


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Expansion of the Idea (continued)

After W orksheet A, the next step is to clarify the speed concept and introduce acceleration.

Tell the students they are now going to explore the question, "How is the speed of the ball/cart changing?"

In other words, they are going to discover exactly how the speed was rising.

Instruct the students to calculate the speed on each run on their lab and record the results in the shaded

rightmost column of the lab's data table. Then have them graph speed vs. time.

Computer Graphing:

If using a computer to form the graph, you might be able to pre-load the fourth and final page of the lab

into the printer so that questions 8-12 appear below the printed graph. Again, check during the approval

process that they have plotted (0,0) and engage them in a discussion if it is missing.

Students may have a tendency to select a quadratic fit with the parabola opening along the x-axis; this is

due to friction. It is crucial that a linear fit be selected, so monitor their graphing and guide them to see

how a linear fit has little, if any, more error than a quadratic fit, so the simpler linear fit should be selected.

Hand Graphing:

Since this data is linear, it should be no great challenge to hand-draw the graph and draw a best-fit line.

You should point out to students that a best-fit line need not hit any of the data points nor go through the

origin; the best-fit line simply runs through the “middle” of the data points as plotted. The students can

then determine the slope and y-intercept of their best-fit line.

After the second graph is made and analyzed, the students are given a brief overview of the types of error

in an experiment and asked to discuss the systematic error in their experiment.

Sample answers for those final pages follow.


answer all questions in complete sentences, except for math formulas

8. What kind of fit (linear, quadractic, inverse, etc.) did you perform on the above speed vs. time graph?

We selected a linear fit.

9. What does this graph indicate about the relationship between speed and time?

Speed is directly proportional to time.

10. Express the relationship you described in question 9 as a proportionality: v % t

11. How was the speed changing as the ball/cart went down the track?

The speed is steadily increasing.

12. Your graph allows you to formulate an equation that fits your data. Write that equation below, substituting the appropriate variableletters for x and y, and rounding off the numbers to the proper significant figures.

v = 6.93 t + 2.07


Emphasize when discussing this graph with the students how the linear graph indicates how the speed is

changing; they should be able to be more specific than simply saying the speed is rising. Soon you will

lead them to realize that the slope of the graph is related to (but not equal to) the acceleration (but they

must first realize that they were graphing average speed, not instantaneous speed, versus time here).


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In discussing the lab before it is submitted, stress to students that a generic statement that there was

“human error” will not suffice. Some human error is random, such as trying to place the ball correctly at

the starting mark or in reading the meterstick to make the marks along the track. That error should not be

discussed. But human reaction time is usually a systematic error in that there is a delay in the triggering

and releasing of the ball. They need to make this clear in their discussion.

The tracks you use may also have obvious imperfections, such as a bowed track, knotholes in a wooden

track, etc. These again systematically bias the data and thus should be discussed. Sometimes students

may find their ball just sits there on a wooden track after release. Obviously that is a systematic error

worth noting. The solution one might employ for this dilemma is moving that starting mark a few

centimeters to get away from the surface imperfection, and altering the data table and graph accordingly.

Types of Laboratory Error

Type Examples Prevention Discussion

personal error(mistakes)

mis-reading a scaleor incorrectly rearranging anequation or calculating afigure

check against lab partners’work; redo parts of lab asneeded when errordiscovered

none; should be correctedbefore lab is submitted

systematic error miscalibration oruncontrolled variables(e.g. friction); includesunavoidable timing errors

calibrate equipment whenpossible; think throughprocedures to minimizeerror

identify any uncontrollablevariables(do not include variablescausing random error)

random error estimating the last digit on ascale reading; minorvariations in temperature orair pressure

eliminate when possible;can never be completelyeliminated

none

13. In a few sentences, discuss the systematic error in this laboratory.

Systematic error would include timing error due to human reaction

time in triggering the stopwatch as well as releasing the ball,

error due to friction on the track, and error caused by imperfections

of the track surface.


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Using the second graph to develop concepts of average speed, instantaneous speed, and acceleration:

After you have discussed the second graph and its interpretation with the students, you need to develop

the concepts of average speed and acceleration.

A Provocative Question for the Class:

W here along the track did the speed you calculated occur? In other words, where along a

given run was the ball/cart rolling at the speed you calculated – at the beginning, middle,

end? (Don't use this terminology yet, but you are asking them at what position the

average speed and instantaneous speed were equal.)

Few students will give the correct answer, that the ball/cart was rolling at the calculated speed when it was

1/4 of the way down the track from its starting point. Lead the class to consider their speed vs. time graph

and observe that distance over time must yield an average speed that occurs at the halfway point in time.

Some will then see that the average speed equaled the instantaneous speed 1/4 of the way down a given

run.

But you must make this concrete. So set up a run on a track on your demo desk and pass out some

stopwatches. Have the students yell out when half of the time has elapsed so that the students can see

where the ball/cart is located when it is moving at its "average" speed.

Now they should be satisfied that d/t yields average speed, and not the speed at an instant. Have them

correct their notes (this will get their attention!) and insert a bar over the v in v=d/t. You can now also use

their speed vs. time graph to show that the average could also be calculated by taking the highest and

lowest speed on the graph, adding them together, and dividing by two. (The graph is linear, so you are

finding the midpoint, which is the average speed.) This yields their second equation for the

year:

Now, you are really playing a game here, since you are finding the average speed on a graph of average

speed vs. time! You need not point this out to the students if it gets too confusing.

Next, have them consider the slope of their v vs. t graph and how it shows how the speed changes with

time. In other words, it shows the acceleration of the ball/cart. Let a student come up with the term

acceleration here; one is bound to think of it. You can then quickly lead them to:

Discuss the units of acceleration (m/s , mi/h , (mi/h)/s, etc.) and also have them note that the acceleration2 2

equation can be algebraically manipulated to predict final speed:

Point out how the best-fit linear equation of the speed vs. time graph matches this format (the acceleration

iis the slope, and the y-intercept is the initial speed). They will note that v is not coming out zero, which

can be attributed to experimental error. You may (or may not) wish to point out that their slope is not really

the acceleration, since they are graphing average speed vs. time and not instantaneous speed vs. time.

Emphasize how the slope of a distance vs. time graph is speed, while the slope of a speed vs. time graph

is acceleration.


UNIT 1: MOTION Worksheet B: Interpreting Motion Graphs

You need to prepare students before giving them W orksheet B. You will be teaching them to interpret

graphs of one-dimensional forward motion with positive or zero acceleration. W ait to cover graphs of

backwards motion, slowing down, and so forth until after the students have assimilated the vector

concepts in unit two, when negative slopes on a motion graph will make much more sense to them.

Now is a good time to set up a motion detector as a demo or group activity and demonstrate various

motions with real-time graphing as you have the students set down into their notes the d, v, and a vs. t

graphs for a motionless object, one moving at a steady speed, one accelerating steadily, and one

speeding up with an ever-increasing acceleration. The students will then be ready for W orksheet B.

Don’t worry if you don’t have a motion detector - here is how the students can be their own motion

detectors:

“Kinesthetic Graphing” Exercise

[Unit 4 includes a version of this for horizontal and vertical velocity vs. time graphs with both

positive and negative slopes.]

Divide your students into several groups. Hand each group a card with a particular d vs. t,

v vs. t, or a vs. t graph on it, and a toy ball. (Large nerf balls work best.) Have them

decide how to move the Nerf ball to create the motion on the graph.

Then have a person from each group present to the class the motion they have decided

upon: he or she will make the ball move in the intended motion, signaling to the class the

time when the graph should begin and end.

Have the students in the other groups individually or as a group sketch a graph of that

motion, telling them the axes that are to be used (d, v, or a vs. time). You may want to

pick a student to sketch the corresponding graph on the board or overhead.

You and the students should expect mistakes. You’ll probably want to walk around the

room, glancing at the sketches or perhaps have the students hold up their sketches to

help you assess how the class is doing.

An easy way to create the cards for this exercise is to cut out shapes A-E from a copy of

W orksheet B and add various axis labels to them.


Unit 1:MotionWorksheet B: Interpreting Motion Graphs

answer questions 1 and 2 in complete sentences

1. What does the slope of a distance vs. time graph indicate about an object’s motion?

Speed

2. What does the slope of a speed vs. time graph indicate about an object’s motion?

Acceleration

Questions 3 - 8 refer to the following generic graph shapes. Write the letter corresponding to the appropriate graphin the blank at the left of each question.

C 3. Which shape fits a distance vs. time graph of an object moving at constant (non-zero) speed?

B 4. Which shape fits a speed vs. time graph of an object moving at constant (non-zero) speed?

A/B 5. Which two shapes fit a distance vs. time graph of a motionless object?

A 6. Which shape fits a speed vs. time graph of a motionless object?

D 7. Which shape fits a distance vs. time graph of an object that is speeding up at a steady rate?

C 8. Which shape fits a speed vs. time graph of an object that is speeding up at a steady rate?

C 9. Which of the following units is equivalent to (meters per second) per second?

a) m b) m/s c) m/s d) m/s2 3

C 10. Which of the following units correspond to the slope of a distance vs. time graph?

a) m b) s c) m/s d) m/s2

C 11. Which of the following units correspond to the slope of a speed vs. time graph?

a) m/s b) m•s c) m/s d) m /s2 2 2



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The table below gives distance and time data for a moving object. Notice the varying size of the time intervals as the distance rises in 20cm increments.

Distance (m)0

20406080

100

Time (s)0

4.56.37.78.910

C 12. Which of the following distance vs. time graphs corresponds to the table data?

C 13. Which of the following descriptions matches the graph you selected in question 12?

a) A motionless object.b) An object moving at a constant speed.c) An object undergoing constant, positive acceleration.d) An object undergoing constant, negative acceleration.

A 14. Which of the following speed vs. time graphs corresponds to the table data?

C 15. Which of the following descriptions matches the graph you selected in question 14?

a) A motionless object.b) An object moving at a constant speed.c) An object undergoing constant, positive acceleration.d) An object undergoing constant, negative acceleration.

BEWARE: If your answers to questions 13 and 15 are different from each other, you are claiming that the same objectcan have two distinct motions simultaneously. Ask yourself, “Is that reasonable?”

16. A woman walks away from a starting point in a straight line.A distance vs. time graph for her motion is shown at right.a. Describe the woman's motion between 0 and 2

seconds.

She is accelerating.

b. Fill out the table below.Time Interval Woman's Speed (m/s)

2 to 4 seconds 34 to 6 seconds 06 to 8 seconds 1



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UNIT 1: MOTION Worksheet C: Combining the Variables of Motion

Next you will have the students invent five additional motion equations, for a total of eight equations they

can use in solving kinematics problems.

W orksheet C reminds them of the three equations they have developed so far. W ork through the first

question with them, to arrive again at , which they saw when you interpreted their speed

vs. time graphs.

Let them work in groups to derive the remaining equations. Give them a hint that number 2 uses the

answer from number 1, 3 uses 2, 4 uses 3, BUT to derive equation 5 they'll need to start over with their

original three equations.

Eventually they will have found the answers, which you should lead them to rewrite into their standard

forms:

1.

2.

3.

4.

5.

(One way to arrive at #5 is to set the first two equations at the top of the page equal to one another and

f isubstitute for t the rearrangement of a = (v - v ) / t.)

It is vital that they note that equation #4 can only be used when the object starts from rest, and that #5 is

especially useful when time is unknown. W ork a couple of examples with them and then give them

W orksheet D.


Worksheet D: 1-Dimensional Motion Problems

1. The head of a rattlesnake can accelerate 50.0 m/s in striking a victim. If a car could do as well, how long would it take for it2

to reach a speed 24.6 m/s (which is about 55 mi/h) from rest? 0.492 s

2. The speed limit on an 86.0 mile highway was changed from 55.0 mi/h to 75.0 mi/h. How much time was saved on the trip forsomeone traveling at the legal speed limit?

0.417 h or 25.0 min or 1500 s

3. In an emergency, a driver brings a car to a full stop in 5.00 seconds. The car is traveling along a highway at a rate of 24.6 m/swhen braking begins.

a. At what rate is the car accelerated? – 4.92 m/s2

b. How far does it travel before stopping?

61.5 m

4. A supersonic jet flying at 200. m/s is accelerated uniformly at the rate of 23.1 m/s for 20.0 seconds.2

a. What is its final speed?

662 m/sb. Physicist Ernst Mach studied the effects of motion faster than sound, and the ratio of a speed to that of sound is

called its “Mach number”. The speed of sound itself is 331 m/s (approx. 740 mi/h) at supersonic airplane altitudes. “Mach 1.00" is the ratio 331/331, or the speed of sound. One of the fastest planes was the SR-71 Blackbird. Itflew at 1059 m/s, so 1059/331 = 3.20; we say it flew at “Mach 3.20.” What is the Mach speed of our jet?

Mach 2.00

5. If a bullet leaves the muzzle of a rifle with a speed of 600. m/s, and the barrel of the rifle is 0.800 m long, at what rate is the

bullet accelerated while in the barrel? 225,000 m/s2

6. What is the acceleration of a racing car if its speed is increased uniformly from 44.0 m/s to 66.0 m/s over an 11.0 s period?

2.00 m/s2

7. An engineer is to design a runway to accommodate airplanes that must gain a ground speed of 360. km/h (approx. 225 mi/h)before they can take off. These planes are capable of being accelerated uniformly at the rate of 3.60×10 km/h .4 2

a. How many kilometers long must the runway be?

1.80 kmb. How many seconds will a plane need to accelerate to take-off speed?

36.0 s

8. A plane flying at the speed of 150. m/s is accelerated uniformly at a rate of 5.00 m/s .2

a. What is the plane's speed at the end of 10.0 seconds?

200 m/s2

b. What distance has it traveled?

1750 m

9. A Tokyo express train is accelerated from rest at a constant rate of 1.00 m/s for 1.00 minute. How far does it travel during2

this time?

1800 m

10. In a vacuum tube, an electron is accelerated uniformly from rest to a speed of 2.60×10 m/s during a time period of 6.50×105 -2

seconds. Calculate the acceleration of the electron. 4.00×10 m/s26

UNIT 1: MOTION Worksheet D: 1-Dimensional Motion Problems

I've shown the answers that did not already appear on the handout. Note that it is important to insist that

students show their givens, equation, substituted numbers with units, and a boxed answer with proper

significant figures. Build good habits early on for showing all work.


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Inquiry Physics: Equipment Suggestions Page 1 of 2

Equipment Suggestions for Unit 1: Motion Equipment for each group (of 3 to 4 students):

Item Suggestions

grooved wooden track OR air track OR other dynamics track with ball or cart, about 1.5 to 2 m long if using a wooden track, a 7 mm groove should run lengthwise along the track for a steel ball (e.g. large ball bearing) to roll along; a wooden block or other stop will be needed at the bottom of the track

I use wooden tracks made locally over two decades ago, with ball bearings from oil field equipment. You could use an air track and glider, available from a variety of sources (we use Daedalon tracks for other labs, but I like the simplicity of the ball on the ramp for this first unit). But if I were starting out, I’d consider buying low-friction dynamics carts with tracks (which can act as long inclined planes) from Pasco. They have various accessories and integrate well with their probeware.

ring stand OR blocks to incline the track

ball for wooden track (diameter ≈ 2 cm and mass ≈ 60 g) OR air track glider OR wheeled toy/cart

meter stick

stopwatch I hate stopwatches with alarms and clocks and the rest, greatly preferring MyChron stopwatches for their simplicity and long battery life: Sargent Welch item WLS77448 Science Kit item 46185M01

masking tape

Inquiry Physics: Equipment Suggestions Page 2 of 2

EQUIPMENT SUGGESTIONS for INQUIRY PHYSICS A Modified Learning Cycle Curriculum

The student handouts often avoid listing specific equipment so that you can be more flexible in your approach. But here I have gathered a listing, current as of the summer of 2010, of the type of equipment I use in each unit for the student labs as well as various demonstrations. There are a number of nationally recognized science education supply houses which offer a vast array of physics education equipment. Do not take their list prices at face value. Contact them about pricing agreements where they might agree to offer a set discount or free shipping, etc. Here is a sampling: GENERAL SCIENCE SUPPLIERS Frey Scientific www.freyscientific.com School Specialty Frey Scientific P.O. Box 3000 Nashua, NH 03061-3000 1-800-225-FREY (3739) Fax: 1-877-256-FREY (3739) Science Kit & Boreal Laboratories www.sciencekit.com P.O. Box 5003 Tonawanda, NY 14151-5003 1-800-828-7777 Fax: 1-800-828-FAXX (3299) Sargent-Welch www.sargentwelch.com P.O. Box 4130 Buffalo, NY 14217 1-800-727-4368 Fax: 1-800-676-2540 Nasco http://www.enasco.com/ 901 Janesville Avenue P.O. Box 901 Fort Atkinson, WI 53538-0901 1-800-558-9595 Fax: 1-800-372-1236 Flinn Scientific www.flinnsci.com P.O. Box 219 Batavia, IL 60510 1-800-452-1261 Fax: 1-866-452-1436 Fisher Science Education www.fishersci.com 4500 Turnberry Drive Hanover Park, IL 60133 1-800-955-1177 Fax: 1-800-955-0740

SPECIALTY SUPPLIERS Pasco Scientific www.pasco.com 10101 Foothills Boulevard Roseville, CA 95747 1-800-772-8700 Fax: 1-916-786-7565 Vernier Software & Technology www.vernier.com 13979 SW Millikan Way Beaverton, OR 97005-2886 1-888-837-6437 Fax: 1-503-277-2440 Design Simulation Technologies, Inc. (Interactive Physics) www.design-simulation.com 43311 Joy Road, #237 Canton, MI 48187 1-800-766-6615 1-734-259-4207 Edmund Scientific http://scientificsonline.com 60 Pearce Ave Tonawanda, NY 14150 1-800-728-6999 Fax: 1-800-828-3299 The Science Source (Daedalon) www.thesciencesource.com 299 Atlantic Highway Waldoboro, ME 04572 1-800-299-5469 Fax: 1-207-832-7281

http://www.freyscientific.com/

http://www.sciencekit.com/

http://www.sargentwelch.com/

http://www.enasco.com/

http://www.flinnsci.com/

http://www.fishersci.com/

http://www.pasco.com/

http://www.vernier.com/

http://www.krev.com/

http://scientificsonline.com/

http://www.daedalon.com/

INQUIRY PHYSICSA Modified Learning Cycle Curriculumby Granger Meador

Unit 1: MotionStudent Papers

inquiryphysics.org

2010

these SAMPLE NOTES, the STUDENT PAPERS, and any PRESENTATIONS for each unit have a creativecommons attribution non-commercial share-alike license; you may freely duplicate, modify, anddistribute them for non-commercial purposes if you give attribution to Granger Meador and referencehttp://inquiryphysics.org

however, please note that the TEACHER’S GUIDES are copyrighted and all rights are reserved so you may NOT distribute them or modified versions of them to others

http://inquiryphysics.org

1: Motion NameLab: Galilean Ramp

There are fundamental principles governing the motion of all objects, from supersonic aircraft toglaciers. This lab is similar to experiments conducted by Galileo Galilei several hundred yearsago which laid the foundations for modern-day physics.

Set up the equipment as shown in the diagram below:

Use the ring stand and ring to raise one end of the track until the distance between the bottom ofthe track and the tabletop is 10.00 cm.

You will be varying distance. If necessary, mark off on masking tape the following distancesfrom the lower end of the track: 25.00 cm, 50.00 cm, 75.00 cm, 100.00 cm, 125.00 cm,150.00 cm, and 175.00 cm. Be sure the tape will not interfere with the motion of the ball or cart. You will measure the time required for the ball/cart to travel each of those seven distances.

1. Begin taking data by placing the ball/cart at the 175.00 cm mark. To start the ball/cart thesame way each time, keep it at the mark on the incline with a pencil until you are ready torelease it and begin timing. Don't push or spin the ball/cart when you pull the pencilaway! Start a stopwatch as the ball/cart is released and stop the watch when it reaches thestop. Make three measurements of time to as many decimal places as possible, ensuringthat the difference between the highest and lowest measurements is no more than 0.10 s. Record those values in the table on the reverse.

2. Gather the same data as before, but start the ball/cart at the mark that will make thedistance equal to 150.00 cm. Make three measurements of time and record them in thetable.

3. Repeat the data gathering process for each of the other distances. Record the data in thetable.

Unit 1: Motion, Lab: Galilean Ramp Page 1 of 5 ©2010 by G. Meador – www.inquiryphysics.org

http://inquiryphysics.org/

Distance (cm) Time (s) Average Time (s)

175.00

150.00

125.00

100.00

75.00

50.00

25.00

The Idea answer all questions in complete sentences

1. Identify the independent and dependent variables in this experiment.

Create a graph of distance traveled along the incline versus average time. Graphs involving time always plot timehorizontally on the x-axis and the other variable vertically on the y-axis. This can violate the usual practice of placing theindependent variable on the x-axis and the dependent on the y-axis.

You need to decide if (0,0) is a valid point to include. Make sure each member of the group has a graph.

2. What is the shape of the line on your graph? (Is it straight or is it curved? If it is curved, state whether it looksparabolic or hyperbolic, etc.)

3. The shape of a graph illustrates the mathematical relationship between the independent and dependent variables. What does your graph specifically show you about the relationship between distance and time?

4. Express the relationship you described in question 3 as a proportionality:

5. According to the graph, what was the ball/cart doing as it went down the track?



6. Apply a best-fit curve to your graph. What type of fit did you perform, or instruct the computeror calculator to perform (linear, quadratic, inverse, etc.)?

7. In question 4, you expressed the basic proportionality between the independent and dependentvariables. Your best-fit curve allows you to now express the precise equation for your group'sdata. Use your graph to fill in the missing values in this equation. Round off the values to theappropriate number of significant figures.

d = ________ + ________ t + ________ t2

Soon we will examine how the speed of the ball/cart was changing.



answer all questions in complete sentences, except for math formulas

8. What kind of fit (linear, quadratic, inverse, etc.) did you perform on the speed vs. time graph?

9. What does this graph indicate about the relationship between speed and time?

10. Express the relationship you described in question 9 as a proportionality:

11. How was the speed changing as the ball/cart went down the track?

12. Your graph allows you to formulate an equation that fits your data. Write that equation below,substituting the appropriate variable letters for x and y, and rounding off the numbers to theproper significant figures.



Types of Laboratory Error

Type Examples Prevention Discussion

personal error(mistakes)

mis-reading a scaleor incorrectlyrearranging anequation orcalculating a figure

check against labpartners’ work; redoparts of lab asneeded when errordiscovered

none; should becorrected before labis submitted

systematic error miscalibration oruncontrolledvariables(e.g. friction);includes unavoidabletiming errors

calibrate equipmentwhen possible; thinkthrough proceduresto minimize error

identify anyuncontrollablevariables(do not includevariables causingrandom error)

random error estimating the lastdigit on a scalereading; minorvariations intemperature or airpressure

eliminate whenpossible; can neverbe completelyeliminated

none

13. In a few sentences, discuss the systematic error in this laboratory.



1: Motion NameWorksheet A: Calculating Motion

1. The Spirit and Opportunity robot rovers landed on Mars in 2004 and explored its surfacefor years. The rovers’ spacecraft and the rovers themselves travelled at wildly differentspeeds.

a. The Spirit rover could move across the Martianlandscape at a maximum of 2.68 m/min. Howmany minutes would it take for it to travel10.4 m, the length of a typical classroom?

2. Spirit journeyed to Mars in a spacecraft thattraveled about 487 gigameters (487×10 m or9

303 million miles) from Earth to Mars,averaging about 27,100 m/s (60,600 mi/h). Usethe SI units to calculate how many Earth days ittook for the spacecraft to complete its journey.

2. A runner in a 1.00×10 meter race passes the 40.0 meter mark with a speed of 5.00 m/s.2

a. If she maintains that speed, how far from the starting line will she be3.00 seconds later?

b. If 5.00 m/s was her top speed, what is the shortest possible time for her entire1.00×10 m run?2

CONTINUED...

Unit 1: Motion, Worksheet A: Calculating Motion ©2010 by G. Meador – www.inquiryphysics.org


3. The graph above describes the motion of a golf ball. Note that it graphs distance from aposition, not distance traveled. The ball is placed on the green at 5 meters from the cupat t=0 seconds.a. How far from the cup was the ball at t = 1 second?

b. What was the speed of the ball at t = 1 second?

c. How far from the cup was the ball at t = 5 seconds?

d. What was the speed of the ball as it moved towards the cup?

e. What happened at t = 7 seconds?

4. Two bicyclists are riding toward eachother, and each has an average speed of10.0 km/h. When their bikes are20.0 km apart, a pesky fly begins flyingfrom one wheel to the other at a steadyspeed of 30.0 km/h. When the fly getsto the wheel, it abruptly turns aroundand flies back to touch the first wheel,then turns around and keeps repeatingthe back-and-forth trip until the bikesmeet, and the fly meets an unfortunate end.

How many kilometers did the fly travel in its total back-and-forth trips?

Unit 1: Motion, Worksheet A: Calculating Motion ©2010 by G. Meador – www.inquiryphysics.org


1: Motion NameW orksheet B: Interpreting Motion Graphs

answer questions 1 and 2 in complete sentences

1. What does the slope of a distance vs. time graph indicate about an object’s motion?

2. What does the slope of a speed vs. time graph indicate about an object’s motion?

Questions 3 - 8 refer to the following generic graph shapes. Write the letter corresponding to the

appropriate graph in the blank at the left of each question.

3. Which shape fits a distance vs. time graph of an object moving at constant (non-zero) speed?

4. Which shape fits a speed vs. time graph of an object moving at constant (non-zero) speed?

5. Which two shapes fit a distance vs. time graph of a motionless object?

6. Which shape fits a speed vs. time graph of a motionless object?

7. Which shape fits a distance vs. time graph of an object that is speeding up at a steady rate?

8. Which shape fits a speed vs. time graph of an object that is speeding up at a steady rate?

9. Which of the following units is equivalent to (meters per second) per second?

a) m b) m/s c) m/s d) m/s2 3

10. Which of the following units correspond to the slope of a distance vs. time graph?

a) m b) s c) m/s d) m/s2

11. Which of the following units correspond to the slope of a speed vs. time graph?

a) m/s b) m•s c) m/s d) m /s2 2 2

CONTINUED...

Unit 1: Motion, Worksheet B: Interpreting Motion Graphs ©2010 by G. Meador – www.inquiryphysics.org


The table below gives distance and time data for a moving object. Notice the varying size of the time intervals as the distance

rises in 20 cm increments.

Distance (m)

0

20

40

60

80

100

Time (s)

0

4.5

6.3

7.7

8.9

10

12. Which of the following distance vs. time graphs corresponds to the table data?

13. Which of the following descriptions matches the graph you selected in question 12?

a) A motionless object.

b) An object moving at a constant speed.

c) An object undergoing constant, positive acceleration.

d) An object undergoing constant, negative acceleration.

14. Which of the following speed vs. time graphs corresponds to the table data?

15. Which of the following descriptions matches the graph you selected in question 14?

a) A motionless object.

b) An object moving at a constant speed.

c) An object undergoing constant, positive acceleration.

d) An object undergoing constant, negative acceleration.

BEWARE: If your answers to questions 13 and 15 are different from each other, you are claiming that the same object

can have two distinct motions simultaneously. Ask yourself, “Is that reasonable?”

16. A woman walks away from a starting point in a straight line.

A distance vs. time graph for her motion is shown at right.

a. Describe the woman's motion between 0 and 2 seconds.

b. Fill out the table below.

Time Interval Woman's Speed (m/s)

2 to 4 seconds

4 to 6 seconds

6 to 8 seconds

Unit 1: Motion, Worksheet B: Interpreting Motion Graphs ©2010 by G. Meador – www.inquiryphysics.org


1: Motion NameWorksheet C: Combining the Variables of Motion

We have already developed three equations for velocity and acceleration:

Using these equations, figure out ways to combine them algebraically to make five otherequations that would enable you to:

f i1. Solve for v when you know v , a, and t.

i2. Solve for d when you know v , a, and t.

i3. Solve for a when you know v , d, and t.

i4. Solve for t when you know d and a, and v=0.

f i5. Solve for v when you know v , a, and d.

Unit 1: Motion, Worksheet C: Combining the Variables of Motion ©2010 by G. Meador – www.inquiryphysics.org


1: Motion NameW orksheet D: 1-Dimensional Motion Problems

1. The head of a rattlesnake can accelerate 50.0 m/s in striking a victim. If a car could do as well, how long2

would it take for it to reach a speed 24.6 m/s (which is about 55 mi/h) from rest?

0.492 s

2. The speed limit on an 86.0 mile highway was changed from 55.0 mi/h to 75.0 mi/h. How much time was

saved on the trip for someone traveling at the speed limit?

0.417 h

3. In an emergency, a driver brings a car to a full stop in 5.00 seconds. The car is traveling along a highway at

a rate of 24.6 m/s when braking begins.

a. At what rate is the car accelerated? – 4.92 m/s2

b. How far does it travel before stopping?

4. A supersonic jet flying at 200. m/s is accelerated uniformly at the rate of 23.1 m/s for 20.0 seconds.2

a. What is its final speed?

b. Physicist Ernst Mach studied the effects of motion faster than sound, and the ratio of a speed to

that of sound is called its “Mach number”. The speed of sound itself is 331 m/s (approx. 740 mi/h)

at supersonic airplane altitudes. “Mach 1.00" is the ratio 331/331, or the speed of sound. One of

the fastest planes was the SR-71 Blackbird. It flew at 1059 m/s, so 1059/331 = 3.20; we say it flew

at “Mach 3.20.” What is the Mach speed of our jet?

5. If a bullet leaves the muzzle of a rifle with a speed of 600. m/s, and the barrel of the rifle is 0.800 m long, at

what rate is the bullet accelerated while in the barrel?

225,000 m/s2

Unit 1: Motion, Worksheet D: 1-Dimensional Motion Problems ©2010 by G. Meador – www.inquiryphysics.org


6. What is the acceleration of a racing car if its speed is increased uniformly from 44.0 m/s to 66.0 m/s over an

11.0 s period?

7. An engineer is to design a runway to accommodate airplanes that must gain a ground speed of 360. km/h

(approx. 225 mi/h) before they can take off. These planes are capable of being accelerated uniformly at the

rate of 3.60×10 km/h .4 2

a. How many kilometers long must the runway be?

b. How many seconds will a plane need to accelerate to take-off speed?

8. A plane flying at the speed of 150. m/s is accelerated uniformly at a rate of 5.00 m/s .2

a. What is the plane's speed at the end of 10.0 seconds?

b. What distance has it traveled?

9. A Tokyo express train is accelerated from rest at a constant rate of 1.00 m/s for 1.00 minute. How far2

does it travel during this time?

10. In a vacuum tube, an electron is accelerated uniformly from rest to a speed of 2.60×10 m/s during a time5

period of 6.50×10 seconds. Calculate the acceleration of the electron.-2

4.00×10 m/s26

Unit 1: Motion, Worksheet D: 1-Dimensional Motion Problems ©2010 by G. Meador – www.inquiryphysics.org


1 Motion NameW orksheet E: Quiz Practice Problems

1. A fly takes off with an acceleration of 0.700 m/s from a wall. How many seconds will it take the fly to reach a2

speed of 12.6 km/h?

2. In the graph:a. What is the speed and acceleration from 0 to 1 seconds?b. What is the speed and acceleration from 1 to 3 seconds?c. What is the acceleration from 3 to 5 seconds?

(assume it is constant)d. What is the object doing from 5 to 7 seconds?

3. The evil Victor Vector has tied poor Velma Velocity to a train track. He is aboard a train which is moving at

5.00 m/s when it is 175 m from the struggling Velma. If the train is accelerating at 3.00 m/s , how much time2

does she have to make her escape?

4. The Hanson brothers are backstage after a concert, sauntering at 0.500 m/s, when a horde of screaming fans

gives chase. The musicians are 12.0 m away from the safety of their dressing room. If they accelerate steadily

and reach the room in 3.20 s, how fast are they traveling as they pass its doorway?

5. Mr. M is a human cannonball in his spare time. If the cannon he uses is 1.75 meters long and he exits the

cannon at a speed of 20.0 m/s, what acceleration does the cannon impart to Mr. M?

Unit 1: Motion, Worksheet E: Quiz Practice Problems ©2010 by G. Meador – www.inquiryphysics.org


Unit 1: Motion Notes Meador’s Inquiry Physics Page 1 of 7

I recommend that you always write out notes,

by hand, on the board for each class. That

allows you to control the pacing and focus,

rather than having students ignore you while

they simply copy down the content of a slide. It

also controls your pacing, so that you don’t race

ahead but instead focus on student

understanding.

Ask frequent questions of students to check

their grasp of the material, and call upon

students to provide the next step when working

examples.

My rule for students is that if I write it on the

board, they must write it in their notes, and I

grade their notes each quarter and take off for

any units with incomplete notes or examples.

Trigonometry-Based Physics (AP Physics B)

These notes apply to both algebra-based

Inquiry Physics and to trigonometry-based

physics. Trig concepts will not be introduced

until Unit 2 on Vectors.

INQUIRY PHYSICS A Modified Learning Cycle Curriculum by Granger Meador

Unit 1: Motion Sample Notes

inquiryphysics.org 2010

these SAMPLE NOTES, the STUDENT PAPERS, and any PRESENTATIONS for each unit have a creative commons attribution non-commercial share-alike license; you may freely duplicate, modify, and distribute them for non-commercial purposes if you give attribution to Granger Meador and reference http://inquiryphysics.org however, please note that the TEACHER’S GUIDES are copyrighted and all rights are reserved so you may NOT distribute them or modified versions of them to others

Unit 1 focuses on development and use of one-dimensional motion equations.



PROBLEM-SOLVING PROCEDURES

1. Write down the givens

2. Show the equation used in original form

3. Show all work, with units

4. Answer must have proper significant figures,

units, and be boxed

Sample Notes for Unit 1: Motion

Unit 1: Motion

speed = distance / time

UNITS: mi/h, km/h, m/s, furlongs/fortnight, etc.

meter = distance light travels in

s

second = defined by vibrations of gas atoms in atomic

clocks

These notes begin after the students have completed the first three pages of the Unit 1 lab. They’ve thus seen that distance is directly proportional to the square of time, and that means the ball was speeding up. So we begin by formally defining those quantities. Note that we do NOT yet distinguish between average and instantaneous speed. That comes after the next part of the lab.

I am very strict with students about these procedures. While I seldom grade on their givens, I take off 2 points for a missing equation, 1 point off for any missing units in their calculations or answer, 1 point off for wrong significant figures in the answer, and award little if any credit unless all work is shown. “All work” means they have to show the plugging in of values from their givens, with units, into their equation. They can then optionally show more work as intermediate steps to the answer. I stress to students that this is how they get partial credit on their work and ensure it is decipherable by both me and by them, including a year or more from now when they use this in college.


Example 1-1

Phluffy the cat was being chased by a lawnmower. She

travelled 10.0 m at 4.00 m/s.

a) What was her time of travel?

v = 4 m/s d = 10.0 m t = ?

b) What was her speed in furlongs/fortnight? 4 m

100 cm

in ft yd furlong 3600 s 24 h 14 day

s m 2.54 cm

12 in

3 ft 220 yd h day ftnight

= 24,051.53901 furlongs/fortnight

=

c) Phluffy hit a wall and stopped in 25.0 μs. How far did

she travel while slamming to a halt? v = 4 m/s t = 25 μs = 25 x 10-6 s d = ? d = vt = (4 m/s)(25 x 10-6 s) = 100. x 10-6 m

= or or

I indicate common errors as I work this example and how many points they would lose. In my scheme, part “a” might be worth 6 points. I’d take off 1 pt. for a wrong or missing equation, 1 pt. for wrong or missing units, 1 pt. for not having 3 sig figs in the answer, and 3 points for a math/algebra error (such as t=dv). So they could make the algebra mistake and still get half credit if their work allowed us both to see that the algebraic rearrangement was where things went awry. But they could also lose half of the credit through careless notational errors.

Part “b” demonstrates unit conversions. I insist they know how to do this, although I will accept work where their calculator made the conversion. I warn them that if they have to borrow a calculator from me, it won’t do those unit conversions.

Next I assign Unit 1: Worksheet A. The next day I walk to each student, assigning points for the number of problems completed, even if they are wrong, and not yet taking off for notational errors. The focus is on them showing work, and I personally point out to them repetitive mistakes he or she has made, such as not showing equations or units or sig figs. Then I go over the worksheet with them on an overhead projector or document camera, pointing out how I showed my work. The final fly problem gives some of them fits, especially in how to show their work. As I walked around the room, I had noted successful solutions to the fly problem using various methods, and I call upon those students to work it on the board.


The next task is to have the students calculate the speeds on the lab, filling in that last column, and then constructing graphs to determine how the speed was changing as the ball went down the ramp. They quickly figure out to graph speed on the y-axis and time on the x, and they use it to complete the lab. When I go over that part of the lab with them, I check that they are seeing how the speed was steadily increasing as time went by. This sets up the disequilibration of what speed really means. I put a track on the demo desk, marking with tape the ¼, ½ , ¾ and full length of a run. I ask them to discuss in their groups this question: “Where along the track was the ball going the speed you indicated in the table?” I then have them vote by hand on where they think that speed occurred. Few will correctly indicate it was ¼ of the way down from the starting point. Instead, most will say it occurred at the halfway point, or ¾ point, or at the end of the track. That lets me then have them consider what distance/time really means. They eventually see that it gives average speed, while my question was about an instantaneous speed. Once they grasp that, I use their linear v vs. t graph to identify that the average speed in the table must have occurred halfway along a run in time, not distance. And then we look on the parabolic d vs. t graph for where a ball is halfway along a run in time…and it is only ¼ of the way along its journey. To prove the point, I hand out a stop watch to each group. The timers time how long a full run of the ball down my demo desk track takes. Then we cut that in half, and they are to yell out as soon as that much time elapses. Everyone else keeps their eye on the ball. They’ll see that the timers yell out when the ball has not yet even reached the halfway point. (Reaction time delay means they won’t yell out precisely when it is ¼ of the way along the journey.) That is the setup for the next part of the notes. First we go back and “fix” the earlier speed equation, adding a bar over the v to indicate that d/t yields average speed. And we use the linear v vs. t graph to understand that average speed is also calculated by simply adding the initial and final instantaneous speeds together and then dividing by two.


Average vs. Instantaneous Speed

yields average speed, . So

where vi = initial speed and vf = final speed

and vi and vf are the speeds at a given instant, or instantaneous speeds.

In the lab the ball was speeding up steadily, so its average speed occurred halfway along the journey in

time:

Graph Slopes

The slope of a distance vs. time graph is the object’s speed.

The slope of a speed vs. time graph is the object’s acceleration.

In our lab the speed increased steadily, so the acceleration was

constant.

acceleration = rate of speed change;

UNITS: m/s2, mi/h2, (mi/h)/s, etc.

Now it is time to introduce the concept of the meaning of the slope of each graph and the concept of acceleration. Some students will have already used the term in the lab.


Graphs of 1-d Motion

I call upon a different student to help me draw each of these graphs, thus hitting at least 12 students during the lecture. I can hit another 8 students by repeatedly asking them what the slope of each graph shape is and what we call that slope. For example, the slope of d vs. t in the steady speed case is a constant positive number, so the object has a constant positive speed. (That will get a few students thinking about what a negative slope would mean, and the golf ball graph on Worksheet A can be a resource for seeing how it indicates whether the object is moving toward or away from you.) By the way, the name for the slope of an acceleration vs. time graph is the “jerk”. For the final set of graphs, I ask all of the students to sketch all three graphs, looking for a pattern in the data. Most will spot how the graph shapes are shifting to the right as you work down the page. The final d vs. t graph has a steeper cubic shape, not a quadratic or parabolic one.

After these notes I assign Worksheet B on graphs. After that they do Worksheet C, using algebra to create five new equations from the existing three equations we have in the notes. That sets the stage for the final examples.


Example 1-2

Phluffy accelerates from 2.50 m/s to 7.00 m/s over 16.0 m.

How much time did this take?

vi = 2.5 m/s vf = 7 m/s d = 16 m t = ?

so thus

t = 3.37 s

Example 1-3 (example 1-2 in AP Physics B)

Phyllis Physics was driving at 90.0 km/h (55 mi/h) when a cat jumped out in the road 40.0 m in

front of her car. Phyllis hesitated 0.750 s before braking at –10.0 m/s2. Did she hit the cat?

COASTING

t = 0.75 s d = ?

so

BRAKING a = –10.0 m/s2 vi = 25 m/s vf = 0 d = ? vf

2 = vi2 + 2ad so

d = 31.25 m

dtotal = dcoasting + dbraking = 18.75 m + 31.25 m = 50.0 m Yes; 50.0 m > 40.0 m

I skip example 1-2 in my trig-based AP Physics B class, so example 1-3 shown here becomes example 1-2 in that course.

Alternatively one can set the total distance to be 40 m and solve for final speed. It will be positive, meaning the car is still moving and the cat is hit. This problem is based on an incident that happened to me near the Crazy Horse monument in South Dakota; happily the cat used up 1 of its 9 lives and survived. I follow this with Worksheet D, then create and administer a quiz over Unit 1.

unit 1: motion - sample

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