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Introduction to Vectors 1

A Hands-on Introduction to Displacement and Velocity Vectors and Vector Operations through the Use of an Inexpensive

“Never Fall” Wind-Up Toy

Gwen Saylor, Department of Physics, State University of New York – Buffalo State College, 1300 Elmwood Ave, Buffalo, NY 14222 [email protected]

Acknowledgements: This paper is submitted in partial fulfillment of the requirements necessary for PHY690: Masters Project at SUNY – Buffalo State College under the guidance of Dr. Dan MacIsaac.


Abstract: This paper presents a set of hands-on activities to be used as an introduction to vector quantities and vector commutation. In these activities students are introduced to basic vector arithmetic through the analysis of the motion of a “Never-Fall” wind up toy. With inexpensive equipment students can visualize the head to tail method of vector addition, determine the horizontal and vertical components of vectors and observe the combination of two concurrent parallel or perpendicular vectors. The importance of the sequencing of these concepts early within in the curriculum is discussed in context of research into the teaching and learning of vectors. These activities introduce students to the characteristics of vector quantities while supporting skills required for vector operations. Through guided activity worksheets students will be exposed to the terminology used frequently on the New York State Regents Physics exam related to vectors.


Introduction:

Vector concepts and vector operations are a central part of the New York States Physics Core Curriculum. Any in depth discussion of motion, forces or field behavior requires knowledge of vector quantities. Experienced physics teachers know that many of the characteristics of vectors that seem obvious to individuals with a background in physics are not at all obvious to many first time physics students. In the same way that physics education has focused on modifying instruction to explicitly address student misconceptions related to forces through the Force Concept Inventory introduced by Hestenes, Wells and Swackhamer (1992), physics instructors must also take steps to address student preconceptions with vector quantities The Vector Knowledge Test, administered to introductory college level physics courses comprised of primarily science majors, revealed that nearly half of the students who reported prior exposure to vectors from high school physics or math entered the class with no useful knowledge of basic vector skills (Knight, 1995).

In probing student preconceptions about vectors, Aguirre presented seven vector characteristics that require explicit instruction (Aguirre and Erickson, 1984). A common theme to the seven characteristics outlined in Aguirre’s research is the role of the reference frame in understanding vectors and the independence of each component vector. Through interviews with students regarding three experimental situations, Aguirre (1988) discovered a number of student preconceptions that require explicit instruction but are often viewed as implicit by instructors.

Specific recommendations from the analysis of the Vector Knowledge Test (Knight, 1995) suggest vectors should be introduced over a course of several weeks prior to introduction of projectiles or Newtonian mechanics. Subsequent investigations using diagnostic testing of 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 vector addition (Nguyen and Meltzer, 2003).

Many physics textbooks present vectors during a unit on forces and then transition quickly to other quantities such as velocity, acceleration, momentum and displacement. From a students’ viewpoint “adding velocity arrows appears very different from adding displacement arrows, and acceleration arrows are totally incomprehensible” (Arons, 1997, p107). Displacement vectors in some scenarios are successive rather than concurrent. As a result displacement vectors are the simplest starting point; however, as instructors transition from displacement vectors to force vectors students are bound to get confused unless the nature of each of these quantities is discussed (Roche, 1997). Care must be taken in curriculum planning to 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 have significant difficulty translating those skills to acceleration. Shaffer and McDermott found in a survey on introductory physics students, graduate students and physics TA’s, that the ability to correctly draw and label a velocity vector


was markedly greater than the number of students who were able to correctly draw and label an acceleration vector (Shafer and McDermott, 2005).

I propose that isolating the initial introduction of vector skills and vector characteristics to non-accelerated motion will help students overcome the prevalent misconceptions documented in aforementioned research. By devoting instructional time to establishing the characteristics of vectors within the context of non-accelerated motion students can gain comfort with vector operations and confront common misconceptions without the compounding difficulty of other complex topics such as projectiles and mechanics. Vectors are an abstract concept for students. A few simple examples at the outset of a discussion are not sufficient to address the level of student difficulty with basic vector skills. The activities presented in this document limit the introduction of vector quantities and vector operations to displacement and velocity scenarios in order to provide explicit instruction of vector characteristics. As instructors introduce new concepts, reference to these common learning experiences can accompany discussion of differences and similarities between quantities.

The two activities, “Activity One: Ladybug Transit” and “Activity Two: Ladybug on a Conveyor Belt” presented in this document attempt to provide instructional tools that make vector characteristics both explicit and highly visual for learners. The motion of the equipment, while not modeled after Aguirre’s design (1988) is similar in intent to two of the experimental situations presented; however, on a more simplistic and cost effective scale. The ability to reason with vectors and apply vector operations in problem solving is essential to understanding of topics such as projectiles, superstition of forces, momentum and force fields. Therefore, the devotion of significant instructional time to basic vector skills early in the year will help students master subsequent vector concepts with less frustration.

Based on interviews and activities, Aguirre concluded that students commonly held preconceptions include the following: (Aguirre, 1998)

Speed and displacement are independent of frame of reference Vectors components act sequentially rather than simultaneously Time is different for the resultant path than for the components Magnitude of component vectors change when two vectors interact

In order to address many of these preconceptions, the second activity of this document is reflective of two of the scenarios presented by Aguirre; Powerboat Crossing a River and Two Orthogonally moving carts. Through observation, data collection and analysis, students confront each of the aforementioned preconceptions.


Vectors in the New York State Regents Physics Curriculum

Many of the vector characteristic outlined by Aguirre (1984, 1988) are addressed by Arons in his discussion of vector skills (Arons, 1997 p106-108) and included in the New York State Physics Core Curriculum. The following chart (See Table 1) summarizes the portions of the Standards from Mathematical Analysis and Scientific Inquiry that relate to vectors in the New York State Physics Core Curriculum. The list includes the required basic skills as well as topics that require the application of vector concepts. .

Table 1: Vector Skills From the NYS Physics Core CurriculumStandard 1: Mathematical analysisKey Idea 1: Abstraction and symbolic representation are used to communicate mathematically

- use scaled diagrams to represent and manipulate 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 determined by vector

addition5.1d An object in linear motion may travel with a constant velocity* or with acceleration*

(Note: Testing of acceleration will be limited to cases in which acceleration is constant.)5.1f The path of a projectile is the result of the simultaneous effect of the horizontal and

vertical components of its motion; these components act independently5.1g A projectile’s time of flight is dependent upon the vertical component of its motion.5.1h The horizontal displacement of a projectile is dependent upon the horizontal component

of its motion and its time of flight.5.1n Centripetal force* is the net force which produces centripetal acceleration.* In uniform

circular motion, the centripetal force is perpendicular to the tangential velocity.5.1q According to Newton’s Third Law, forces occur in action/reaction pairs. When one

object exerts a force on a second, the second exerts a force on the first that is equal in magnitude and opposite in direction.

5.1p

The impulse* imparted to an object causes a change in its momentum

5.1q

According to Newton’s Third Law, forces occur in action/reaction pairs. When oneobject exerts a force on a second, the second exerts a force on the first that is equal in magnitude and opposite in direction

5.1s

Field strength* and direction are determined using a suitable test particle. (Notes:1)Calculations are limited to electrostatic and gravitational fields. 2)The gravitational field near the surface of Earth and the electrical field between two oppositely charged parallel plates are treated as uniform.)

(NYSED, 2008) Full text available at http://www.p12.nysed.gov/ciai/mst/pub/phycoresci.pdf


The three basic vector skills listed in the outline above as Key Ideas 5.1a-c provide a foundational framework for topic specific skills that follow. While representative of many of the required vector skills and concepts, the list of Key Ideas is not intended as a lesson plan. Many intermediate skills must be introduced in order for students to successfully. In order for a student to transition from the basic level of functionality listed in Key Ideas 5.1a-c in the table above to mastery of skills and concepts in the remainder of the list, learners must be able to demonstrate the following skills and understanding:

Define terms such as displacement, velocity, resultant, equilibrant and component. Establish the relationship between component vectors and the resultant vectors including

the concept of additive inverse (Arons, 1995, pg 107) Define the meaning of a negative vectors in relation to the Cartesian coordinate grid and

rectangular vector components Understand that vector quantities are not fixed to a location (Brown, 1993)

The activities presented in this document are designed to introduce students to vectors involving motion in two dimensions. Through the use of hands-on activities, students can easily visualize vector problems. Each activity is conducted in the space of a student desktop, making it visible to each group member. Students will observe the motion of the toys and apply prior knowledge from the kinematics unit, geometry and algebra. Activity One “Ladybug Transit” allows students to resolve vectors into horizontal and vertical components by graphical methods and introduces the equations necessary for the analytical commutation of vectors. During activity two, “Ladybug on a conveyor belt”, students can observe the combination of parallel or perpendicular vectors using a moving toy and a moving grid to model two dimensional motions. Guided questions are used throughout to support concept development, vocabulary acquisition and skill application.

From professional experience and anecdotal conversations with colleagues, one of the added difficulties when it comes to vector skills is students’ inability to understand some of the terminology used in the assessment of vectors on the New York State Physics Regents Exam. Many colleagues express the need to teach vectors and the need to separately teach an additional series of lessons on how to help students answer Regents questions. To that end, the activities and student worksheets presented in this document support skills required for success on the Regents Physics exam. Wherever possible questions asked on the student worksheet are correlated to past Regents Exam questions. (See Appendix C: Vector Related Regents Questions). Regents questions that correlate to items included in the activities are referenced within the document according to test date and item number. Appendix C lists selected questions in descending date order and then ascending numerical item number order. Full versions of exams are available online from the NYSED at http://www.nysedregents.org/physics/ . The Regents questions selected are just a small sample of representative vector problems that have been included on Regents Exams between 2003 and the 2011. Terminology that is a common

http://www.nysedregents.org/physics/


source of difficulty for students as evidence by class discussion and test item analysis is applied throughout the “Activity Two - Student Worksheet” in order to address and resolve student issues with terms as early as possible in the course. For example, the word “magnitude” is not a common term for adolescents in every day conversation. As a result this term can cause a great deal of frustration for students throughout the year when working on test preparation items. While this paper is primarily focused on displacement and velocity, an occasional example of a force question is included in Appendix C in an attempt to provide connection between the activity in this document and the student assessment at the conclusion of the year.

While some of the activity content goes beyond the requirements of the Physics Regents exam, the skills established during the investigations are important to understanding subsequent vector quantities.

Required prior knowledge:

The majority of students enrolled in Regents Physics have been exposed to many of the underlying math skills necessary for computation with vectors. Basic trigonometry and geometry are covered in the New York State P-12 Common Core Standards for Mathematics in grades 7, 8 and in the High School Geometry, Algebra and Trigonometry Core. Vectors are presented for the first time in the NYS Algebra 2 and Trigonometry course. Depending on the sequencing of math and science courses within a district, some students will first encounter vectors during physics without formal prior instruction in math. Many students will simultaneously encounter vectors for the first time while enrolled in math and in physics class. Based on experience, the addition of directionality to problems involving SOH CAH TOA and the Pythagorean Theorem can pose a substantial learning hurdle to many students.

A quick review of SOH CAH TOA is recommended if a large time gap has occurred since earlier student exposure. Students should also be familiar with the polar coordinate system and Cartesian coordinates system for the purpose of collecting data during the exploration phase. Taking time to establish models for the connection between Cartesian coordinate, polar coordinates and directionality of rectangular components at the outset of the activity can be a useful discussion point for struggling students during the investigation. Two methods I developed to help students are summarized here:

1. Students can easily relate to maneuvering around a computer screen given only four directional keys. This can also be easily demonstrated for students by superimposing and arrow over a grid in a word processing program and directing them to use the arrow keys to negotiate from the tail to the head of the vector.

2. Another activity that is very relatable to students and serves the purpose of introducing the idea of vertical and horizontal components is exploring how an Etch-a-sketch works. Each knob on an Etch-a-Sketch controls either horizontal


or vertical motion, and each knob is able to move in either a positive or negative direction. The possible combined motions are useful in applying student knowledge of “x,y” on an axis to a Cartesian grid.

The “Vector-Addition” simulation provided by Phet provides good reinforcement and practice for connecting angles in the polar coordinate system to “x” and “y” values in the Cartesian coordinate system (http://phet.colorado.edu/en/simulation/vector-addition). To use this animation to link the two coordinate systems, place a single vector on the screen and rotate it about a fixed point. The angle of the vector and the corresponding horizontal and vertical components will appear at the top of the screen.

TEACHER BACKGROUND

Activity 1: Displacement vectors with a “Never Fall” wind-up toy – Ladybug Transit

Purpose:

Students will observe the motion of a Never Fall wind-up toy as it moves around the board as an introduction to the method of head to tail vector addition. Students will break-down 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 have determined the overall horizontal and vertical displacement students will determine the resultant displacement of the toy using basic trig and geometry. Students will be introduced to the basic mathematical steps used in solving vector problems.

Equipment:

Never fall wind-up toys are widely available on-line and in toy stores for a cost of approximately $3-4 each. The toys pivot when they reach an edge of a surface or they change surfaces. White dry-erase boards with an edge are a perfect platform for the Never Fall toy. In the first stage of the activity a dry erase board with a gridded surface is used to introduce the idea of components without the need for equation. A plain dry erase board is required for the final activity in which equations are introduced. Note that all surfaces must be level and clean or the toys will pivot. Basic measurement such as a ruler and protractor are needed.

Procedure and Instructional notes:

Students should label the dry erase board with a Cartesian axis at the outset of the activity to avoid confusion as they proceed. With a fully wound toy, students should place the toy in the center of the board and track the motion of the toy on the board with a dry erase marker. A toy should be able to make a minimum of three distinct vectors but more are possible. Students should add arrows to their diagram to indicate the direction of motion for each vector. The start and finish point should be labeled (See Figure 1).

http://phet.colorado.edu/en/simulation/vector-addition


Figure 1- Sample student work steps A-D Figure 2- Sample of student data steps E-F

Students will determine the horizontal and vertical component of each vector by counting squares on the grid. At this stage many students do not grasp the purpose of breaking down vectors into components and some questioning may occur. The main goal during this phase is to learn the method for graphically determining components. Once students determine individual components they will find the sum of all horizontal and all vertical motion. (See Figure 2)

Students should be led to understand through guided questioning that it is possible to add the horizontal components because they are on the same axis or parallel vectors. The same can be done for the vertical components. In contrast, adding the lengths of the original vectors will not result in the displacement of the object, but rather the distance traveled which is a scalar quantity and does not have an associated direction. This helps also helps students distinguish between the terms vector and scalar quantities (Appendix C: RE 6/2011 #1). This concept can take a bit of time for some students to comprehend. This single exercise will not be sufficient practice for many students to master the skill or vocabulary, but it will provide a common experiential reference for further discussion and skill practice.

If time is an issue students may need to copy the work onto a piece of graph paper for use a second day. This can be a beneficial step as it requires students to count out the horizontal and vertical components of each line in order to make an accurate map. This is essentially the same skill as the activity requires but is easier for students to comprehend.

In the second stage of the activity students will draw the resultant vector with a directional arrow directed from the start point of the toy to the end point of the toy (See Figure 3). Students will determine the horizontal and vertical component of the resultant and compare those values with the overall horizontal and vertical motion found during phase one steps A-F. The idea of the resultant should be restated or reinforced at this point. In this scenario, in an effort to draw conclusions students will describe the resultant with terms such as the short-cut or the quickest path. This is helpful for developing the idea of the equilibrant as the path home later in the process. Those descriptions are appropriate only for the concept of displacement. When

Horizontal X

Vertical Y

A 2 3B 3 -5C -3 -3D -5 3∑ -3 -2


introducing other vector quantities such as velocity or force, the idea of the “quickest path” will not describe the resultant. Instructors should take time to present additional ideas such as the sum of all motions.

Figure 3: Drawing resultant

Students will then re-write the vectors from their list onto individual index cards without indicating the order the order in which the vectors were “mapped”. Groups will exchange card piles and using graph paper attempt to solve for the other teams resultant displacement vector using the skills learned during the prior stage.

Through a guided class discussion student will analyze the outcomes of the “vector card exchange”. In the process, the idea that vectors are not fixed to a single location is developed. This is an important characteristic of vectors that is difficult for many students to comprehend. The presentation of displacement vectors as fixed in space can lead to misconceptions. Students need to recognize that one characteristic of vectors is that they exist independent of location (Hoffman, 1975 Ch1, I know I need actual page). One way of presenting this concept to students is to explain the distinction between a statement such as “50 km to the West” and “50 km to the West of the high school” (Brown, 1993). The first statement is in fact a vector quantity while the second is not. Questions that compare distance traveled on a map with the overall displacement of an object, directly contradict this vector characteristic. One notable example appeared on the January 2006 Physics Regents Exam (Appendix C, 1/2006, #12). Past exam questions are widely used in NYS classrooms for test preparation. This is not an isolated case; students are regularly presented with vector problems that represent displacement in a fixed location on a map in the form of practice Regents questions and in text books. Teachers must present this scenario without strenuous objection due to the commonality of the presentation; however, it is worth taking time to create a distinction between these scenarios for the purpose of transitioning beyond displacement vectors. By presenting students with the opportunity to randomize the order of the displacement vectors through the exchange of vector cards, the impact of this misconception can be minimized.

Based on conversations with colleagues, the idea of the concurrent displacement is not explicitly discussed since it is seldom assessed formally; however, students should understand

Resultant X

Resultant Y

-3 -2


concurrent displacement in order to apply the idea when studying projectiles, the “boat in a river” scenario and perpendicular force problems.

At this point the idea of the equilibrant as the shortest return path can be developed. Students should recognize that the return path is equal in magnitude but opposite in direction to the original resultant. The idea that opposite direction means a difference in direction of 180 degrees requires explicit instruction from the teacher. The equilibrant is a concept most often applied in force problems such as objects at rest on an inclined plane. The development of the concept at this point will allow students to see its universal application to vector quantities rather than to attribute this term solely to the concept of forces.

Follow up activity and introduction to equations:

During the concept development portion (Maier and Marek, 2006) of this topic, students are exposed to vectors without the grid. This is accomplished by drawing a vector on a dry erase board or by tracing a single line of the ladybug toy’s motion (vector should not be on an axis). Students measure the length of the displacement vector with a meter stick and find the angle relative to the horizontal axis using a protractor. Some review of protractor skills may be necessary.

After completion of steps L-O of the activity on the Activity One Student Worksheet, students are introduced to four resource equations based on SOH CAH TOA and the Pythagorean Theorem that will be used for vectors throughout the year. (R stands for the resultant of any vector quantity)

Rx = Rcosθ Ry = Rsinθ θ=tan-1(Ry/Rx) R2=Rx2 + Ry

2

The New York State Physics Reference table uses the variable “A” in place of R. At this point “R” is used to reinforce the term resultant. Throughout the year other variables will be used in place of R, such as “vi” for initial velocity, “F” for force or “d” for displacement. While this activity uses displacement vectors, the use of the “R” is preferred as it aids in vocabulary acquisition of the term resultant.

Activity 2: Two-dimensional motion or “Ladybug on a conveyor belt”

Purpose:

Students will explore the simultaneous motion of a wind-up toy and the surface on which it moves. This exploration will be representative of concurrent motion such as a boat on a river or a person on a moving sidewalk. The purpose of this exploration is to help students develop the concept of vector addition for parallel and perpendicular vectors. Students will also explore the concepts of displacement and velocity in depth and obtain practice in transitioning between these quantities to further develop understanding of vector addition and the underlying concepts associated with each of these vector quantities.


Equipment:

This activity can utilize any wind-up toy that reliable maintains a straight path and constant velocity for a minimum of 10 seconds. “Never fall” toys described in the prior activity are a good option. Speeds for the toys should be less than 10 cm/s in order for students to be able to complete the activity without high levels of frustration. In addition to ladybug themes, they offer robots, bulldozers and other fun objects. While for adults this may seem like an unimportant factor, students find the toys to be a fun and motivational aspect of the activity.

The movable surface should be 50-60 cm in length and 25-35 cm in width and have a smooth finish. (The toy should be able to traverse the width in 10 seconds while undergoing constant speed). A surface cut from a gridded dry erase poster board provides the ideal surface as it allows you to verify the straight-line horizontal and vertical motion of the toy throughout the lab and allows students to mark-up surface as they see fit. A stopwatch will be required to record the time of motion for the toy and the paper.

In addition to the wind-up toy, students will require a fixed dry erase surface with meter sticks will be attached. This board and the meter sticks will be the reference frame for the motion of both the movable surface and the toy. (See Figure 4)

Figure 4: Activity Two Equipment Set-up

The equipment in this lab provides an inexpensive alternative and small scale alternative to a demonstration commonly used in many high school physics classrooms (Mader and Winn, 2008, p101-5). While the velocity of the paper is not truly constant speed it is a close enough representation for students to comprehend the learning goals for the investigation.


Procedure and Instructional Notes:

It should be discussed with students that they need to practice several times with the equipment before they begin taking data. Several actions need to be coordinated and every member of the group must be prepared to make the appropriate measurements and recognize events which might lead to error. If errors are recognized, students should repeat the scenario. Decisions should be made by the group relating to measurement techniques. For example, a common point on the toy should be agreed upon for determining displacement of the ladybug relative to the grid and relative to the frame of reference. Students could select the front, the wind up knob or the back feet as long as they agree upon a common reference. Strategies such as marking starting points should be decided upon by each group. Each student should be assigned a role within the group. Here is a possible division of tasks for a group of three students:

- One student pulls paper at a constant speed- One student times the motion- One student winds the ladybug, releases the ladybug and stops the ladybug. This

individual is responsible for clearly communicating start and stop to the other members of the group.

- At the conclusion, one student should use a dry erase marker to indicate each of the following end points 1) the grid relative to the fixed frame 2) the ladybug toy relative to the grid and 3) the lady bug relative to the fixed frame. This task could be assigned to the student with the timer or a fourth group member if possible.

Students will then take measurements and record in the data tables provided for each scenario.

At the outset of the activity, each student group should define the horizontal and vertical axis. Since the activity requires 3-4 participants it should not be assumed that each individual has the same view point, but they must agree on the same definition for the positive axes. The gridded surface moves between the meter sticks on the horizontal axis in either the positive or negative direction. The toy can move in any direction but will be limited in this activity to motions parallel or perpendicular to the surface. The grid is useful in verifying the toys horizontal or vertical motion. If the toy deviates from straight line motion, the trial should be repeated. If a slight deviation is unavoidable due to a malfunction of the toy’s mechanism, students should make note on their data and move on. The fixed dry erase board and the movable grid must be clean and flat so as not to interfere with the toys turning mechanism.

Students must first determine the average speed of the toy on the gridded surface. This value will be used as reference throughout the lab. After several trials with the toy, students should find a fixed distance reached while toy still maintains a constant speed and determine the amount of time it takes for the toy to reach that point. (This will vary depending on the type of toy). In this discussion the toy will be referred to as a ladybug, which was the toy used in the activity at our school.


The student worksheet guides the learner through the activities with the intention of introducing vector concepts, vector vocabulary and terminology important to understanding subsequent vector quantities. The “Activity 2: Student Worksheet” is attached as Appendix B.

During the first phase of the activity students observe the addition of parallel vectors by moving the toy and paper in the same direction and then compare with motion in opposing direction. In addition to collecting data and observing the resultant motion, students are asked to translate information into a graphical representation of the components and the resultant. (Student questions #2 and 3). Students are led to make a conclusion about how the relative direction of motion impacts the size of the resultant (Student question #4).

The inspiration for this line of inquiry is multifold, including helping students with visualizing scenarios for problem solving on state assessment questions and in developing the idea of the additive inverse (Arons, 1997 p107) for force vectors to be applied during subsequent laboratory activities and discussion. In many topics throughout the year, students are presented with parallel vectors. Based on experience working with students, this concept is not intuitive to all learners. The idea that vectors which agree in direction add to a larger resultant seems obvious, but students struggle with this idea when presented with graphical representation. The representations and terminology used on the Physics Regents add some level of complication. When discussing errors on this type of questions with students they usual express difficulty in making a connection between the phrasing of the question and class discussion or practice. This line of questioning is in part inspired by observing student difficulty with Regents Physics questions which connect magnitude of a resultant with direction or angle of the components (See sample question 1/2008, #38, Appendix C and 1/2003, #38).

In Activity One, displacement vectors were presented as sequential quantities. During this portion of the activity students observe concurrent displacement for an object undergoing two dimensional motions. Many students can visualize displacement vectors with minimal difficulty but find velocity vectors challenging (Roche, 1997). Concurrent displacement due to constant velocity is equally challenging. Many students think of displacement vectors as sequential values occurring during time intervals. This is in fact the case for displacement problems presented on the New York State Physics Regents (Appendix C, 6/2008, #57-59; 1/2003, #51-52). The distinction between the scenarios presented in “Ladybug Transit” and “Ladybug on a conveyor belt” should be made explicit for students.

Student questions 2 and 3 ask students to distinguish between displacement and velocity and require them to transition between the two quantities given a time variable. This encourages students to not only apply variables to an equation but to further demonstrate an understanding of the difference between each quantity. At this stage of the course, students are still adjusting to the use of units to describe values and this step encourages them to practice using units to differentiate between quantities. This is a habit which is invaluable throughout the rest of the year.


Questions 5 through 11 apply to scenarios in which the components are perpendicular. Two questions require students to collect data and analyze in a similar method as previously used with parallel vectors. By directly observing a single object experience both horizontal and vertical motions, students can grasp the idea of simultaneous vectors. This activity is a visual that is necessary for the visual learners in the group. The intention of having students work through this series of problems is to provide a common experience for reference during future problem solving and to apply previously introduced mathematical skills. It is important to check student diagrams of head-to-tail or parallelogram addition to be certain that the resultant drawn in the diagram matches the actual path of motion for the toy. Some students struggle with the rules of drawing resultants and despite observations they will continue to arbitrarily draw resultant that form a triangle but that do not represent the direction of resultant motion. This activity is an opportunity to provide a connection between the addition rule sets and observed motion. Students are assessed on their ability to combine perpendicular vectors through many question formats include word problems, graphical representations and scaled diagrams (See Appendix C: 6/2010, #2; 1/2008, #56; 1/2004, #2).

Introducing concurrent displacement and velocity vectors at this early point in the curriculum prepares students for projectiles by establishing in advance the independence of the components motion. In addition, practice with concurrent components of velocity prepares students for the first step of problems in which projectiles are launched at an angle. If students are already comfortable with the initial vector problem for such a scenario, they can focus on learning the new tasks involved in projectile problem solving. When the skills are all combined into a topic, many students have difficulty ordering the steps, which shows that they do not fully understand the concepts.

Throughout this series of activities, students are asked to determine resultants given the horizontal and vertical components. In the process of determining the magnitude and direction they apply the reference equations repeatedly. The various scenarios allow students to practice with using the Cartesian coordinates. By drawing the various diagrams for questions 5, 9, 10 and 11 students can further practice connecting the Cartesian coordinates with a polar coordinate system. The polar coordinate method of labeling vectors can be useful when introducing or discussing the concept of the equilibrant as opposite or equivalent to a 180 degree difference relative to the resultant. If you only discuss vectors in terms on the angle between the horizontal and the resultants students will have a difficult time considering a 360 degree plane of motion. While questions on the Regents exam rarely deal with a polar coordinate system, the idea is necessary for facilitating a complete discussion of force on an inclined plane.

At the initial stages of this investigation, students are asked to make the horizontal and vertical velocities a similar value. Several questions contained in this portion of the activity introduce the idea of inequality between the horizontal and vertical in order to assess the students’ ability to visualize the resultant angle based on the relative size of the components.


This skill is important when students explore projectiles and forces. A prediction stage is added during this step to encourage students to move beyond pure drill practice with equations and to assess student ability to apply reasoning behind the equations.

After a significant number of practice questions asking students to find the resultants from the horizontal and vertical components it is advisable to change the pattern. This ensures that students are able to identify the variables and apply the concepts rather than just plug in values and solve. Questions 12-14 require students to apply the same set of equations that have been used throughout the lab but in an alternate method. Many students fall into the trap that the equation that fits the situation is the one in the form of “unknown equals….”. This is an important misconception to rectify early in the course, not only in the case of vector applications but all problem solving. Instructors should be prepared to field a significant number of questions from students who are struggling with mastery at this part of the activity. Students’ weak grasp of vector concepts makes it difficult for them to understand creative presentations of two dimensional data. Evidence of student difficulty with this type of task was observed during the grading of a recent regents exam that presented students with a graph of initial vertical velocity versus initial velocity and tasked students to connect information to equations to find launch angle the horizontal component of motion (Appendix C, 6/2010, # 51-53). While quantitative item analysis was not available, test graders noted that only a handful of students out of 300 received full points for all questions. That poor completion rate indicates a difficulty with interpretation, but also hints to the fact that students are able to complete problems due to drill and practice but not able to apply concepts. Questions 12-14 were included in this activity based on the above observation.

Throughout the entirety of Activity Two students confront several of the vector misconceptions. Following is a table of the misconceptions as stated by Aguirre (1988) and the strategy by which Activity Two addresses the misconceptions.

Table 1: Vector Misconceptions (Aguirre, 1988) addressed in Activity Two: Ladybug on a Conveyor Belt

MISCONCEPTION ACTIVITY CONNECTIONPath is an intrinsic property of a moving body; that is, it is independent of any reference frame

In each scenario students measure the displacement relative to a fixed frame in order to reinforce the idea of relativistic motion

The magnitude of the component velocities increases/decreases due to the interaction with the other component

Students reference the average velocity found at the outset of the lab against the component velocity of the bug in each stage. Students observe that the motion of the paper has no impact on the component velocity of the toy. Students can also directly observe the independent motion of each component.

Speed is an intrinsic property of a moving body, and it is

For each scenario students determine the velocity relative to the fixed frame in order to reinforce the idea of relativistic


independent of any reference frame.

motion.

Time along the resultant path is shorter than time it takes for the vertical or horizontal component.

Students only record a single time interval that applies for all pieces of data.

Extensions:

At the conclusion of the activities presented in this activity students can begin the concept application phase of the learning. Several relatable scenarios can be used for this stage. The boat traveling in a river with current is an example that students can understand. An additional visual aid for this scenario is to envision a cork placed in the river following the south motion of the river while the boat travels to the east (Vandegrift, 2008). While the cork was added to this problem for the purpose of exploring relativistic motion originally, it can also be a visual aid for students just starting to learn about vectors. Another application scenario that is relatable to daily life is person riding on an escalator. Nearly every student has given in to the temptation of running up the down escalator. Students can easily apply this experience to the concept of vector addition and in particular the concept of the “additive inverse” (Robinett, 2003)

Alternative Vector Instructional Strategies:

A number of activities are widely used by teachers to introduce vectors to students. From personal experience and a literature review I have compiled a list of common strategies for introducing vectors. Some of these activities represent valuable experiences for students but would be more effective during the concept application of a learning cycle rather than during the exploratory or conceptual development phases (Maier and Marek, 2006).

A force table is one of the more common introductory experiences for teaching vectors, mechanical equilibrium and the vector triangle (Greensdale, 2002). From personal experience, this method of introduction is counterproductive for many students. While students are engaged in the exploration phase, the concept development stage is ineffective for many learners because they cannot assimilate the data from the force table with other vector tasks. If preceded by the activities outlined earlier in this document, a force table might serve the role of concept application.

A variation on a method I have tried with a past class is outlined in Erdman’s “Outdoor Vector Lab” (Erdman, 2004). Erdman also expresses frustration with the prevalence of the introductory force table lab. In this lab activity students are tasked to “Find the distance between the two points shown to you” using strings, protractors and measurement devices such as meter sticks or measuring tapes. The points in this lab are on opposite sides of a large structure such


as the school building. As students engage in the problem solving they must decide whether they should focus on resultants or components of the vectors to a common point. This method leads students to a discovery of the orthogonal components of vectors. While the approach provides an exploratory activity it is logistically impractical for many school buildings due to size or student supervision requirements. Similar activities can be completed with the aid of a GPS with great accuracy (Larson, 1998).

Vector treasure hunts are a popular method used to introduce vectors. In this investigation students use a compass to create a treasure map using vectors. The map is then passed to another group for them to follow (Windmark, 1998). This method requires prior knowledge of head to tail addition. Whereas the method used for with the Never Fall toy allows students to discover head to tail addition in the process of exploration.

Another common way of addressing two dimensional vectors is to demonstrate a constant velocity car moving on a piece of long thin craft paper (Mader and Winn, 2008, p101-5). The orientations of the car and paper are varied to illustrate the effect of direction and magnitude of the component vectors on the resultant displacement and velocity of the buggy. This approach provides a good visual but does not allow for effective exploration and data collection by all members of the group. In the past I have found this demonstration effective for some students; however, it has not served well as a common experience to reference with students. While this demonstration is useful in reinforcing the independence of horizontal and vertical motion for the application phase of vectors it does not explicitly address many of the other aforementioned misconceptions.

Concept Application

Once students have developed a comfort with vector representations for displacement and velocity vectors, students will build upon that knowledge with vector representations of acceleration and forces. After these activities students have the requisite knowledge to determine the horizontal and vertical components for the initial velocity of an object launched at an angle. This application should be held onto until students have familiarity with acceleration due to gravity. Doing so will ensure that they understand how the vertical velocity will change after launch.

The math skills and conceptual understanding required for free body diagrams and other force problem applications are the same as for displacement and two-dimensional motion, but students are not always able to make the transition between vector types without discussion of how the vectors represent different physical quantities (Arons, 1997, 108)


Outcomes:During the activities students were engaged and actively questioned observation. At the

outset they had some difficulty managing the materials but once they started to picture the situations they moved swiftly through each of the exercises.

Activity 1: Lady Bug transit

Students traced the motion of the “Never-Fall” Lady bug as it moved around a gridded surface. After tracing the motion, students analyzed each vector in terms of horizontal and vertical displacement. In the process students use basic SOH CAH TOA and trigonometry to compare components of the path vectors to the resultant displacement. Students were able to successfully determine the horizontal and vertical components of each vector without much difficulty. With guidance they were able to determine the total horizontal and total vertical displacement for the toys motion. Students easily applied the Pythagorean Theorem to determine the length of the hypotenuse but only a small portion of students understood the significance of that step in determining the new resultant. Equations involving the determination of the angle were introduced for the first time at this point, so while many were able to manipulate and solve, students were not at a stage at which they could replicate the process. From experience this is a more advanced stage than most students reach upon initial introduction to vectors.

The idea that the summation of the horizontal and vertical components of the motion was equal to the horizontal and vertical components of the resultant displacement was retained by many but not all students as we proceeded through other stages of concept development.

Even though students were previously instructed that the order of vectors does not impact the resultant, the process of switching component values from one group in a random order of vectors was instructional. Some students tried to sneak a look at the partner groups diagram so they could replicate the order and be the ones who get it right. Those students were completely shocked to find that the order of addition is irrelevant to the outcome. While some students struggled with the idea of a fixed starting point, they did not attach to an order for adding one vector before another.

The primary student inquiry during skill focused worksheets, in which resultants were found by analyzing horizontal and vertical components on a grid, was “but where does this answer go?” In other words, students were able to determine the horizontal and vertical components and find the resultant but they were uncertain of how to determine the “starting point” as termed by some of them. After some further guidance many students accepted the idea that vectors do not have a fixed starting point. This is in fact a challenging set of explanations to provide to students, on the one hand we give them rules to construct diagrams in a specific ways to graphically add vectors using either the “tip to tail” or parallelogram method, and then proceed to tell them that the resultant while drawn this way in the vector diagram is


really just a length with direction. It takes a while for students to gain comfort with the idea that the methods are just tools for working with vector values.

The final phase of the activity was a largely successful activity for introducing the reference equations. Some guidance was still required for guidance with identifying variables to substitute into equations. Most students were able to recognize the difference between the addition of parallel vectors and perpendicular vectors by the following day. More than half of the students were able to determine the resultant and angle based on horizontal and vertical components in a introductory problem the following day and during the first portion of “Activity Two: Lady bug on a conveyor belt.” In light of past experience with the force table, lab this activity was highly successful despite the fact that students did not all achieve mastery at this point. A higher number than in previous years demonstrated a functional skill level as evidenced by test outcomes.

Activity 2: Ladybug on a Conveyor Belt

At the outset of Activity Two students showed some frustration with the coordination of the materials; but, this was quickly resolved with some individual instruction for each group. The nature of the data in this lab invites some error but is sufficient to demonstrate the major ideas to students.

This activity serves not only to reinforce the idea of a vector but also to further develop the idea of displacement, speed and velocity. As students work through the lab, instructors must keep in mind that students are in the early stages of concept application in terms of displacement and velocity quantities. Regular checks of understanding are critical. As I walked around the room I heard many conversations between students in a lab group which demonstrated real progress with the application of the Cartesian coordinate grid. While students initially struggled interpreting some of the terminology in the instructions, they were able to give adequate explanations when asked to describe what occurred in a given scenario or to predict an outcome.

Student questions throughout the early phase of the lab revealed a prevalence of a misconceptions highlighted by Aguirre (1988) “The path followed by a moving object depends on the reference frame in which the observer is located.” While the frame of reference for the lab question was in relation to the fixed dry erase board and meter sticks, the concept of the frame of reference was poorly understood by some groups. The activity provided a useful visual for students to establishing a model for future discussions. The method I used to help students understand this idea of reference frame was to query students as to what would be observed from various view points. Examples of queries included: “what would lady bug moving parallel on the vertical path to the first ladybug observe?”; “what would a ladybug moving only due to the horizontal motion of the paper observe?”; and “how are these motions different from a lady bug watching from a completely fixed position near the starting point?”. Each of these could be directly observed through demonstration.


Some students required additional visual reinforcement of frame of reference. Many video clips and animations exist to help with developing this concept. The film entitled “Frames of Reference” (1960) with Professors Hule and Ivey, presented by the Physical Science Study Committee is a classic. Clips are widely available on-line with a simple search (see http://www.archive.org/details/frames_of_reference). Another engaging video series is available free of charge at the Discovery Channel’s website for the MythBuster episode entitled “Vector Vengeance”. The page contains a number of post-show videos for a ball launched from a moving truck. For purposes of juxtaposition, videos from the frame of reference of the camera on the truck and videos from a stationary camera are provided. (see http://dsc.discovery.com/videos/mythbusters-vector-vengeance-high-speed.html) [Since I make reference do I need to provide a link? Is this the proper location for the link? Do these links also get included on the reference page]

A typical activity at this point of concept development is to have students complete a force table investigation as I have done in the past. After introducing the lady bug activity on the moving grid, I developed a new appreciation for the difficulty prior students experienced when attempting to connect observations from the force table lab to motion problems. Force tables do not encourage students to confront preconceptions involving frame of reference or help students develop the characteristics of vector quantities. Understanding the frame of reference is critical to visualizing two dimensional velocity problems.

As students proceeded through the guided worksheet it was evident that they were beginning to make connections to concepts presented during the lecture phase of the introduction. The first part of the activity deals with the parallel velocities. Students are tasked with determining the resultant displacement and velocity of the ladybug toy if both the lady bug and the paper are moving in the same direction. Students recognize quickly that the displacement of the paper with respect to the frame of reference and the displacement of the lady bug on the grid add to equal the total displacement of the bug relative the fixed frame of reference.

After completion of this activity, I noticed a significant decrease in difficulty with practice Regents Questions that involve the influence of the “difference between vector angles” the magnitude of the resultant vector when compared with previous years. (See Appendix C; 6/2003, #2). The level of frustration observed with students from prior years as they struggled with this type of question is what originally inspired the creation of these activities. While a small percentage of students still struggle when presented with this type of scenario, a few prompts relating to this activity are all that is necessary to correct their thinking. Students still struggle with the phrasing of questions and struggle to select the correct multiple choice answer, but with prompts they demonstrate an understanding of the concept.

http://dsc.discovery.com/videos/mythbusters-vector-vengeance-high-speed.html

http://www.archive.org/details/frames_of_reference


The activity also allowed for repeated transition between the concepts of displacement and velocity. At first students had significant difficulty distinguishing between the two vectors quantities but after further prompting they were able to explain the difference between the two values. It became clear that for many students this is an idea that needs to be made more explicit. The difference between displacement and velocity is considered implicit to many instructors including myself; however for this activity the number of inquiries regarding this connection were very revealing regarding student difficulty in separating these two concepts at this stage of learning.

The ultimate goal of this activity was for students to recognize the concurrent nature of the components and to visualize both parallel and perpendicular vectors. It was particularly rewarding to observe the conceptual connection made when students explored perpendicular velocities. This was the first time students showed a true recognition of the idea that a single object was experiencing two different velocities. The resultant displacement and the resultant velocity were represented by the actual path of the toy. While this is a statement that I make frequently in relation to projectiles, this is the first time that students seemed to be able to verbalize the idea to me rather than nod with vague recognition. This realization is an important idea to establish before beginning a discussion of projectiles.

Once students established an idea for independence of horizontal and vertical motion, they moved quickly through the phase of the activity in which the horizontal velocity vector was changed in order to determine the impact on angle of motion. This provides an important framework for many subsequent vector skills. After this activity I was able to introduce the idea of initial velocity components for objects launched at angle without progressing the projectile phase after launch. By the time we address projectile motion several weeks later, students will have had significant practice with two dimensional vectors and some of the difficulty separating the two tasks will be minimized.

Conclusions:

Vectors and vector arithmetic are crucial skills for a successful physics student. The literature supports my personal finding that students find vectors very challenging. The background literature on how students learn vectors and student level of understanding contains many conclusions that instructors at the college and high school level do not realize how much difficulty students have with learning vectors. (Shafer and McDermott, 2005; Knight, 1995; Nguyen and Meltzer, 2002; Arons, 1997). While teachers may in fact gloss over some concepts mistakenly assuming the ideas to be implicit understandings, many teachers attempt to address the various preconceptions referenced within this document. In my opinion many teachers are very aware of the degree of difficulty student encounter when learning vectors, but lack of time and effective tools minimize the number of students who will reach mastery of these skills. Additionally, planning adequate curricular time for this introduction strategy is difficult but necessary for future skills. One of the difficulties in convincing teachers to expand time spent


on vector quantities is the seemingly minimal treatment of complex vector problems on the New York State Regents; however, vector skills underlie a number of questions that might be categorized under a different topic. More investigation would be needed to find out whether student difficulty with some of these concepts arises from a lack of vector knowledge, a lack of understanding of the specific concept or a combination of both.

Despite the challenges with equipment manipulation posed by these activities, both were very effective tools for developing both vector understanding and vector computation skills. Students often referred back to the activities when explaining answers in subsequent drill exercises demonstrating that they served the intended role of a highly visual set of touchstone activities. With each new addition of a vector quantity throughout the curriculum, thought be given to how students visualize the combination of the new vectors quantity, but the characteristics of vectors presented in this unit can be applied. A higher functional skill level with vectors allows students to focus on new challenges while perfecting existing skills rather than feel overwhelmed by the compounding effect of the challenges.

References:

Aguirre, J. (1988). Student Preconceptions about Vector Kinematics. The Physics Teacher, 26, 212-216.

Aguirre, J., & Rankin, G. (1989). College students' conceptions about vector kinematics. Physics Education, 24, 290-294.

Arons, A.B. (1997). Teaching Introductory Physics. New York: John Wiley & Sons, Inc.

Brown, N. (1993). Vector Addition and the Speeding Ticket. The Physics Teacher, 31, 274-276.

Flores, S. (2004). Student use of vectors in introductory mechanics. American Journal of Physics, 72, 460.

Hestenes, D., Wells, M., & Swackhamer, G. (1992). Force Concept Inventory. The Physics Teacher, 30, 141-158.

Hoffman, B. (1975). About Vectors. New York: Dover.

Knight, R. (1995). Vector knowledge of beginning physics students. The Physics Teacher, 33, 74-77.

Larson, R. F. (1998). “Measuring Displacement Vectors with GPS”. The Physics Teacher, 36, 161


Mader, J & Winn, M. (2008). Teaching Physics for the First Time. College Park, MD: The American Association of Physics Teachers.

Maier, S. and Marek, E. (2006). “The Learning Cycle: A Reintroduction.” The Physics Teacher, 44, 109-113

New York State Education Department (a). 2008 Core Curriculum for the Physical Setting/Physics. Retrieved October, 2011 from: www.emsc. nys ed.gov/ciai/mst/pub/phycoresci.pdf

Nguyen, N.-L., & Meltzer, D. (2003). Initial understanding of vector concepts among students in introductory physics courses. American Journal of Physics, 71(6), 630.

Robinett, R.W. “It’s, Like, Relative Motion at the Mall.” The Physics Teacher, 41. 140.

Roche, J. (1997). Introducing Vectors. Physics Education, 32, 339-345.

Shafer, P. S., & McDermott, L. (2005). A research based approach to improving student understanding of the vector nature of kinematic concepts. American Journal of Physics, 73, 921-931.

Vandegrift, G. (2008). “The River Needs a Cork.” The Physics Teacher, 46, 440.

Zollman, D. (1981). “A Quantitative Demonstration of Relative Velocities.” The Physics Teacher, January, p. 44

Widmark, S. (1998). “Vector treasure hunt.” The Physics Teacher. 36, 319

Never Fall Wind-up toy pricing retrieved from http://www.officeplayground.com/Wind-Up-Toys-C35.aspx . September 2011.

How do I reference the many REGENTS EXAMS Included in the APPENDIX and throughout the document?

Do the Videos and Phet Animations recommended to the reader get included here?

http://www.officeplayground.com/Wind-Up-Toys-C35.aspx

http://www.officeplayground.com/Wind-Up-Toys-C35.aspx

http://www.emsc.nysed.gov/ciai/mst/pub/phycoresci.pdf


Appendix A:

Activity One - Student Worksheet: Introduction to Vector Components

Procedure: Map the path of the wind-up toy as it moves around the board.

A. Assign directions on the board representing the directions of +x, -x, +y and -y

B. Fully wind the “Never Fall” ladybug and place at a location on the dry erase board.

C. Trace the motion with a dry erase marker. Use an arrow to indicate the direction of the toy. Each line is a vector. [Optional: Copy the motion of the toy onto a piece of graph paper indicating scale of original grid in centimenters (ie, 1 block equals)]

D. Label each vector with a letter.

E. Determine the horizontal and vertical component of each vector by counting grid blocks. Note the sign of each motion according to grid set-up in Step A. Record in table provided below.

F. Determine the total horizontal and vertical components by finding the sum of each column.

G. Create a triangle using the total horizontal and vertical components from the table (Draw in the space below, horizontal first). Using your knowledge of geometry and trigonometry, what is the length of the hypotenuse of the triangle? What is the angle between the hypotenuse and the horizontal component? Based on the X and Y values, in what quadrant is this vector?

Horizontal X Vertical Y

ABCDE

∑ ∑


H. Draw a line from the start point to the end point of the ladybug motion with an arrow pointing toward the end point. This line is called the resultant.

I. Determine the horizontal and vertical components of the resultant vector by counting on the grid.

X = Y =

J. Draw a triangle with horizontal and vertical components from the previous step keeping direction of the components in mind. Determine the length of the hypotenuse using Pythagorean Theorem. Provide a scale based on the length of one block on the reference grid.

K. Write each of the vector pairs from the table in step E onto an individual card. Mix up order of the cards and exchange with another group. On a piece of graph paper, diagram the motion of the other teams toy using vector skills. Determine the resultant. Compare the values with the other group.

L. Motion without a grid - Starting the ladybug from near the center of a blank dry erase board, allow it to move one single path to the edge of the board. The lady bug must travel a path that is entirely horizontal or vertical. Trace the vector and indicate direction with an arrow. Measure the length in cm of the vector and determine the angle of the vector relative to the horizontal axis. Record values in the table below.

M. On the dry erase board, draw the horizontal and vertical components of the vector indicating appropriate direction. Measure the length of each vector and use a + or – to indicate appropriate direction on either axis. Record values in table below.

Magnitude or length of Vector (cm)

Angle or direction of vector relative to x axis


N. Draw a scaled version of the resultant displacement and the horizontal and vertical components in the space provided below: [draw horizontal and then vertical when making triangle (ie ). This is a standard form that will allow us all to have the same reference angle)

O. Using your knowledge of SOH CAH TOA determine the angle between the resultant and the horizontal. Show your work.

Questions

1. How do the hypotenuse from step H and the hypotenuse for step J compare?

2. What conclusion(s) can you draw from this finding?

3. In Step K you were not provided with any order for adding vectors. Did the random order have any impact on the magnitude or direction of the resultant vector?

Magnitude of length of the horizontal component (cm)

Magnitude or length of the vertical component (cm)


4. What is the vector term for the hypotenuse of triangle that results from combining the horizontal and vertical vectors?

5. What type of vector quantity is represented in this activity?

Conclusion (Complete in the space below): Compare the distance traveled to the displacement of the lady bug toy. Explain the difference in process for finding each value. In your comparison, explain the terms vector and scalar in terms of the concepts of distance and displacement.


Appendix B: Student Worksheet #2: Ladybug on a conveyor belt

Purpose: To investigate and represent the motion of an object experiencing two simultaneous (concurrent) velocities. Vector addition and vector resolution will be used to analyze the motion of a wind-up toy.

Materials:

Wind-up ladybug 3 Meter sticks Dry erase grid Large dry erase boardStop watch

Procedure: A. Motion of the toy is due to wind up device. Before all experiments the toy will be fully

wound. B. The paper moves along the horizontal x axis by pulling it at roughly constant speedC. The relative motion of the objects is found by moving both the paper and the toy

between stationary meter sticks

Prior to all data collection students should practice coordinating materials and establish a realistic division of labor. See your instructor if you need assistance with this step

Data and Calculations

1. Determine the average speed of the ladybug as it moves toward the right across a stationary grid. (Take the average of 3 trials). You will use this average speed as a reference value throughout the lab.

Toy DataDistance (cm) Time Speed

AVERAGE


2. Determine the resultant velocity of the toy and the paper if both have a rightward velocity. Attempt to move the paper at a speed similar to the toy. Fill in measurements on the table below

Time of Interval

Displacement of paper relative to meter stick

Displacement of ladybug relative to the grid

Displacement of the ladybug relative to the meter sticks (resultant)

Velocity of the paper

Velocity of the ladybug on the paper

Resultant velocity of the ladybug

a) Show your work used to determine the velocity values for the chart above

b) Construct a vector diagram that shows how the displacement of the toy and the displacement of the paper add to the resultant displacement of the toy. (Label each vector with a magnitude including units. Does not have to be drawn to scale but should show relative size.)

c) Construct a vector diagram that shows how the velocity of the toy and the velocity of the paper add to the resultant velocity of the toy. (Label each vector with a magnitude including units. Does not have to be drawn to scale but should show relative size.)

d) Since the paper and the lady bug are both moving in the same direction, how would we define the angle between their motions?


3. Move the paper to the left at a similar constant speed to that of the rightward moving lady bug for several seconds. As a group record the following. Note the direction of motion with a positive or negative sign.

Time of experiment

Displacement of paper relative to meter stick

Displacement of ladybug relative to the grid





a) Construct a labeled displacement diagram

b) Construct a labeled velocity diagram

c) What is the difference in direction (angle) between the papers velocity and the ladybugs velocity?

4. Vector Rule: The maximum resultant occurs when the vectors are arranged at an angle of ____________ or similar direction. The minimum resultant occurs when the vectors are arranged at an angle of ______________ or opposite direction. Howcould get a different resultant without changing the magnitude (size) of the velocity vectors.


5. With the toy starting at the bottom left corner of the paper and pointed upward, pull the paper to the right at a pace close to that of the toy. Let it run until the toy reaches a point at the top. On the dry erase board, draw a line that connects the start and end point (ie, the resultant displacement) Fill in data on the chart below

Time of experiment

Horizontal Displacement of paper relative to meter stick

Vertical Displacement of ladybug relative to the grid





Magnitude:

Angle:

Magnitude:

Angle:

Construct displacement and velocity vector diagrams showing components and resultants.

6. Predict what would happen to the angle of motion if you pull the paper at twice the speed to the right while the lady bug moves upward. Roughly draw the vectors (Hint: think of the paper as the x vector and the toy as the y vector)


7. Complete the scenario presented in the previous question. (ie lady bug upward and paper 2X velocity right)

Time of experiment

Horizontal Displacement of paper relative to meter stick

Vertical Displacement of ladybug relative to the grid





Magnitude:

Angle:

Magnitude:

Angle:

a. Draw a labeled vector diagram of the resultant displacement and horizontal and vertical components.

b. Determine the resultant velocity (magnitude and direction). Show work

8. Predict what would happen to the angle of motion if the velocity of the paper were moving to the right at a velocity about half that of the lady bug. Draw the predicted


vector diagram. Complete this scenario with the materials. Does the actual motion match the prediction?

9. Draw the observed resultant path (and components) for the lady bug moving upward and the paper moving to the left (use similar speeds for both the paper and the ladybug). Label the components with the appropriate sign (+ or -).

10. Draw the observed resultant path (and components) for the lady bug starting from the top of the paper and moving downward while the paper is moving rightward. (use similar speeds for the paper and the ladybug)

11. Draw the observed resultant path (and components) for the lady bug starting from the top of the paper and moving downward while the paper is moving leftward at about twice the speed of the lady bug.

12. a) Determine the horizontal component of the motion of the paper if the lady bug travelled with a resultant speed of 14.2 m/s and a vertical speed of 8 m/s. Show your work.

Hint: V= Vx = Vy =


b) What was the angle of the resultant for the previous problem if the lady bug was moving upward and the paper was moving left? (show work with equation and substitution)

c) What will be the resultant displacement of the lady bug after 4 seconds? Show work (include magnitude and directions)

13. When an object is launched at an angle, the initial or start velocity can be broken down into two components, velocity directed horizontally and velocity directed vertically. What is the launch velocity of an object with a horizontal component of 40 m/s and a vertical component of 30 m/s?

14. What are the initial horizontal and vertical components of an object’s velocity if it is launched at 35 m/s at an angle of 60 degrees?

Conclusions: (to be completed on a separate paper and attached)


Explain how the addition of vector quantities is different from addition of scalar quantities. Differentiate between the terms speed and velocity. Explain why vectors are an important concept in motion and how they can be useful in other aspects of physics (Think about real life scenarios in which this applies). Summarize the methods of combining horizontal motion and vertical motion to determine a resultant and the methods for finding the horizontal and vertical components of a resultant vector. Explain how the angle between the vectors influences the magnitude of the resultant vector. Be sure to define terms used in your explanations.

APPENDIX C: Sample Vector Problems on the NYS Regents 2003 to 2011

NYS Regents Exam

NYS Core Ref

Some of the assessment items listed here deal with force vectors. They were included to illustrate a problem


type. Most can be easily modified to apply to either displacement or velocity. Scalar is to vector as(1) speed is to velocity (3) displacement is to velocity (2) displacement is to distance (4) speed is to distance

June 2011 #1

St 4, KI 5.1

A model airplane heads due east at 1.50 meters per second, while the wind blows due north at 0.70 meter per second. The scaled diagram below represents these vector quantities.

a) Using a ruler, determine the scale used in the vector diagram. [1]b) On the diagram above, use a protractor and a ruler to construct a vector to represent the resultant velocity of the airplane. Label the vector R. [1]c) Determine the magnitude of the resultant velocity. [1]d) Determine the angle between north and the resultant velocity. [1]

June 201166-69

St 1, KI 1St 4, KI 5.1

A baseball player runs 27.4 meters from the batter’s box to first base, overruns first base by 3.0 meters, and then returns to first base. Compared to the total distance traveled by the player, the magnitude of the player’s total displacement from the batter’s box is(1) 3.0 m shorter (3) 3.0 m longer(2) 6.0 m shorter (4) 6.0 m longer

June 2010#1

St 1, KI 1St 4, KI 5.1

A motorboat, which has a speed of 5.0 meters per second in still water, is headed east as it crosses a river flowing south at 3.3 meters per second. What is the magnitude of the boat’s Resultant velocity with respect to the starting point?(1) 3.3 m/s (3) 6.0 m/s(2) 5.0 m/s (4) 8.3 m/s

June 2010 #2

St1, KI 1St 4, KI 5.1

A machine fired several projectiles at the same angle, , above the horizontal. Each projectile was fired with a different initial velocity, vi.

June 2010#51-53

St 1, KI1St 4, KI


The graph below represents the relationship between the magnitude of the initial vertical velocity, viy, and the magnitude of the corresponding initial velocity, vi, of these projectiles.

a) Determine the magnitude of the initial vertical velocity of the projectile, viy, when the magnitude of its initial velocity, vi, was 40. meters per second. [1]b)Determine the angle above the horizontal at which the projectiles were fired. [1]c)Calculate the magnitude of the initial horizontal velocity of the projectile, vix, when the magnitude of its initial velocity, vi, was 40. meters per second. [Show all work, including the equation and substitution with units.] [2]

5.1

The speedometer in a car does not measure the car’s velocity because velocity is a(1) vector quantity and has a direction associated with it(2) vector quantity and does not have a direction associated with it(3) scalar quantity and has a direction associated with it(4) scalar quantity and does not have a direction associated with it

June 08 #1

St 1, KI 5.1

June 08#41

St 1, KI 1St 4, KI 5.1


A dog walks 8.0 meters due north and then6.0 meters due east.

a) Using a metric ruler and the vector diagram, determine the scale used in the diagram. [1]

b) On the diagram above, construct the resultant vector that represents the dog’s total displacement. [1]

c) Determine the magnitude of the dog’s total displacement. [1]

June 08 #55-57

St 1, KI 1St 4, KI 5.1

Two forces act concurrently on an object. Their resultant force has the largest magnitude when the angle between the forces is(1) 0° (3) 90°(2) 30° (4) 180°

Jan 2008#38

St 4, KI 5.1

A student on her way to school walks four blocks east, three blocks north, and another four blocks east, as shown in the diagram.

Jan 2006 #12


Compared to the distance she walks, the magnitude of her displacement from home toschool is(1) less(2) greater(3) the same

June 2006#3

St 4, KI 5.1

A 3-newton force and a 4-newton force are acting concurrently on a point. Which force could not produce equilibrium with these two forces?(1) 1 N (3) 9 N(2) 7 N (4) 4 N

June 2006 #6

St 4, KI 5.1

Velocity is to speed as displacement is to(1) acceleration (3) momentum(2) time (4) distance

June 2004 #1

St 4, KI 5.1


June 2004 #2

A vector makes an angle θ, with the horizontal. The horizontal and vertical components of the vector will be equal in magnitude if angle θ is(1) 30° (3) 60°(2) 45° (4) 90°

June 2003#2

A hiker walks 5.00 kilometers due north and then 7.00 kilometers due east.What is the magnitude of her resultant displacement?What total distance has she traveled?

June 2003 # 51 and 52


The diagram below shows a worker using a rope to pull a cart.

The worker’s pull on the handle of the cart canbest be described as a force having(1) magnitude, only(2) direction, only(3) both magnitude and direction(4) neither magnitude nor direction

January 2003#2

A car travels 90. meters due north in 15 seconds. Then the car turns around and travels 40. meters due south in 5.0 seconds. What is the magnitude of the average velocity of the car during this 20.-second interval?(1) 2.5 m/s (3) 6.5 m/s(2) 5.0 m/s (4) 7.0 m/s

January 2003 #2

Which pair of forces acting concurrently on an object will produce the resultant of greatest magnitude?

1/2003#36

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