Unit 1: Motion - SAMPLE

INQUIRY PHYSICS A Modified Learning Cycle Curriculum by Granger Meador

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INQUIRY PHYSICS

inquiryphysics.org

A Modified Learning Cycle Curriculum by Granger Meador, ©2010

Unit 1: Motion

Teacher’s Guide

these TEACHER’S GUIDES are copyrighted and all rights are reserved so you may NOT distribute them or modified versions of them to others However, the STUDENT PAPERS, SAMPLE NOTES, 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

1 Motion

Teacher's Guide

Inquiry Physics Key Concepts Speed is the m easurable rate of change in the position of an object. Acceleration is the m easureable rate of change in speed. Both graphical and num erical representations of position, speed, and acceleration can be utilized to describe and predict m ovem ent.

Student Papers Lab: Galilean Ram p (analyze m otion of ball/cart down a track) W orksheet A: Calculating Motion (initial m otion problem s and a graph) W orksheet B: Interpreting Motion Graphs W orksheet C: Com bining the Variables of Motion (form ulating additional equations) W orksheet D: 1-Dim ensional Motion Problem s 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 com pelled to introduce the velocity concept here; that will com e later in unit two. Tell the students that the sym bol 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 tim e, or

Acceleration is the change of speed with respect to tim e, or

If acceleration is constant, average speed is also given by

IN Q U IRY P H YSICS T EACH ER 'S G U ID E

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UNIT 1: MOTION Lab: Galilean Ramp, pages 1-3 Pre-Lab Preface this lab with definitions, exam ples, 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 m etric system (SI) prefixes. Also refresh your students on the basic equations and graphical form s 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 Equipm ent for each group (of 3 to 4 students): grooved wooden track OR air track OR other dynam ics track with ball or cart, about 1.5 to 2 m long if using a wooden track, a 7 m m 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 (diam eter $ 2 cm and m ass $ 60 g) OR air track glider OR wheeled toy/cart m eter stick - show the students how to read the m eter stick to as m any decim al places (precision/tolerance) as possible; typically they'll be reading 3 to 4 significant figures stopwatch m asking 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. tim e graph (or students can put a hand-plotted graph in that space). The fourth page is sim ilarly designed for a speed vs. tim e graph, and will not be used until a day or two after the lab begins. Don’t worry if your students can’t use com puters or calculators - later in this guide you’ll be shown how to handle a parabolic relationship without such equipm ent. For the data collection, each group should set up the equipm ent in exactly the sam e way so that the class data can be com piled, com pared, and used to invent the concepts. Each student in a group should have an assigned role (e.g. ball/cart release, tim ing, recorder, and ball/cart stop and return). For this first lab, stress that students should check each other on their m easurem ents. Before the students collect data, discuss with them the various experim ental errors to be avoided or m inim ized, such as: which part of the ball or cart is to be held at the m ark on the tape (the front, not the m iddle or back), how should the release and tim ing be coordinated (verbal cue from tim er), 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 m easured first (because it is the one least affected by tim ing errors; the shorter runs where tim ing errors are m ore significant are saved for the end when the experim enters are m ore 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 m otion using Microsoft Excel or another spreadsheet, or to graph the data by hand. I do NOT advocate using probeware to analyze the m otion, as this can easily short-circuit the developm ent 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 m otion graphs collected as they m ove in front of a m otion detector. If using a calculator or com puter to graph, first show the students how to use the m achinery. Have them m ake 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.


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UNIT 1: MOTION Lab: Galilean Ramp, pages 1-3 W hat 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 sim ply 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 determ ining the num erical relationship between distance and tim e. They’ll just need to draw two different graphs - one to determ ine the type of relationship and the second to find the num erical values in it. You will probably need to lead them through the process with sam ple 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. tim e (tim e 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 m athem atical m eaning of that shape, or lead them to realize that it im plies that distance is directly proportional to the square of tim e.

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 com pute the square of the tim e for each part of the experim ent. 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 m eans. It m eans that distance is directly proportional to the square of tim e. Have them eyeball a best-fit line for the second graph. (Here it is useful to rem ind 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 “m iddle” 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 com pute its slope. The resulting equation will be d = k t2 + b where d is the distance, k is the slope, t2 is the square of the tim e, and b is the graph’s y-intercept. Voila! They now have about the sam e equation a fancy com puter or calculator would have calculated when form ing a best-fit parabola to the original distance vs. tim e graph (the only difference is that the t term in the quadratic autom atically has a coefficient of zero: d = k t2 + 0 t + b).

5.

Use the results. Now the students can fill in the equation for the graph and m ove on. If you have them m ake any predictions about distance or tim e, have them use the second graph (the linear one of d vs. t2) since the best-fit line will likely be m ore true to the data than the best-fit parabola.

Do NOT assum e students will easily follow all of this! A com m on error is thinking that the second graph indicates distance and tim e (instead of tim e squared) are directly proportional. Som e students will sim ply go through the drill of m aking the graphs without thinking about what they are doing, unless you question them verbally or in written form .

Sam ple answers for those two pages are shown next.


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

Distance (cm)

Time (s)

Average Time (s)

5.20

175.00

HAVE THE STUDENTS LEAVE THIS COLUMN BLANK UNTIL LATER, WHEN THEY WILL CALCULATE THE SPEED AND RECORD IT HERE

4.38

150.00

4.08

125.00

3.53

100.00

2.74

75.00

2.55

50.00

1.93

25.00

The Idea answer all questions in complete sentences 1.

Identify the independent and dependent variables in this experiment.

The answer is not available in this online variable, sample. and the time is the dependent The distance is the independent variable. Create a graph of distance traveled along the incline versus average time. Graphs involving time always plot time horizontally on the x-axis and the other variable vertically on the y-axis. This can violate the usual practice of placing the independent 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 looks parabolic or hyperbolic, etc.)

The answer is not available in this online sample. It is curved, like a parabola. 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?

The answer is not available in this online sample. The distance is directly proportional to the square of the time. 4.

Express the relationship you described in question 3 as a proportionality: Not d%

5.

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

2 in sample. tavailable

The answer is not available in this online sample.faster. It was speeding up/accelerating/moving


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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.)?

The answer is not available fit. in this online sample. We used a quadratic 7.

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

-3.74

12.4

4.54

d = ________ + ________ 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 Com bine the student graphs either using transparencies or, m ore effectively, by inserting each group's data into a single Graphical Analysis or other program file and having the com puter 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 em phasize on question 3 how the graph shows that distance is proportional to the square of tim e, and NOT tim e is proportional to the square of distance. Students often read too m uch or too little into answering question 5. Help them realize that the graph indicates distance is rising at a faster rate than tim e: that the ball/cart is covering m ore and m ore distance during each tim e interval as it rolls down the track. The use of the term acceleration should be neither discouraged nor encouraged; do not attem pt to define acceleration yet. Ask the students what units distance over tim e would have. (They should respond m /s.) Now you can tell them that distance divided by tim e is called speed and is sym bolized by a v to form their first equation:

Note that the bar over the v has been om itted 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. Em phasize how speed is always expressed in units of distance over tim e (e.g. m /s, m i/h, furlongs/fortnight).

Expansion of the Idea Instruct the students on the proper approach to problem -solving and showing all of their work. For exam ple: 1. State givens 2. Show original equations being used 3. Show all work, including units (show num bers plugged into the equation, and always show the units on any m easurem ent) 4. Round the answer to the proper significant figures and box it W ork an exam ple out with them in their notes, and then assign W orksheet A, their first problem set. Sam ple answers for W orksheet A are shown next.


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UNIT 1: MOTION Worksheet A: Calculating Motion

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 d = 10.4 m t=?

b.

t=d/v = (10.4 m)/(2.68 m/min) = = 3.8806 min 3.88 min

Spirit journeyed to Mars in a spacecraft that traveled about 487 gigameters (487×109 m or 303 million miles) from Earth to Mars, averaging about 27,100 m/s (60,600 mi/h). Use the SI units to calculate how many Earth days it took for the spacecraft to complete its journey.

d = 487× 10 9 m ( or 4.87 × 1011 m) v = 27,100 m/s t = ? days t = d / v = (487 × 10 9 m) / (27,100 m/s) = 17,970,480 s = 17,970,000 s 208 days

= 2. A runner in a 1.00×102 meter race passes the 40.0 meter mark with a speed of 5.00 m/s. a.

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

tThe = 3.00 s is not availabledinrunthis = vonline t = (5sample. m/s)(3 s) = 15 m answer v = 5.00 m/s dstart = ? dstart = 40 m + drun = 40 m + 15 m = b.

55.0 m

If 5.00 m/s was her top speed, what is the shortest possible time for her entire 1.00×102 m run?

dThe = 100 m is not available in this online sample. answer v = 5.00 m/s t = d / v = (100 m) / (5 m/s) = t=?


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UNIT 1: MOTION Worksheet A: Calculating Motion

3.

The graph above describes the motion of a golf ball. Note that it graphs distance from a position, not distance traveled. The ball 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 meters Answer not available in online sample. 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?

0Answer meters/second; it is moving yet not available in not online sample. 2Answer metersnot available in online sample. d = 5 m;not t =available 5s v = sample. d/t = 5 m / 5 s = 1 m/s Answer in online Answer available online sample. the ball not reached the in cup 4.

Two bicyclists are riding toward each other, and each has an average speed of 10.0 km/h. When their bikes are 20.0 km apart, a pesky fly begins flying from one wheel to the other at a steady speed of 30.0 km/h. When the fly gets to the wheel, it abruptly turns around and flies back to touch the first wheel, then turns around and keeps repeating the back-and-forth trip until the bikes meet, and the fly meets an unfortunate end. How many kilometers did the fly travel in its total back-and-forth trips?

The dflyanswer = ? is not available indthis vflytfly =sample. (30 km/h)(1 h) = 30.0 km fly = online vfly = 30.0 km/h dbicycle = 10.0 km vbicycle = 10.0 km/h tfly = tbicycle = dbicycle/vbicycle = 10 km / 10 km/h = 1 h There are several variants to solving this problem; consider having students show their differing solutions on the board.


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UNIT 1: MOTION Lab: Galilean Ramp, pages 4-5 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 rightm ost colum n of the lab's data table. Then have them graph speed vs. tim e. Computer Graphing: If using a com puter to form the graph, you m ight 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 m issing. Students m ay 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 m onitor their graphing and guide them to see how a linear fit has little, if any, m ore error than a quadratic fit, so the sim pler 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 sim ply runs through the “m iddle” of the data points as plotted. The students can then determ ine the slope and y-intercept of their best-fit line.

After the second graph is m ade and analyzed, the students are given a brief overview of the types of error in an experim ent and asked to discuss the system atic error in their experim ent. Sam ple answers for those final pages follow.


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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?

Answer availableproportional in this online to sample. Speednot is directly 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 steadily Answer not is available inincreasing. online sample. 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 the proper significant figures.

v = 6.93not t + available 2.07 Answer in online sample.

Em phasize when discussing this graph with the students how the linear graph indicates how the speed is changing; they should be able to be m ore specific than sim ply 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 m ust first realize that they were graphing average speed, not instantaneous speed, versus tim e here).


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UNIT 1: MOTION Lab: Galilean Ramp, pages 4-5 Types of Laboratory Error Type

Examples

Prevention

Discussion

personal error (mistakes)

mis-reading a scale or incorrectly rearranging an equation or calculating a figure

check against lab partners’ work; redo parts of lab as needed when error discovered

none; should be corrected before lab is submitted

systematic error

miscalibration or uncontrolled variables (e.g. friction); includes unavoidable timing errors

calibrate equipment when possible; think through procedures to minimize error

identify any uncontrollable variables (do not include variables causing random error)

random error

estimating the last digit on a scale reading; minor variations in temperature or air pressure

eliminate when possible; can never be completely eliminated

none

13.

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

Answer not available in online sample. 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.

In discussing the lab before it is subm itted, stress to students that a generic statem ent that there was “hum an error” will not suffice. Som e hum an error is random , such as trying to place the ball correctly at the starting m ark or in reading the m eterstick to m ake the m arks along the track. That error should not be discussed. But hum an reaction tim e is usually a system atic error in that there is a delay in the triggering and releasing of the ball. They need to m ake this clear in their discussion. The tracks you use m ay also have obvious im perfections, such as a bowed track, knotholes in a wooden track, etc. These again system atically bias the data and thus should be discussed. Som etim es students m ay find their ball just sits there on a wooden track after release. Obviously that is a system atic error worth noting. The solution one m ight em ploy for this dilem m a is m oving that starting m ark a few centim eters to get away from the surface im perfection, and altering the data table and graph accordingly.


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UNIT 1: MOTION Lab: Galilean Ramp, pages 4-5 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, m iddle, end? (Don't use this term inology 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. tim e graph and observe that distance over tim e m ust yield an average speed that occurs at the halfway point in tim e. Som e will then see that the average speed equaled the instantaneous speed 1/4 of the way down a given run. But you m ust m ake this concrete. So set up a run on a track on your dem o desk and pass out som e stopwatches. Have the students yell out when half of the tim e has elapsed so that the students can see where the ball/cart is located when it is m oving 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. tim e 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 m idpoint, which is the average speed.) This yields their second equation for the year:

Now, you are really playing a gam e here, since you are finding the average speed on a graph of average speed vs. tim e! 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 tim e. In other words, it shows the acceleration of the ball/cart. Let a student com e 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 /s2 , m i/h2 , (m i/h)/s, etc.) and also have them note that the acceleration equation can be algebraically m anipulated to predict final speed: Point out how the best-fit linear equation of the speed vs. tim e graph m atches this form at (the acceleration is the slope, and the y-intercept is the initial speed). They will note that v i is not com ing out zero, which can be attributed to experim ental error. You m ay (or m ay not) wish to point out that their slope is not really the acceleration, since they are graphing average speed vs. tim e and not instantaneous speed vs. tim e. Em phasize how the slope of a distance vs. tim e graph is speed, while the slope of a speed vs. tim e graph is acceleration.


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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-dim ensional forward m otion with positive or zero acceleration. W ait to cover graphs of backwards m otion, slowing down, and so forth until after the students have assim ilated the vector concepts in unit two, when negative slopes on a m otion graph will m ake m uch m ore sense to them . Now is a good tim e to set up a m otion detector as a dem o or group activity and dem onstrate various m otions with real-tim e graphing as you have the students set down into their notes the d, v, and a vs. t graphs for a m otionless object, one m oving 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 m otion detector - here is how the students can be their own m otion 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 m ove the Nerf ball to create the m otion on the graph. Then have a person from each group present to the class the m otion they have decided upon: he or she will m ake the ball m ove in the intended m otion, signaling to the class the tim e when the graph should begin and end. Have the students in the other groups individually or as a group sketch a graph of that m otion, telling them the axes that are to be used (d, v, or a vs. tim e). You m ay want to pick a student to sketch the corresponding graph on the board or overhead. You and the students should expect m istakes. 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 .


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UNIT 1: MOTION Worksheet B: Interpreting Motion Graphs

Unit 1:Motion Worksheet 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?

Answer Speed not available in online sample. 2.

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

Acceleration Answer not available in online sample. 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.

N/A 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/s2 d) m/s3

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/s2 d) m2 /s2


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UNIT 1: MOTION Worksheet B: Interpreting Motion Graphs

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) Time (s) 0 0 20 4.5 40 6.3 60 7.7 80 8.9 100 10

N/A C

12.

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

N/A 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.

N/A A

14.

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

N/A 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:

16.

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?”

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.

Sheavailable is accelerating. Not in online sample. b.

Fill out the table below. Time Interval 2 to 4 seconds 4 to 6 seconds 6 to 8 seconds


Woman's Speed (m/s)

3 Not available in 0 sample. online 1

FO R

U N IT 1: M O TIO N

P AG E 16 O F 18

UNIT 1: MOTION Worksheet C: Combining the Variables of Motion Next you will have the students invent five additional m otion equations, for a total of eight equations they can use in solving kinem atics problem s. W orksheet C rem inds 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. tim e graphs. Let them work in groups to derive the rem aining equations. Give them a hint that num ber 2 uses the answer from num ber 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 form s: 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 substitute for t the rearrangem ent of a = (v f - v i) / 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 tim e is unknown. W ork a couple of exam ples with them and then give them W orksheet D.


FO R

U N IT 1: M O TIO N

P AG E 17 O F 18

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 im portant to insist that students show their givens, equation, substituted num bers with units, and a boxed answer with proper significant figures. Build good habits early on for showing all work.

Worksheet D: 1-Dimensional Motion Problems 1.

The head of a rattlesnake can accelerate 50.0 m/s2 in striking a victim. If a car could do as well, how long would it take for it to reach a speed 24.6 m/s (which is about 55 mi/h) from rest?

2.

0.492 s N/A

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 legal speed limit?

0.417 h or 25.0 min or 1500 s N/A 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. N/A a. At what rate is the car accelerated? – 4.92 m/s2 b. How far does it travel before stopping?

61.5 N/A m 4.

A supersonic jet flying at 200. m/s is accelerated uniformly at the rate of 23.1 m/s2 for 20.0 seconds. a. What is its final speed?

662 N/A m/s 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?

Mach 2.00 N/A 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? N/A 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 N/A 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 3.60×104 km/h 2 . a. How many kilometers long must the runway be?

1.80 N/A km b.

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

36.0 s N/A 8.

A plane flying at the speed of 150. m/s is accelerated uniformly at a rate of 5.00 m/s2 . a. What is the plane's speed at the end of 10.0 seconds?

N/A m/s2 200 b.

What distance has it traveled?

N/A m 1750 9.

A Tokyo express train is accelerated from rest at a constant rate of 1.00 m/s2 for 1.00 minute. How far does it travel during this time?

1800 N/A m 10.

In a vacuum tube, an electron is accelerated uniformly from rest to a speed of 2.60×105 m/s during a time period of 6.50×10-2 N/A seconds. Calculate the acceleration of the electron. 4.00×106 m/s2


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U N IT 1: M O TIO N

P AG E 18 O F 18

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

Suggestions 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 lowfriction dynamics carts with tracks (which can act as long inclined planes) from Pasco. They have various accessories and integrate well with their probeware. 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 1 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

SPECIALTY 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)

Pasco Scientific www.pasco.com 10101 Foothills Boulevard Roseville, CA 95747 1-800-772-8700 Fax: 1-916-786-7565

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)

Vernier Software & Technology www.vernier.com 13979 SW Millikan Way Beaverton, OR 97005-2886 1-888-837-6437 Fax: 1-503-277-2440

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

Inquiry Physics: Equipment Suggestions

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

Page 2 of 2


Unit 1: Motion Student Papers

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

1: Motion

Name

Lab: Galilean Ramp There are fundamental principles governing the motion of all objects, from supersonic aircraft to glaciers. This lab is similar to experiments conducted by Galileo Galilei several hundred years ago 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 of the track and the tabletop is 10.00 cm. You will be varying distance. If necessary, mark off on masking tape the following distances from 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 the same way each time, keep it at the mark on the incline with a pencil until you are ready to release it and begin timing. Don't push or spin the ball/cart when you pull the pencil away! Start a stopwatch as the ball/cart is released and stop the watch when it reaches the stop. Make three measurements of time to as many decimal places as possible, ensuring that 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 the distance equal to 150.00 cm. Make three measurements of time and record them in the table.

3.

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

Unit 1: Motion, Lab: Galilean Ramp

Page 1 of 5

©2010 by G. Meador – www.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 time horizontally on the x-axis and the other variable vertically on the y-axis. This can violate the usual practice of placing the independent 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.

W hat 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 or hyperbolic, etc.)

3.

The shape of a graph illustrates the mathematical relationship between the independent and dependent variables. W hat 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?


Page 2 of 5


6.

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

7.

In question 4, you expressed the basic proportionality between the independent and dependent variables. Your best-fit curve allows you to now express the precise equation for your group's data. Use your graph to fill in the missing values in this equation. Round off the values to the appropriate number of significant figures. d = ________ + ________ t + ________ t2

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


Page 3 of 5


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 the proper significant figures.


Page 4 of 5


Types of Laboratory Error Type

Examples

Prevention

Discussion

personal error (mistakes)

mis-reading a scale or incorrectly rearranging an equation or calculating a figure

check against lab partners’ work; redo parts of lab as needed when error discovered

none; should be corrected before lab is submitted

systematic error

miscalibration or uncontrolled variables (e.g. friction); includes unavoidable timing errors

calibrate equipment when possible; think through procedures to minimize error

identify any uncontrollable variables (do not include variables causing random error)

random error

estimating the last digit on a scale reading; minor variations in temperature or air pressure

eliminate when possible; can never be completely eliminated

none

13.

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


Page 5 of 5


1: Motion

Name

Worksheet A: Calculating Motion 1.

2.

The Spirit and Opportunity robot rovers landed on Mars in 2004 and explored its 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?

2.

Spirit journeyed to Mars in a spacecraft that traveled about 487 gigameters (487×109 m or 303 million miles) from Earth to Mars, averaging about 27,100 m/s (60,600 mi/h). Use the SI units to calculate how many Earth days it took for the spacecraft to complete its journey.

A runner in a 1.00×102 meter race passes the 40.0 meter mark with a speed of 5.00 m/s. a.

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

b.

If 5.00 m/s was her top speed, what is the shortest possible time for her entire 1.00×102 m run?

CONTINUED... Unit 1: Motion, Worksheet A: Calculating Motion


3.

4.

The graph above describes the motion of a golf ball. Note that it graphs distance from a position, not distance traveled. The ball 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?

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?

Two bicyclists are riding toward each other, and each has an average speed of 10.0 km/h. When their bikes are 20.0 km apart, a pesky fly begins flying from one wheel to the other at a steady speed of 30.0 km/h. When the fly gets to the wheel, it abruptly turns around and flies back to touch the first wheel, then turns around and keeps repeating the back-and-forth trip until the bikes meet, 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


1: Motion

Name

W orksheet B: Interpreting Motion Graphs answer questions 1 and 2 in complete sentences 1.

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

2.

W hat 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. W rite the letter corresponding to the appropriate graph in the blank at the left of each question.

3.

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

4.

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

5.

W hich two shapes fit a distance vs. time graph of a motionless object?

6.

W hich shape fits a speed vs. time graph of a motionless object?

7.

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

8.

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

9.

W hich of the following units is equivalent to (meters per second) per second? a) m b) m/s c) m/s 2

10.

11.

d)

m/s 3

W hich of the following units correspond to the slope of a distance vs. time graph? a) m b) s c) m/s d)

m/s 2

W hich of the following units correspond to the slope of a speed vs. time graph? a) m/s b) m•s c) m/s 2 d)

m 2/s 2

CONTINUED ...

Unit 1: Motion, Worksheet B: Interpreting Motion Graphs


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) Time (s) 0 0 20 4.5 40 6.3 60 7.7 80 8.9 100 10 12.

W hich of the following distance vs. time graphs corresponds to the table data?

13.

W hich 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.

W hich of the following speed vs. time graphs corresponds to the table data?

15.

W hich 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. 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?”

BEW ARE:

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

W oman's Speed (m/s)

2 to 4 seconds 4 to 6 seconds 6 to 8 seconds Unit 1: Motion, Worksheet B: Interpreting Motion Graphs


1: Motion

Name Worksheet 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 other equations that would enable you to: 1.

Solve for vf when you know vi, a, and t.

2.

Solve for d when you know vi, a, and t.

3.

Solve for a when you know vi, d, and t.

4.

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

5.

Solve for vf when you know vi, a, and d.

Unit 1: Motion, Worksheet C: Combining the Variables of Motion


1: Motion

Name

W orksheet D: 1-Dim ensional Motion Problem s 2

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

b.

4.

2

A supersonic jet flying at 200. m/s is accelerated uniformly at the rate of 23.1 m/s for 20.0 seconds. a. W hat is its final speed?

b.

5.

How far does it travel before stopping?

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.” W hat is the Mach speed of our jet?

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? 2 225,000 m/s

Unit 1: Motion, Worksheet D: 1-Dimensional Motion Problems


6.

W hat 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 2 rate of 3.60×10 4 km/h . a. How many kilometers long must the runway be?

b.

8.

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

2

A plane flying at the speed of 150. m/s is accelerated uniformly at a rate of 5.00 m/s . a. W hat is the plane's speed at the end of 10.0 seconds?

b.

W hat distance has it traveled?

2

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 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 time -2 period of 6.50×10 seconds. Calculate the acceleration of the electron. 6 4.00×10 m/s 2

Unit 1: Motion, Worksheet D: 1-Dimensional Motion Problems

5


1 Motion

Name

W orksheet E: Quiz Practice Problem s 1.

A fly takes off with an acceleration of 0.700 m/s2 from a wall. How many seconds will it take the fly to reach a speed of 12.6 km/h?

2.

In a. b. c.

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/s2, how much time 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?

the graph: W hat is the speed and acceleration from 0 to 1 seconds? W hat is the speed and acceleration from 1 to 3 seconds? W hat is the acceleration from 3 to 5 seconds? (assume it is constant) d. W hat is the object doing from 5 to 7 seconds?

Unit 1: Motion, Worksheet E: Quiz Practice Problems


Unit 1: Motion Notes

Meador’s Inquiry Physics

Page 1 of 7


Unit 1: Motion Sample Notes

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.

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

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

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.

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



Page 2 of 7

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

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

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=?

Page 3 of 7

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.

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

100 cm m

in

ft

yd

furlong

3600 s

24 h

14 day

2.54 cm

12 in

3 ft

220 yd

h

day

ftnight

= 24,051.53901 furlongs/fortnight

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.

= 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

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.



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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.



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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/s

2

, 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.



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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”.

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.

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.



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?

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

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

BRAKING

t = 0.75 s d=?

a = –10.0 m/s2 vi = 25 m/s vf = 0 d=?

so

vf2 = 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

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.