Introductory labs on the vector nature of force and accelerationStephen E. Kanim and Keron Subero

Citation: American Journal of Physics 78, 461 (2010); doi: 10.1119/1.3373923 View online: http://dx.doi.org/10.1119/1.3373923 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/78/5?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Physics Labs with Flavor II Phys. Teach. 49, 295 (2011); 10.1119/1.3578425 Pile Driver Exercise Phys. Teach. 44, 4 (2006); 10.1119/1.2150748 Inertia and Acceleration Phys. Teach. 43, 332 (2005); 10.1119/1.2033513 Acceleration of a pulled spool Phys. Teach. 39, 481 (2001); 10.1119/1.1424598 Work and variable force: A classic chain problem Phys. Teach. 39, 37 (2001); 10.1119/1.1343429

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Introductory labs on the vector nature of force and accelerationStephen E. Kanima� and Keron Suberob�

Department of Physics, New Mexico State University, Las Cruces, New Mexico 88003

�Received 16 September 2009; accepted 7 March 2010�

We discuss the use of long-exposure digital photography in introductory mechanics laboratories.Students at New Mexico State University use inexpensive digital cameras to record the motion ofobjects with attached blinking light emitting diodes. These photographs are used to make inferencesabout the velocity and acceleration of the moving object. We use the analysis of these photographsto promote student understanding of the vector nature of kinematics quantities. In subsequentlaboratories we build on this understanding to help students relate the acceleration vector for amoving object to the net force vector for that object. We give details about the equipment we use anddescribe the sequence of activities that we have developed for a two-dimensional motion laboratoryand for a laboratory on Newton’s second law. Finally we present some pre- and post-test data onquestions related to the concepts underlying these laboratories. © 2010 American Association of PhysicsTeachers.

�DOI: 10.1119/1.3373923�

I. INTRODUCTION

The laboratories that we describe in this paper were moti-vated by a desire to improve students’ conceptual under-standing of Newton’s second law as an equation relating twovector quantities. In addition, we wanted students’ initial ex-posure to physics laboratories to exemplify physics as theapplication of a small number of general principles.

In a study of concept interpretation in physics, Reif andAllen1 outlined five steps that are necessary to apply thedefinition of acceleration to the motion of a particle: �1�Identify the velocity v� of the particle at the time of interest t;�2� identify the velocity v�� of the particle at a slightly latertime t�; �3� find the small velocity change �v� =v��−v� of theparticle during the short time interval �t= t�− t; �4� find theratio �v� /�t; and �5� repeat the preceding calculation with atime t� chosen progressively closer to the time t. They notethat understanding and applying the definition of accelerationhave many complexities that are hidden by the seeminglysimple definition a� =dv� /dt, for example, the procedure forsubtracting vectors.

Reif and Allen suggested that the procedural knowledgenecessary to interpret the definition of acceleration is usuallynot adequately taught. They recommended that the presenta-tion of operational definitions of concepts such as accelera-tion be followed by the opportunity for practice with a vari-ety of specific cases. This emphasis on operationaldefinitions has been reiterated by Arons2 and by Shaffer andMcDermott.3

As part of their study, Reif and Allen asked students in anintroductory physics course for scientists and engineersqualitative questions about the directions and relative mag-nitudes of accelerations. Even though these students hadbeen using acceleration as part of the course for at least twomonths, none were able to answer more than three of 13questions correctly. In answering the same questions, physicsfaculty members rarely used the definition of accelerationand often used information about forces to answer questionsthat were purely kinematical. Both experts and novices haddifficulty applying the definition of acceleration to objectsmoving in a curved path. They tended to use case-specificknowledge about acceleration rather than a more general

definition, and both groups often used this case-specific

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knowledge incorrectly. Shaffer4 found similar difficulties.Asked to indicate the direction of the acceleration of a swingat five points along its path, none of the 124 students in acalculus-based introductory physics course answered cor-rectly for all points. Of the 22 physics teaching assistantswho were asked the same question, only three drew the ac-celeration correctly for all five points. Similar results wereobtained on physics comprehensive exams for graduate stu-dents.

Our goal for the two laboratories were for students to gainpractice at using the definition of acceleration to determineits direction and to reason about its relative magnitude atvarious points for objects moving in two dimensions; recog-nize that Newton’s second law requires that the direction ofacceleration and of the net force be the same; and be able touse kinematics to make inferences about the relative magni-tudes of forces acting on an object. Our epistemological goalwas to have the laboratories contribute to students’ sense ofthe coherence of the kinematics and dynamics they werelearning.

II. PREVIOUS CURRICULA ONTWO-DIMENSIONAL MOTION

Because we wanted to design a laboratory experience thatallows students to apply the definition of acceleration, ourMotion in Two Dimensions laboratory was modeled after thepencil-and-paper “motion diagram” approach described byVan Heuvelen5 and found in Physics Active Learning Guide6

and in the tutorial Motion in Two Dimensions from Tutorialsin Introductory Physics.7 Huggins8 recently described a simi-lar technique starting with stroboscopic photographs. In allof these curricula, students first draw velocity vectors thatshow the direction and relative speed of an object as it movesalong a path. Change-in-velocity vectors are then constructedfrom the velocity vectors, and the acceleration vectors aredetermined from the change-in-velocity vectors.

There have also been efforts to strengthen students’ under-standing of kinematics and dynamics in two dimensionsthrough laboratory activities. For example, laboratories atNew Mexico State in the 1960s included analysis of two-dimensional motion with strobe photographs of dry ice pucks

on a glass surface. Subsequently, spark timers were used to

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analyze motion on an air table.9 Others have used frame-by-frame analysis of video recordings �including peak storage10

and video capture11 techniques�. Although the student exer-cises described in video capture papers typically includeanalysis of motion in component directions, in principle amotion diagram approach could be applied to data collectedthrough video capture.

III. LABORATORIES USING LONG-EXPOSUREDIGITAL PHOTOGRAPHY

The laboratories require a digital camera with a manualsetting or a “shutter priority” setting.12 Students photographa moving flashing light emitting diode �LED� using exposuretimes of between 1 and 4 s. By using flashing LEDs withonly short “off” periods, the students create photographswith light streaks that look very much like motion diagrams.Terzella et al.13 described a similar photographic techniquefor measuring the acceleration of gravity.

We use lower resolution used cameras14 that we buy on-line for about $60. Although we now have one camera foreach laboratory group, we have run the laboratories with asfew as three cameras for 20–24 students without causingsignificant wait times because the photographs do not takevery long to make and download. There are instructions forhow to use the cameras at each laboratory station. Studentsare usually not familiar with using a shutter priority setting,but have no difficulty downloading the pictures and sendingthem to the black and white laser printer. The cameras useAA batteries, which often need to be recharged during a 1week laboratory cycle. Software at each laboratory station�currently Adobe Photoshop Elements� allows students to in-vert the photographs before they print them so that the darkbackground is printed as light gray and the light streaks areprinted as dark lines. This inversion saves toner and makes iteasier to write on the prints. Additional equipment includescables for downloading the pictures from the camera, an in-expensive gooseneck lamp at each laboratory station �so thatstudents can read and write while other groups are takingphotographs�, and several tripods.

We tried a variety of flashing light sources15 before decid-ing to build our own so that we can choose a flash rate andcontrol the fraction of each period that the LED is on �theduty cycle�. Each blinkie has an on/off switch, a secondswitch to change the flash rate, and a trimpot variable resistorto control the duty cycle �Fig. 1�. In the future we will prob-ably replace the variable resistor with two fixed resistors thatset the duty cycle at about 90%. The circuit is powered bytwo CR2032 3-V disk batteries installed in a holder on theback of the board. We have not yet had to replace a batteryexcept when the circuits were accidentally left on. The cir-cuits are about 2.4 cm wide and 4.8 cm long.

For the Motion in Two Dimensions laboratory, circuits aremounted on a toy hovercraft, on a Pasco roller coaster cart,and in a PVC plumbing pipe cap used as a spherical pendu-lum bob, as shown in Fig. 2.16 For the Newton’s Second Lawlaboratory, the blinkie circuits are also mounted onto a thinboard that is used as a physical pendulum and onto a blockof wood cut to fit a 1.2 m Pasco track. The Pasco track wasmounted on a hinge so that we can adjust its angle to the

horizontal.

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A. Two-dimensional motion laboratory

Because the laboratory is used in an algebra-based course,students find average accelerations over short time intervalsrather than instantaneous accelerations. In the Motion in TwoDimensions laboratory we guide students through the appli-cation of an operational definition of average acceleration,a�ave=�v� /�t. After finishing this laboratory, we would likestudents to be able to subtract one vector from another anduse information about the velocity of an object along a pathto find the direction of the change in velocity for a short timeinterval. We also want them to recognize that this direction isthe direction of average acceleration.

Fig. 1. Schematic for blinkie circuit. Switch S1 turns the blinkie circuit on;S2 increases the time constant. The potentiometer controls the fraction oftime in each cycle that the LED is on.

Fig. 2. Equipment for Motion in Two Dimensions laboratory. Clockwisefrom lower left: Blinkie circuit boards, roller coaster cart, spherical pendu-

lum bob, and toy hovercraft.

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As described in Ref. 5, the procedure for using motionmaps begins with drawing velocity vectors along the object’spath. The change-in-velocity vectors for the points of interestare found by subtracting the vector just before a point fromthe vector just after that point, which allows a qualitativedetermination of the direction of acceleration at that point.We modified this procedure for our laboratories as follows. Along-exposure photograph of an object with a blinkie at-tached shows the path of the object. The photograph is in-verted and printed, and students draw vectors with tails at thebeginning of each light stripe and heads at the end. Eachvector they draw represents the displacement s� during thetime that the blinkie was on. Because each of these vectorsrepresents a displacement for the same duration, their lengthsare proportional to the average speed of the object and can beused as average velocity vectors for the time intervals thatthe blinkie is on. Two adjacent vectors can be used to deter-mine the change in the �average� velocity from one lightstripe to the next: We assign this change-in-velocity vector�and the associated average acceleration� to the space be-tween adjacent vectors. The lengths of these vectors allow aqualitative comparison of the magnitudes of the accelerationfor various points along a path.

In the first part of the laboratory the operation of theblinkie circuit is described, and an example photograph isgiven. A drawing is shown of what a blinkie trail looks likefor a hovercraft that is kicked so that it changes direction andspeed, and students are asked to make inferences about thespeed of the hovercraft based on measurements taken fromthe drawing. Students draw scaled vectors to represent thehovercraft’s velocity before and after the kick. They areguided through the procedure for graphical vector subtrac-tion and then determine the change-in-velocity vector for thehovercraft. The direction of the acceleration of the hovercraftdue to the kick is determined. Students then find the change-in-velocity vector for a portion of a drawing based on ablinkie photograph, and are asked some qualitative questionsabout this vector.

Following this extended introduction, students take long-exposure digital photographs of three objects and determinethe acceleration direction for each motion following thesame procedure. Because we have only one station for eachexperiment and there are up to eight laboratory groups, welet groups take these photographs in any order. The motionsof a spherical pendulum, a hovercraft on a ramp, and a toyroller coaster are intended to illustrate that the same proce-dure and definition are applicable across a range of motions.For each of these motions students are asked to print aninversion of the photograph they obtained and find the accel-eration vector for various points along the path. We brieflydescribe details of each laboratory activity.

1. Central force motion: Spherical pendulum

A spherical pendulum approximately 2 m long is sus-pended from the ceiling. The LED protrudes through a holein the center of the bottom surface of the pendulum bob. Thecamera is placed on the floor �about 1.5 m below the pendu-lum bob� pointing up toward the suspension point of thependulum. Students push the pendulum bob so that it movesin an ellipse. Figure 3 is a photograph of this motion; with anexample of the vector construction we expect our students todo to find the direction of the change in velocity. The point

of suspension can be seen in the photographs, and when

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asked to give a description for the acceleration direction thatis true for all points on the path, students generally state thatit is toward the center of the ellipse.

2. Parabolic motion: Hovercraft on a ramp

Students push a toy hovercraft so that it travels in a pa-rabola along a piece of plywood with one edge raised so thatit is at about a 4° angle. The camera is mounted on a tripodthat is placed at the bottom of the plywood ramp. The ramphas a grid painted onto it so that the pictures that are takenhave horizontal and vertical reference lines. A long-exposurephotograph taken of this motion is shown in Fig. 4.

For this laboratory activity the camera is further from thetop of the plywood ramp than from the bottom, and hence alight stripe in the photograph will be smaller for a hovercraftin motion near the top of the ramp than it will be for ahovercraft moving at the same speed near the bottom. If thevector generated from a light stripe at one location is sub-tracted from one generated at another location, an error in themagnitude and direction for the change-in-velocity vector isintroduced. Usually, these errors are small enough that theydo not prevent the students from concluding that the accel-eration direction is down the ramp, parallel to the grid linesin the photograph. We intend to modify this laboratory sothat the camera is mounted above the plywood and is pointeddirectly down to minimize this distortion.

Fig. 3. Photograph of spherical pendulum moving in an ellipse. The blinkieflash rate is about 10 Hz, and the exposure is 1.5 s, less than the time of acomplete orbit to avoid overlap of stripes. Superimposed on the photographare examples of vector differences based on the length and direction of thelight streaks just before and just after the point of interest.

Fig. 4. Long-exposure photograph of the toy hovercraft moving in a para-

bolic path. The blinkie flash rate is about 3 Hz, and the exposure is 3 s.

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3. General two-dimensional motion: Roller coaster

A toy roller coaster cart travels down a flexible track andis launched into a container filled with packing peanuts. Thecart has a blinkie circuit mounted on the top with the LEDplaced through a hole in the side of the cart close to thecenter of mass. The resulting photograph �see Fig. 5� is thebasis for the initial homework problems associated with thislaboratory.

Because the laboratory generally takes less than 2.5 h,many student groups choose to do this part of the homeworkin the laboratory room. Subsequent homework problems givestudents practice with graphical vector subtraction, withfinding a change-in-velocity vector from two velocity vec-tors, and with determining the direction of an accelerationvector for objects moving along various paths.

B. Newton’s second law laboratory

The next laboratory, Forces, introduces weight, normalforces, tension, and friction. The directions of these forcesand the factors affecting these forces are explored, and stu-dents practice drawing free-body diagrams. The fifth labora-tory, Addition of Forces, provides practice with vector addi-tion, reasoning about the relative magnitudes of forces forsituations where the net force is zero, and culminates in ex-ercises using the force table. Newton’s Second Law, the sixthlaboratory in the sequence, requires students to reason aboutrelative force magnitudes for situations where the accelera-tion is nonzero.

The initial exercises for the laboratory are also pencil-and-paper. Students use a photograph of the two light stripesadjacent to the bottom of the roller coaster to determine thatthere is a large change in velocity and therefore a large ac-celeration upward at that point. They draw a free-body dia-gram of the coaster at the bottom of the track and are guidedthrough the reasoning required to compare the magnitudes ofthe normal force on the cart and the weight at that point.

Students take photographs of circular motion with chang-ing speed, linear motion with acceleration both in and oppo-site to the direction of motion, and motion with no accelera-tion. For each motion they determine the direction ofacceleration, draw a free-body diagram for the object at oneor more points, and use the direction of the acceleration toreason about the relative magnitudes of the forces in the

Fig. 5. Long-exposure photograph of a roller coaster cart traveling down thetrack and then launched into a catch basin. The blinkie flash rate is about 20Hz, and the exposure is 2.5 s. Students are asked to find the accelerationdirection for points A-E as a homework problem.

free-body diagrams. By focusing on a common procedure,

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we emphasize that Newton’s second law is a unifying prin-ciple that applies to all motions and unites much of what theyhave studied up to this point.

1. Physical pendulum

The physical pendulum is a board approximately 2 m longwith a blinkie circuit at one end. From a photograph of theswing �see Fig. 6�, students determine the direction of theacceleration at two points. They draw a free-body diagram ofthe blinkie circuit for these points and are asked to determinethe relative lengths of the tension and the weight vectors thatresults in a net force direction consistent with the accelera-tion.

2. Constant velocity: Hovercraft on a level surface

The toy hovercraft is pushed across a plywood sheet thatis level on the floor. Students take a 2 s photograph of thehovercraft and determine that there is no change in velocityand therefore no net force. From a free-body diagram of thehovercraft, they reason about the relative magnitudes of theforces on the hovercraft.

3. Linear motion: Block on a ramp

For the last two activities of this laboratory, a woodenblock with a blinkie attached slides down a 1.2 m long Pascotrack used as a ramp. In the first case, the wood is in directcontact with the track, and the block speeds up after it isreleased. In the second case, the block is turned over so thata cork sheet attached to it is in contact with the ramp. Withthe same ramp angle, the larger coefficient of friction be-tween the cork and the track causes the block to slow downafter students give it an initial push �see Fig. 7�. For thesetwo cases students determine the relative magnitude of thethree forces in the free-body diagram consistent with a netforce in the direction of the acceleration. They then comparethe forces for the case when the block is speeding up to the

Fig. 6. Photograph of a single swing of the physical pendulum. The blinkieflash rate is about 10 Hz, and the exposure is 1.5 s. Students are asked tofind the acceleration at points A and B.

forces when the block is slowing down.

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The homework for this laboratory asks students to deter-mine the acceleration direction for a variety of situations andto adjust the vector lengths for forces on the free-body dia-gram so that the direction of the net force is the same as thedirection of the acceleration.

IV. EFFECTIVENESS OF LABORATORIES

The introductory physics laboratories at New MexicoState are not strongly tied to the lecture instruction for thecourse: Students receive separate grades for laboratory andlecture. The results we report here are for students enrolledin lecture sections with traditional instruction. Free-responsepretests are given at the beginning of each laboratory to pro-vide a sense of students’ initial understanding of the labora-tory topics. We use qualitative questions on the multiple-choice laboratory final as a crude measure of students’understanding at the end of the semester. The laboratory finalaccounts for 20% of students’ laboratory grade. It is givenduring the last laboratory meeting and for most semesters hashad 36 questions �three per laboratory�. Students may takeup to the full 3 h laboratory period to complete the final, butmost are done in 0.5 h.

There are only a few questions on nationally administeredassessments that ask for information about force based onknowledge of acceleration. We report results from two ofthese questions in the following. Most questions about forceand acceleration on these assessments are for one-dimensional motion or are for one component of parabolicmotion and do not test for understanding of force and accel-eration as vector quantities. Because the focus of the labora-tories we have described is on the vector nature of force andacceleration, most of the assessment questions we haveasked were written by us or are questions that have beenasked as part of other research projects into student under-standing of two-dimensional kinematics and dynamics.3,17,18

A. Questions about acceleration in two dimensions

When questions about the direction of acceleration for anobject in two-dimensional motion are asked on a pretest,

Fig. 7. Photograph of the wooden block as it slows down on an inclinedramp. The blinkie flash rate is about 10 Hz, and the exposure is about 2 s.Here the cork surface is in contact with the metal track. When the block isplaced with the wooden side in contact with the track instead, the blockspeeds up.

about 4% of our students answer correctly, which is consis-

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tent with pretest data reported in Ref. 3. The question shownin Fig. 8�a� from the Mechanics Baseline Test asks for thedirection of the acceleration of the block when it is at theposition shown. Reported results19 range between 12% and18% correct after standard university instruction. About 39%of the 135 students in our algebra-based course and 51% ofthe 88 students in our calculus-based course answered thisquestion correctly on the laboratory final. Nagpure17 gaveresults for a question about a dog that is speeding up as itmoves along a curved path �see Fig. 8�b��. After instructionusing a modified version of the Motion in Two Dimensionstutorial, 44% of the students in an algebra-based course and69% of the students in a calculus-based course at the Uni-versity of Maine answered correctly. We asked a multiple-choice version of this question on our laboratory final: About20% answered correctly in both algebra- and calculus-basedsections. For a similar question about an object changingboth direction and speed, about 60% of the students in acalculus-based course at the University of Washington an-swered correctly after tutorial instruction, as reported in Ref.3.

B. Questions relating force direction to accelerationdirection

One question on the Force Concept Inventory20 asks for acomparison of forces based on information about accelera-tion. For an elevator moving upward at a constant speed �seeFig. 8�c��, 61% of students recognized that the tension in thecable would be equal to the weight of the elevator after stan-dard instruction at Harvard University. About one-third ofthe 223 students in both courses answered this question cor-rectly on our laboratory final exam. For an object that isspeeding up as it moves in a curved path �see Fig. 8�d��,about one-quarter of the students in our laboratory coursescorrectly chose the direction of the net force �about one-thirdgave a direction toward the center of the curve�. We obtainedsimilar results after extensive practice in the lecture portionof a course at New Mexico State.18 We do not have compa-rable data from other institutions.

Our results about acceleration directions are not encourag-ing. Our pretest results are consistent with results reportedelsewhere and suggest that without intervention, only a small

Fig. 8. �a� Question from Mechanics Baseline Test on acceleration in two-dimensional motion. �b� Question about acceleration direction for a dogspeeding up. �c� Question from Force Concept Inventory requiring reason-ing about forces based on information about kinematics. �d� Question aboutdirection of net force for an object changing direction and speed.

fraction of students are able to determine the direction of the

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acceleration for general two-dimensional motion. There issome improvement in student performance from pre- to post-test. However, most students do not seem to apply the pro-cedural knowledge we are attempting to encourage withthese laboratories.

Results are also poor for questions requiring reasoningabout forces based on knowledge about acceleration direc-tion. Our post-testing indicates that most students in ourlaboratory do not develop a functional understanding ofNewton’s second law as an equation relating two vectorquantities.

V. CONCLUSION

Previous research indicated that most students cannotreadily determine the direction of an acceleration after stan-dard instruction. The laboratories we have developed incor-porate motion diagrams into the laboratory in an effort tostrengthen students’ understanding of the procedure for find-ing acceleration. In addition, we want to encourage the useof this procedure for determining acceleration direction aspart of reasoning about force directions and magnitudes.

Are these laboratories an improvement over video motioncapture techniques? The same procedure for finding the di-rection of acceleration could be incorporated into a labora-tory that uses stroboscopic photographs or video motion cap-ture. However, the long-exposure photographs described inthis paper seem to be much more directly amenable to treat-ment as motion diagrams: Even before the picture is down-loaded, the entire path of the object appears in a single pho-tograph, with equal time intervals marked. In addition, theequipment required is relatively inexpensive.

Students do not have significant difficulties with the labo-ratory procedures or with the equipment, and most like mak-ing the photographs and are able to perform the requiredanalysis. However, post-testing of student understanding ofthe underlying concepts has yielded disappointing results. Asa stand-alone intervention, the laboratories seem to be onlymarginally successful, as measured by our multiple-choicelaboratory final.

Before these laboratories were developed we attempted toaddress the same instructional goals by extensive use of mo-tion maps as part of lecture instruction.18 This use also metwith limited success: In the near future we hope to teach alecture section that emphasizes the vector nature of kine-matic and dynamic quantities with these laboratories in placeas reinforcement of this approach.

Newton’s second law is central to most introductory me-chanics courses. Yet it seems that most students in thesecourses leave without an understanding of how to apply thedefinition of acceleration except in special cases and withoutrecognition of the vector nature of force and acceleration.Our results suggest that reasoning about vector quantities isdifficult for most students. We hope that further refinementof the tools we have developed can contribute to improvingstudent understanding of Newton’s second law.

ACKNOWLEDGMENTS

We would like to thank our collaborators, Luanna Gomezat Buffalo State College and Michael Loverude at California

State University, Fullerton, for their ongoing assistance with

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the development of these laboratories. In addition, we aregrateful for the patience and for the assistance of the NewMexico State Laboratory Coordinator, Christine Pennise.This curriculum development project was supported througha CCLI grant from the National Science Foundation CCLIProgram �Grant Nos. DUE-0341333, DUE-0341350, andDUE-0756909�.

a�Electronic mail: skanim@nmsu.edub�Electronic mail: suberok@nmsu.edu1F. Reif and S. Allen, “Cognition for interpreting scientific concepts: Astudy of acceleration,” Cogn. Instruct. 9 �1�, 1–44 �1992�.

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4P. Shaffer, “Research as a guide for improving instruction in introductoryphysics,” Ph.D. dissertation, University of Washington, Seattle, 1993.

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11 See, for example, P. Laws, “Calculus-based physics without lectures,”Phys. Today 44 �12�, 24–31 �1991�; W. Benenson and W. Bauer, “Framegrabbing techniques in undergraduate physics education,” Am. J. Phys.61 �9�, 848–851 �1993�; R. Beichner, “The impact of video motionanalysis on kinematics graph interpretation skills,” ibid. 64 �10�, 1272–1277 �1996�; P. Laws and H. Pfister, “Using digital video analysis inintroductory mechanics projects,” Phys. Teach. 36 �5�, 282–287 �1998�;D. Brown and A. Cox, “Innovative uses of video analysis,” ibid. 47 �3�,145–150 �2009�.

12A good source of up-to-date information about available cameras is Digi-tal Photography Review, �www.dpreview.com/�. This website allows fil-tering so that only cameras with manual modes and adequate exposuretimes are listed.

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14We use Canon Powershot A60 �2 megapixel� and A70 �3.2 megapixel�cameras. We prefer older cameras because lower resolution pictures takeless memory and less time to print.

15The most promising of these sources were the novelty “blinkies” avail-able for less than a dollar. However all of the blinkies that we tried hadtwo or more LEDs that flashed alternately, and although each single LEDflashed at a steady rate, the time interval from one flash to the subsequentone alternated. There are also LEDs available with built-in flashing cir-cuits, but the flash rate is typically about 3 Hz, a bit slow for our experi-ments.

16The toy hovercraft is Kick-Dis, manufactured by Estes. The roller coastercart and the track on which it travels are from Pasco.

17B. Nagpure, “The effects of reasoning about vector components on stu-dent understanding of two-dimensional acceleration,” M.S. thesis, Uni-versity of Maine, Orono, 2008.

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