Introduction to Vectors 11

A Hands-on Introduction to Displacement/Velocity Vectors andFrame of Reference through the Use of an Inexpensive Wind I IF

Toy QP a \f@RiB8:1ttH9 WQrizgptal GriV

Gwen Saylor, Department of Physics, State University of New York - Buffalo State College,1300 Elmwood Ave, Buffalo, NY 1422~SaYIOr@acSdny.or~

Acknowledgements: This paper is submitted in partial fulfillment of the requirements necessaryfor PHY690: Masters Project at SUNY - Buffalo State College wlder the guidance of Dr. DanMaclsaac.

Introduction to Vectors I 2

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terminology common in the New York State Regents Ph~>ics curriculwn through a focus ondisplacement and velocity vectors:\Vith inexpensive equipment student are able to visualize thetip to tail method of vector addition, deter line the horizontal and vertical components of vectorsand observe the combination of two COl lIrrent parallel or perpendicular vectors. Studentsobserve the motion of a toy all a mov' g grid to establish the concept of frame of reference forrelative motion. The terminology d level of difficulty are geared toward a Iligh school Regentsclass. Through guided activity IVO sheets, student groups explore scenarios that frequently ,. Lappear on the New York State Regen~Pi4 IVam related to vectors. ~ /5'~ "'

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Introduction: _.J IT"'" £I C eo., Ie r ( c,tIf . Throughout the year r~dents a.!!,..confromed with problems that require combiningI'--t. ct ....,~ e.. ., \ parallel or perpen ICU ors quantities and resolving vectors into horizontal and vertical

. Vectors are a HPl'w§"1 sktlW8( fill t tN~; sf ~hlilCS Students I,ave greater dIfficulty Ie.~ components. Students are also tasked to graphically solve for the resultant oftlVo vectors usingwIth vectors than IS reflected mmany lugh school cumculums. ~ure to address the central .C { • ~ lIther head-to-tail or parallelogram methods for scaled vectors that are not necessanly alIgned torole of vectors in problem solving early on in the curriculum lea s sludellt diilictUty wIth C Co,... I t e the honzontal 01 the vertIcal axesproblem solving throughout the entire year. This paper summarizes a method for i ~~I

vectors early m the Regents PhYSICScumculum a - ce erated motIon The _ \ While questIons m the student worksheet are occasIOnally above the dIfficulty of theltles mtroduce the vector SkIlls and vector tenmnology that are appropnate to the ~York Co ~ t.k - Regents cUfflculwn they are deSIgned to confront common mlsconceptlOns that lead to

ts PhYSICSCore CUfflculum The mtroductlon of vector quantitIes and vect~r ~ challenges as students develop a greater understandmg of vectors m over the course of the year.oper 0 0 I P acement and velOCIty scenanos wbile~romotIng skills that appl \ ,. , If vector presentations are linlited solely to those presented on past Regents Physics exams,proJecttl erposltIon off<tces, momentum and fO~ ft~ i\ ~ students may develop misconceptions. In order for a student to transition from the basic level of

\ 0\~ or lI.')t ~ 1/ It rt :\,'f S,J functionality listed in Key Ideas 5.1a-c in~Table ~o mastery of skills and concepts in the\; \' ~tt.( The two tIv S p~ this docwnf~t, "ACtIVIty One Ladybug TranSIt" and c: ~.' J. or Is . remainder of the curriculmn, learners must be able t~ aemonstrate the following skills and

4)""r ,~~ "AchvltyTwo:Lad aConveyorBelt"p~mstructIonaltools~~~makevector '" rf? ¥ ctI wlderstanding:~C<~ characteristics both expl it an ighly vis or learners. Each activity~mducted in tbe t I " -' pat ~.'I'., space of a student desktop, ~ing ~aCh group member. Students observe the '1 ~ ""at . Define terms such as dIsplacement, velOCIty, resultant, equilibrallt and componenJ,.s\-;--1' • motion of the toys and apply Prio\~~ge from the kinematics unit, geometry and algebra. f~ 71 . Establish the relatIonship between compone,£ctood the resultant vectors Itfi:dmg

r;..e.J. ~ Activity One "Ladybug Transit" a1YSwsstudents to resolve vectors into horizontal and vertical ~ - # 7'J7 the concept of addItIve lllverse (Arons, 1995, g 107 ----f ~ e« r-eo. ~ \ components by graphical meth,ods and introduces the equations necessary for the analytical \ ~ ~ rlJ'Jlf>t , .• . Define the meanmg ofa negatIve vectors m re aMn to the honzontal and vertIcal axes

commlltation of vectors. During activity two, "Ladybug on a conveyor belt", students observe ~ M • Understand that vector quantItIes are not fixed to a locatIOn (Brown, 1993)~\JY\. the combination of concurrent parallel or perpendicular vectors using a moving toy and a moving ~ . <.\ k I:\ .. ~~. grid to model two dimensional motions. By observingthetoy relative to the grid and relative to /' ~.Rrylew: J:>4 r ~(O>l "1&-

A "'~\ ~ the fixed frame students WIll learn that both path and velOCIty are dependent on the frame of In the same way that physics education has focused on modifying instruction to explicitly'v "C"..::r \ reference. GUIded quesl10ns are used throughout the actIVitIes to support concept development, address student misconceptions related to forces through the Force Concept Inventory introduced l---

vocabulary acqUISItIon and skill application. by Hestenes, Wells and Swackhamer (1992), physics instructors must al,so t~steps to address J" S

;Y Vectors in the New York State Regents Physics Curriculum student preconceptIons With vector quantIl1es. The Vector Know/edge lest, mll1lstered to \ J 'introductory college level physics courses comprised of primarily science ma rs, revealed that 'fh rS

. The following chart (See Table 1) summarizes the portions of the Standards from nearly half of the students who reported prior exposure to vectors from hi 'h sc 001 physics or k,/1Mathematical Analysis and Scientific Inquiry that relate to vectors in the New York State math entered the class with no useful knowledge of basic vec~r skill Knight, 1995 High e IC,~J..Physics Core Curriculwn. scbool teacbers no doubt struggled to present the matelial {Ju~st.:'?en\S did not develop any long (~

('~e~:~~e:;;'1l0~ Q./f r"l/~re slvl,.lr ~ Hf p4 fler $1";" LJIn probing student preconcepti6'ns about vectors, Agwrre presented seven vec'"lr V~~O J"

characteristics that require explicit instruction (Aguirre and Erickson, 1984). A common theme r-to the seven characteristjo:s outlined in Aguirre's research is the role of the reference frame in ~ If'W'unde"ding ve~n ~e independence of each component vector. Based on interviews and • IA9;)'ac~~i~, Aguirre :McI~~ that students commonly held preconceptions regarding vectors '\ _clude the folio mg: (.~&i; I,~ \0\\""-

Speed and displacement are independent offrame of reference ~ .~-It e J t{ II' Vector components act sequentially rather than simultaneously I '"'C ,\.t-_µ .Time is different for the resultant path than for the components vo v Sd ~r ( .)~\"

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Table 1: Vector Skills From the J'I.'YSPhysics Core CurriculumStandard 1: Mathematical analysisKey Idea 1: Abstraction and symbolic representation are used to communicate mathematically

use scaled diacrams to reoresent and manioulate vector QuantitiesStandard 4: Scientific InQuiryKey Idea 5:Energy and matter interact through forces that result in changes in motion.5.1a Measured Quantities can be classified as either vector or scalar.5.1b A vector may be resolved into perpendicular components.5.1c The resultant of two or more vectors, acting at any angle, is detemlined by vector

addition(NYSED, 2008) Full text available at http://www.pI2.nysed.gov/ciaiimstJpub/phycoresci.pdf

Introduction to Vectors I 5

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~ m":~o",,~.~"b ,~~Lm~:.~o;!-Fm::::.:~::::::I ,I ~.the two points shown to you" using strings, protractors and measurement devices such as metersticks or measuring tapes. The points in this lab are on opposite sides of a large structure suchas the school building. As students engage in the problem solving they must decide whether theyshould focus on resultants or components of the vectors to a common point. This method leadsstudents to a discovery of the orthogonal components of vectors. While the approach providesan exploratory activity it is logistically inlpractical for many school buildings due to size orstudent supervision requirements. Similar activities can be completed with the aid of a GPS withgreat accuracy (Larson, 1998) .

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~r':!flmmendations from~ analysis of the Vector Knowledge Test ~ 'Yc;.;' f~ suggest ~~rs should be introduced over a course of several weeks prior to introduction /..J2~ ~of projectiles or Newtonian mechanics. Subsequent investigations using diagoostic testing of 7.- r -~-

introductory college students noted that students demonstrated some intuitive knowledge of ~vectors but lacked the ability to apply skills such as tip-to tail and parallelogram methods of rvector addition (Nguyen and Meltzer, 2003). U~.t d.o

.Many physics textbooks present vectors during a unit on forces and then transition l/'

quickly 'iOofuer quantities such as velocity, acceleration, momentWl1 and displacement. From a 'fl:) v uk r .JJ. 7students' viewpoint "adding velocity arrows appears very different from adding displacement ~;f c/. n--1 •arrows, and acceleration arrows are totally incomprehensible" y\fO~S' 1997, P 107). , I• J r. . . t l,k.( +. bU.-t. (! ,K1i)

DIsplacement vectors In some scenarIos are succe lve rather an concurrent. As aresult displacement vectors are the simplest starting poi t; however, as instructors transition from f~t.displacement vectors to force vectors, students are bol~onfused unless the nature ofeach oftbese quantities is discussed (Roche, 1997).~usibe taken in curriculum planningto allow adequate discussion and exploration with the addition of each new vector quantity.Students who grasp the idea of vectors as they relate to velocity p~e..signjf1'\ant difficultytranslating those skills to acceleration. Shaffer and McDermott\'oflmjl~~ s6rvey onintroductory physics students, graduate students and physics TA's, that the ability to correctlydralV and label a velocity vector was markedly greater than the number of students who wereable to correctly draw and label an acceleration vector (Sh;l&ilrad 1.!cBcilholl. 2~

A number a1ivities are widely used by teachers to introduce vectors to students.Some of these acti It~S represent val uable experiences for students but would be more effe,during the conce t application of a learning cycle rather than during the exploratory orconceptual dey. lopment phases (Maier and Marek, 2006).

Vector treasure hunts are a popular method used to introduce vectors. In tlus ~ • :./.nvestigation students use a compass to create a treasure map using vectors. The map is ~ Ipassed to another group for them to follow (Windmark, 1998). This method requlf~orknowledge of head to tail addition. Whereas the method used for with the ~y allows •tudents to discover head to tail addition in the process of exploration. f' ('V· 0 uS

Another corrunon way of addressing two dimensional vectors is to demetant tJJ' <velocity car moving on a piece oflong thin craft paper (Mader and Winn, 2008, 101-5). he /I P,4orientations of the car and paper are varied to illustrate the effect of direction an . e of fthe component vectors on the resultant displacement and velocity of the bug lIS ach D .. ;'",$.j.e./~provides a good visual but d low for e ec I . ata collection by allmembers of the gro . n the past I have found this demonstration e ective for some students;however, it h ot served well a c~mmon expenence to reference with students. While thisdemonstra on is useful in reinforcingne Independence of horizontal and vertical motion for theapplication ase of vectors it does not explicitly address man of the other aforem n .misconceptIons ~

Required Student Prior Knowledge and Curricular Rationale for Activities e'" ~ ~ r ~

The aclIvltles presented III this document are Illtended as an mtroduclion to vectors for ~ S' ~Regents PhYSICSstudents WIth little to no prIor exposure to vector quanti lies ActIVitIes are d~S' c.~ j

deslgoed to mtroduce vector vocabulary as It IS applied on the NYS Regents Exam and pro Ide a'-OO ~

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VISUalIntroducnon to the charactenstlcs of vector quantllIes to be used tllfoughout the ye ~~ We ..(Ie; I After several years oft s InstructIOnal strategies utlIned pre ously lD thIS ument() ( I ~

l' WIthout great succ , I deSIgned esrcrltle,s te.~c lyaddres rob] s t~a~ rjc.~ b •$&iI t,/~ 1Vl... were frustratmg dents ,,' /VI) /111.1 ~

j ~ In our school, eighty-five percent of students are concurrently e ed in the NYS ~ }" Algebra 2 and Trigonometry course, ten percent of students are co rently enrolled in the NYS J-[e..,tr.-I Geometry course and five percent are enrolled in a higher lev math course. The Algebra 2 and 5'4\ J TrIgonometry course ISthe first occurrence of vectors 1I1 thej{llath cumculum A pnorI~ ...:sWlodgOOf; CAll TOA ~ .,=ol. bo,"""," ... ",~ro""d<'" ,d"0)

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v"" dlrectIOnaltty as part of geometry or p~n solvmg wIth tng70metIY functIOns While this I~ \'1v.rI worksheets require stIldents to compare and contrast the two vector quantities under I~ seems lIke a SImple added step, many' students struggle IVlth asslgnmg a dIrectIOn to a resultant. ~ 't \ investigation: displacement and velocity. It is important to encourage students to make /.;J ~~., 1\ )C J distinctions between each vector quantity that is introduced subsequently in the curriculum. .//_V These activities are intended to be utilized just after the introduction of the tenns 1'\- ~

displacement, velocity, vector and scalar. Students should already be able to distinguish between '\ '- Activity One requires roughly 45 minutes for introduction, student work and discussion.!PG.L.. -t~ tile paired vector/scalar terms of distance/displacement and speed/velocity. Students should also ll\,f"'If ;ecessary some of the discussion can occur during a later time frame. Activity Two requires ,,~.?

be familiar with equations for constant velocity. Basic vector terminology, such as resultant, t- I roughly 90 minutes for introduction, student activities and discussion. The completion of the £J 1(j'" equilibrant and horizontal and vertical components should be presented in the introduction to the Il follow-up questions may require additional clas~e 'f /ian be assigne,d as homework. ~ ~~ );

\. actIVIty along Wltll a revIew of basIc geometry The equations lIsted on the PhysIcs Reference <C-" L I .. tJ} cln~ "\.. ~">Table [A,=AcosO and A,=Asm8) are tntroduced as part of ActIVIty One and appiJed througnout The termillology used III the act IS c nSlstent IVlth presentatIOns ofmatenal on theA7 ty T 'ilt 'h -.II ~ New York Slate Physics Regents Exa . S e of the questions are specifically modeled afterCIVI wo '- \ S'""O.J-tCl' =- l( ~. rz;...,. S ~ Q., Q commontypesofquestiollSthatappearo the state assessment. This is done with the intentioll

The actIVItIes are deSIgned to speCIfically address vector mlsconce~s outlined by of drawing parallels between experience and phrasing that frequently causes frustration on theAguirre (1988) upon the introduction to vector quantities. The foUowing table outlines th+;o /J _ part of studems. ~misconceptions and the strategy utilized within the activity to address the misconception. ~ ~ . \5 .J J. . -rt._.. 7 . (\It ..,(a "/ J

;- I / EqUIpment: , r..........J "I'l~ ,,,\" 1.-- If. "'c....~ ~i~ ar~vi1e~~~ atf~-1fut ~d~ ~Jy~ttes for a cost of X",cr., ~

~ approximately $3-4 each. The toys pivot wh 1 t e reach an edge of a surface or they change v9' J~f -._/}o"? itfaces. A dry erase surface with a grid appl to ster board or thick cardboard is ideal for/ ~'~U 1/1 ~ aking tile toy pivot and allowing students t m' rk t e motions. The movable surface should be , •""', 4. ~ 0-60 cm in lengtll and 25-35 em in width d h ve smooth finish. (The toy should be able to / ...,

r. ' iJ ~ traverse the width in 10 seconds while u ergoi g c nstant speed). A surface cut from a gridded 'le,L,-.' ,f l,Jt, d I' dry erase poster board provides the ide surface as' allows you to veritY the straight-line /.1l_ i~..,.,....'. L ? horizontal and vertical motion of the y throug ou the lab and allows ~ents~ark-up . r~... ~-,- t.A i) surface as they see fit '1? "CI~t:v' .Yo ~I' c:.. ::;.::::::

It is advisable to purchase ra toys to 10 for malfWlctions or breakage. The cost of L _each lab set-u consisting of rids \ ith ster ar and to s canle to approximately $10 C~he \......,.. • ""ost would increase ifbasic la~ ies such as s~-watch s met~ ~ • reV ~J ~ /

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>.., C~ .. ~."1,c;;::t-t..w./?KStudents will observe the motion of a ~ever Fal wind-up toy as it moves aroWld the I ~

board as an introduction to the method of he a al vector addition. Students will break-down "f'''-neY .

each vector into horizontal and vertical components and tabulate in a table as a method to ~ ~determine the relationship between the components of vectors and the resultant. Once they havedetenllined the overall horizontal and vertical displacement students will detennine the resultant ~ /J.displacement of the toy using basic trig and geometry. Students will be introduced to the basic I ~m"I,,~"o' ,,~,." '" >01.,", .OOM proWom,. =r ., ~~

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Table 1: Vector Misconceptions (Aguirre, 1988) addressed in Activity Tw0:tadybug on aConveyor Belt

MlSCONCEPTION ACTIVITY CONNECTIONPath is an intrinsic property of a In each scenario students measure the displacement relativemoving body; that is, it is to a fIxed frame in order to reinforce the idea ofrelan:E.independent of any reference motIOn.;' { /..franle .... ""C of"". - rt5

The magnitIlde o~component Students reference the average velocity fOWld at the outset ofvelocities increas /d creases the lab against thecomponent velocity of the bug in eachdue to the interacti with tlle stage. Students observe that the motion of the paper has noother component '9r '¢ impact on the component velocity of the toy. Students can

also directly observe the independent motion of each".1~(, t S component

Speed is an intrinsfc Itoperty of For each scenario students detennine the velocity relative toa moving body, and I is the fixed frame in order to reinforce the idea of relativisticindependent of any reference motion.frame.~e along tile ~nt path is Students only record a single time illterval that applies for allshortenrnm-tim it es for the p~ata.vertical or horiz component Purpose:

Each activity utilizes a guided worksheet [~], but requires activemanipulation of materials and critical thinking. Students will work in small groups to problemsolve throughout the guided activities. Due to the fact that this is an initial introduction tovectors, students would have great difficulty creating the scenarios for themselves. The purposeof these activities is for students to continually reason through problems involving displacementand velocity so that they can draw conclusion about the nature of each quantity. The guided

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label the dry erase board with a an axis at the outset of the activity r:<\.1'kas they proceed. With a full)' und toy, students should place the toy in the ett es-. II .

'ard and track the motion of toy on the board with a dry erase marker. A toy tJ J s ents are introduce four resource equations based or\2.0H CAl;! TOA angee to make a minimWTI of ee distinct vectors but more are possible. Students eLi) S~ ythagorean Theorem that ill be used for vectors throughout the year. (R stands for the

should altl arrows to their diagraI indicate the direction of motion for each vector. The start of any vec or quantity.,) :r!:.. ~ Ti: ~(See Figure 1), C\.. /l J"-~-~W- I L ~ jj:."L t:.r ~ ~I r 1.> f..t.../ R, = I)cos9 Ry = Rsin9 9=tan·I(RylR,) R-=R} + R,.-

fj'f/c! 'is (L A- &\ '\A.Start. Horizontal Vertical Y 1=-- The New York State Physics Reference table uses the variable "A" in place ofR At thisFinish * X r~r

7 point "R" is used to reinforce the teml resultant. Throughout the year other variables will beused in place of R, such as "vi' for initial velocity, "F" for force or "d" for displacement. While

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A 2 3 ~ this activity uses displacement vectors, the use of the "R" is preferred as it aids in vocabulary JY+ B 3 -5 acquisition of the term resultant. .rF ",/,., iT ....UJ" JD"-e.tkre v Sf' oJ (-

C -3 -3 -t::t.}te.s- / lIot:~J,sD -5 3 :: Activity 2: Two-dimensional motion or "Ladybug on a conveyor belt" I .I ,_

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Purpose: L,/'IriD .. .s el~ 0 .. "1)- ..t'~"''''l'Cr1- Sample student work steps A-D Figure 2- Sample of student data stepi'E-F - / '7 ?

11 '0 L)". 1"1". • Students will explore the simultaneous motion of a wind-up toy and the surface on whichStudents will determine the horizontal and vertical component of each vector by'counting it moves. This exploration will be representative of concurrent motion such as a boat on a river

'es on the grid. The main goal during this ph"ase is to learn the method for graphically ~ or a person on a moving sidewalk.detmlining components. Once students detennine individual components they will find the SWll 1tror/all horizontal aIld all vertical motion, (See Figure 2) C.~ In addition to the wind-up toy, students will require a fixed dry erase surface with meter

'- .....t"" S''' sticks attached. This board and the meter sticks will be the reference frame for the motion ofIf time is an issue students may need to copy the work onto a piece of graph paper for use I both the movable surface and the toy. (See Figure 4) J

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using graph paper solve for the ¢ teams resultant displacel ent vect\lf using the skills leamedduring the prior stage, ./

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In the second stage of the activity students will draw the resultant vector with arectional arrow directed from the start point of the toy to the end point of the toy (See Figure

tudents will determine the horizontal and vertical component of the resultant and comparee values with the overall horizontal and vertical motion fOWld during phase one steps A-F,

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~re and Instructional Notes:

7'lA er:f-Students will then re-write the vectors from their list onto individual index cards without

indicating the order in which tile vectors were "mapped". Groups will exchange card piles and

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