I. : Introduction

The physicist in each of us wants to explain the natural world we observe. However, before any explanations can be attempted, accurate observations must be made of what we wish to explain. An observation is a record of what your senses tell you about the natural world. Very often, these observations are expressed as measurements. When a measurement is made, a number with a unit is associated with the observation. For example, when someone says “1.2 seconds,” “1.2” is the number, and “seconds” is the unit. Consider another example: a car is going very fast, but a measurement of speed is needed to determine if a speeding ticket is appropriate. The importance of measurement for understanding the natural world is nicely expressed by the following:

Throughout this teacher resource, many activities will involve measurement of kinematics quantities. The natural world can be thought of as consisting of space, time, and matter. Space describes where you are in the natural world. Because it is possible to move [up or down]; [left or right]; and [forward or backward], space is said to have three dimensions or to be three-dimensional. Activities #5, 6, 7, and 8 in this teacher resource deal with ideas concerning where something is located in space.

Time is associated with where you are in the sequence: past, present, and future. Unlike space, where you have freedom of movement, time sweeps you along as it passes. You have your “place in time,” but are free to move about in space. Activities #1, 2, 3, and 4 deal with ideas of where something is located in time.

With only space and time, the natural world would be empty. “Matter” is the physical material that exists in space and time. All tangible objects are made of matter. Some of the characteristics or properties of matter are mass, volume, density, color, and temperature.

In the rest of this teacher resource, you will deal not only with space and time, but also with motion, which involves all three of the quantities space, time, and matter. When something (i.e., matter) moves, it changes its position in space during a certain time interval. The quantities used to describe motion are time, position, distance traveled, displacement, speed, velocity, and acceleration. Activities #9 to 22 deal with concepts related to speed and velocity. Activities #23 to 49 deal with concepts related to acceleration. The AAPT/PTRA’s Teaching About Newton’s Second Law teacher resource (in process of publication) continues this story by introducing the effect of forces and mass on motion.

The Teaching About Kinematics teacher resource has a variety of activities. Students in middle school can do many of them, while others are appropriate only for high school students. The following descriptions indicate the grade level(s) that are typically appropriate for each activity. The activities are arranged in a logical developmental sequence, so teachers can select from the activities as listed or pick and choose activities that fit into their own sequence of lessons for students.

The following is a possible scope and sequence for the introduction and development of the topic Kinematics/Motion using Teaching About Kinematics teacher resource.

• 1.

  Discuss the meaning of Newton’s quote about motion.¹ (See page 3.) This quote expresses the idea that motion is a fundamental concept that underlies the understanding of many other topics.

• 2.

  Ask students to imagine and describe a world with no MOTION—describe and discuss (Pair Share).² No motion = “frozen” world (i.e., like a photograph), thus no time passes. From this discussion the need to develop an understanding of time and space emerges.

• 3.

  Following this introduction, the following possible scope and sequence can help to answer the question, “Where are you in time?”

  • Activity #1, “Significant Times” What does the word “time” mean? (Pair Share)² Discuss time as an instant vs. time as an interval. When we ask, “What time does class end?” we are referring to an instant in time. When we ask, “How long (i.e., much time) does it take to drive home?”—we are referring to time as an interval. A time interval is always the difference between two instants in time (i.e., a time interval = Δt = t_f – t_i). The symbol Δt represents the time interval while t_f represents the instant of time at the end of the interval, and t_i represents the instant of time at the beginning of the interval. Time intervals can be equal for different initial times (e.g., the time interval can be the same for several class periods even if the class periods start at different times.) For studying motion, we are often interested in the time interval, which is generally just called “time.” This activity is appropriate for middle or high school students, and has a goal of distinguishing between time as an instant and time as an interval. The activity also introduces use of significant digits.

  • Activity #2, “Making a One-Second Timer” This activity, appropriate for middle or high school students, is an example of an inquiry activity. Provide the learner with the question, but not with the method or solution. See teacher’s notes on page 28. This activity is designed to introduce the student to the measurement of time as an inquiry activity. The operational definition for the period of a pendulum is developed. Students will gain experience with the use of a stopwatch and learn how to minimize measurement error.

  • Activity #3, “Pendula on Parade” This activity is appropriate for middle or high school students and is designed to introduce the relationship between the period of a pendulum and the length of the pendulum. Other variables, such as mass of a pendulum bob and the size of swing, could also be explored. Most people are surprised to learn that the mass has no effect on the period, and that the angle of the swing has only a small effect. If the angle is 30 degrees, you must add 2% to expected value for the period, for 60 degrees, 11%, and for 90 degrees, 45%. Thus if the angular effect on the period is to be ignored, the angle should be less than 30 degrees. At this point a full-scale pendulum laboratory activity can be done to investigate the effect of the mass of the bob, the length of pendulum, and the angle of swing on the period of a simple pendulum. This is a nice activity in which to use whiteboarding³ or other public presentation technique. The teacher’s notes on page 34 discuss finding the period of a pendulum by graphical analysis.

  • Activity #4, “Investigation of a Flashing Light” This activity is appropriate for students who are familiar with the slope-intercept equation of a straight line. It can be done using the Modeling Method of Physics Instruction to demonstrate the relationship between frequency and period of a flashing light. See activity teacher’s notes on page 36 for a description of the activity based on the Modeling Method of Physics Instruction. Also see http://modeling.asu.edu, referenced 20 August 2008.

• 4.

  The following activities help to answer the question, “Where are you in space?”

  • Activity #5, “Traveling Washer in One Dimension” This activity is appropriate for middle or high school students and is designed to point out the differences among initial position, final position, distance traveled, and displacement. This distinction is important when you are pointing out the difference between speed and velocity, and when you are using kinematics equations.

    Describe and discuss the question, “How do you know where you are in space?” (Pair Share).⁴ Establish that the position of a point on an object is described relative to some reference point. The location of the reference point is arbitrary. Examples of reference points are the origin of a graph, the prime meridian, the equator, and sea level.

    Position is the length (and direction) of a straight line from a reference point. The displacement is the final position (d_f ) minus initial position (d_i). Displacement = d_f – d_i or Δd = d_f – d_i, “as the crow flies.”

    Distance traveled is the length of the path traveled (i.e., like an odometer reading in a car), and is often not the same as the displacement.

  • Activity #6, “Traveling Washer in Two Dimensions” This activity is appropriate for middle or high school students and is designed to point out that direction is an important part of position and displacement. The concept of a vector quantity begins to emerge. This activity also points out that position can involve more than one dimension in space. Students will review the use of a magnetic directional compass and how directional angles are measured (e.g., azimuth angle and position angle).

  • Activity #7, “Position vs. Time Graphs Using a Motion Probe” with a Graphing Calculator or Computer. This activity is appropriate for middle and high school students. Students produce graphs of position vs. time in a very kinesthetic manner.

  • Activity #8, “Where am I?” This worksheet is appropriate for high school students. On this worksheet the teacher can have different groups of students do questions #1, 3, 4, 6, and 7. Teachers can elect to have students do these questions by scale drawings and/or by calculations. Questions #1, 2, 3, 5, 6, 9, and 10 are rectangular-coordinate to polar-coordinate questions, while questions #4, 7, and 8 are polar-coordinate to rectangular-coordinate questions. See notes on coordinate systems on page 49. This is a good opportunity to use whiteboards⁵ or other class presentation and discussion techniques.

    Discuss coordinate systems used for specifying location of objects in space (e.g., rectangular, polar, spherical), and how these relate to 1-, 2-, and 3-dimensional space.

• 5.

  The following activities help to answer the question, “How fast are you going?”

• A.

  Activity #9, “Measurement of Speed on a Smooth and Level Surface” This activity is appropriate for middle or high school students and is used to develop the equation for constant speed $(d=vΔt)$ by graphical analysis. Notes on graphing techniques are included in the teacher’s notes on page 69.

• B.

  Turnpike story This is an example of a story to illustrate the difference between average speed and instantaneous speed.

• C.

  Activity #10, “Comparison of Average Speed and Final Speed” This activity is appropriate for middle or high school students and is designed to illustrate that when an object moves with constant speed the instantaneous speed at any time is equal to the constant average speed. Note that constant speed is like cruise control.

• D.

  Activity #11, “Comparing Linear and Circular Speed” This activity compares the speed of a toy car moving in a straight line with its speed while moving in a circle. This inquiry activity is appropriate for middle or high school students.

• E.

  Activity #12, “Making Graphs of Constant Speed Using Vibrating Timer Tape” In this activity students make a graph using vibrating ticker tape timer. The tape is cut into intervals and pasted on graph paper to make a speed vs. time graph. This activity is appropriate for high school students. See Activity #34 (Teacher’s Notes) for information about making a graph with a ticker tape.

• F.

  Activity #13, “Constant Speed Problem” This sample problem can be solved both graphically and algebraically. This activity reviews results of previous activities and is appropriate for high school students. Use whiteboarding⁶ technique.

• G.

  Activity #14, “Pace Yourself” This activity provides a tactile experience of constant speed, and an introduction to the wave equation $v=fl$⁠. This activity is appropriate for high school students.

• H.

  Activity #15, “Worksheet – Constant Speed” Sample questions can be used as practice. Appropriate for high school students.

• I.

  Activity #16, “Measurement of Speed on an Inclined Plane (Average vs. Final Speed)” This activity is designed to develop the relationship among initial, final, and average speeds for an object moving with constant linear acceleration. Measure average, initial, and final speed of a toy car rolling down an inclined plane. The activity is very similar to Activity #10 except that the car is accelerating down an inclined plane. The activity is appropriate for middle school and high school students.

  Develop the formula for $v(average)=vf+vi2$⁠. Note that this equation is only true if the object is moving in a straight line and has a constant acceleration. It is called the “Average Equation.”

• J.

  Activity #17, “Velocity vs. Time Graphs Using a Motion Probe” with a Graphing Calculator or Computer. This activity is appropriate for middle and high school students. Students produce graphs of velocity vs. time in a very kinesthetic manner.

• K.

  Activity #18, “Relative Motion and Related Rate Activities” This activity is appropriate for high school students. For example, consider the motion of astronauts working on the space station.

  • Domino Method for adding relative motion vectors. $v→1−3=v→1−2+v→2−3$

  • Walking on a people-mover or escalator.

  • Boat traveling down river (with current) or up river (against current).

  • Boat traveling perpendicular to current.

  • How to move a boat straight across a flowing river.

• L.

  Activity #19, “Finding Speed & Velocity of a Car Traveling with Uniform Circular Motion” Measure the average and instantaneous speed and velocity of an object (e.g., tethered toy car, flying airplane, phonograph turntable, and so on) moving in a circle at a constant speed. This activity is designed to illustrate that the average speed and average velocity are not always numerically equal, and that the magnitude of the instantaneous velocity is equal to the value of the instantaneous speed. Teachers can introduce the value of π as “How much greater is the distance around a circle than the distance across the circle?” See teacher’s notes on shoestring and tennis-ball can example. (“Among friends π = 3.”)⁷ This activity is appropriate for high school students.

  • Measure and compare the average speed for quarter, half, three-quarters, and full circle.

  • Measure the instantaneous speed.

  • Measure and compare the average velocity for quarter, half, three-quarters, and full circle.

  • Measure the instantaneous velocity at several points on the circle.

  Discuss instantaneous speed as the average speed measured over a very small period of time. Ideally the instantaneous speed is the limit of the average speed as the time interval approaches zero.

  $v(instantaneous)=limΔt⇒0v(average)$

  Discuss the difference between speed and velocity. Discuss use of words and symbols. See notes on pages 285 and 286.

• M.

  Activity #20, “Flying in Circles” This is an alternative way of measuring the speed of an object moving with uniform circular motion. This activity is appropriate for middle or high school students.

  Discuss average speed as the distance traveled divided by the time of travel. This equation is called the “speed equation.” In Algebra I classes this is often called the rate equation.⁸

  $average speed=distance traveledtime intervalor in symbolsv¯=dΔtord=v¯Δt$
  Average velocity is the displacement divided by the time interval of travel. This equation is called the “velocity equation.”

  $v→(average)=d→f−d→itf−tiord→f=d→i+v¯(average)Δt$
• N.

  Activity #21, “Kinematics of a Student – Speed” This activity provides a tactile experience of constant speed. It is appropriate for high school students.

• O.

  Activity #22, “Worksheet – Motion with Constant Speed” This is a worksheet to review the basic ideas of speed using graphs and equations. It includes a review of using slope of a position vs. time graph to determine velocity, and of using the area under a velocity vs. time graph to find displacement.

• 6.

  The following activities help to answer the question, “What is your acceleration?”

  • A.

    Activity #23, “Position, Velocity & Acceleration vs. Time Graphs Using ‘Moving Man’” This activity uses a computer simulation titled “Moving Man” and is appropriate for both middle and high school students to investigate motion graphs.

  • B.

    Activity #24, “Freely Falling Object I, Relationships between Velocity and Time of Falling and Distance Fallen” This activity uses a PASCO ME-9207B freefall adapter apparatus and graphical analysis of data to determine kinematics relationships. Part 1 explores the relationship between final velocity and the time of falling for a freely falling object. Part 2 explores the relationship between final velocity and the distance of falling for a freely falling object. Part 1 analysis of the data results in the equation “final velocity is equal to acceleration due to gravitational force multiplied by time interval.” In symbols $vf=ag(Δt)$⁠. Part 2 analysis of the data results in the equation “final velocity squared is equal to twice the acceleration due to gravitational force multiplied by distance fallen.” In symbols $vf2=2ag(Δd)$⁠. This activity is appropriate for high school students who have studied algebra up to the quadratic equation. If the apparatus for measuring time for a falling object is not available, then substitute Activity #28, “The Case of the Slope Shifter,” or a computer simulation such as Activity #27 “Freely Falling Object IV, Free Fall Simulation.”

  • B.

    Activity #25, “Freely Falling Object II, Relationship between Distance Fallen and Time of Falling” This activity uses a PASCO ME-9207B freefall adapter apparatus and graphical analysis of data to determine the relationship between distance of fall and time of falling for a freely falling object. Analysis of the data results in the equation, “Displacement is equal to 1/2 the acceleration multiplied by time interval squared.” In symbols Δd = 1/2 a(Δt)². This activity is appropriate for high school students who have had algebra up to the quadratic equation.

  • C.

    Activity #26, “Freely Falling Object III, Relationship between Distance Fallen and Time of Falling” If a PASCO ME-9207B freefall adapter apparatus is not available, this is an alternative activity for Activity #24 and #25 for the development of the Δd = 1/2 a(Δt)² equation, and is appropriate for high school students who have had algebra up to the quadratic equation.

  • D.

    Activity #27, “Freely Falling Object IV, Relationship between Distance Fallen and Time of Falling” This is a Freefall Simulation Activity. It is a second alternative to Activity #24 and #25. Students must have access to the Internet to do this simulation.

  • E.

    Activity #28, “The Case of the Slope Shifter” This is a “tongue in cheek” series of four activities called chapters, with a story line to develop the kinematics equations from experimental data using a vibrating timer. The first chapter is about the equation “change in velocity is equal to acceleration multiplied by time interval.” In symbols, $Δv=a(Δt)$⁠, or $vf=vi+a(Δt)$⁠. The second chapter is about the equation “displacement is equal to 1/2 acceleration multiplied by time squared.” In symbols, Δd = 1/2 a(Δt)².

    This equation is also developed in Activity #25, 26, or 27. The third chapter is about the equation “average velocity is equal to (final velocity plus initial velocity) divided by two.” In symbols

    $v(average)=vf+vi2.$

    This equation is also developed in Activity #16. The last chapter is about the equation “Final velocity squared is equal to initial velocity squared plus twice the product of acceleration and displacement.” In symbols, $vf2=vi2+2a(Δd)$⁠.

  • F.

    Activity #29, “Worksheet on Straight Line Motion Equations and Graphs”

  • G.

    Activity #30, “Measurement of Acceleration Due to Gravitational Force in a Variety of Ways” This activity is appropriate for high school students. It lists several different techniques that can be used to determine the acceleration of an object near the surface of the Earth.

  • H.

    Activity #31, “Measurement of Acceleration on an Inclined Plane” This activity is appropriate for middle school and high school students, and is designed to give students practice calculating the value of a constant acceleration using the basic kinematics equations. Students calculate acceleration of a toy car on an inclined plane and compare the results gotten, using different equations. This calculation can be done three ways depending on the background of the student/class.

  • I.

    Activity #32, “Measurement of Reaction Time with a Falling Ruler” This activity is an indirect method to determine reaction time using a dropped ruler. Note d = 1/2 a(Δt)² (“mechanical drawing” equation). This activity is appropriate for middle or high school students.

  • J.

    Activity #33, “Foot and Hand Reaction Times” This activity is an indirect measurement and comparison of reaction time of the hand and the foot. This activity reviews Activity #31 and is appropriate for middle school and high school students.

  • K.

    Activity #34, “Finding Speed and Acceleration with a Vibrating Timer” and/ or #36, “Acceleration due to Gravitational Force Using a Vibrating Timer” These activities use stroboscopic (e.g., spark gap, vibrating timer, and ticker tape timer) data to make graphs of position, velocity, and acceleration vs. time. These activities are appropriate for high school students.

  • L.

    Activity #35, “Worksheet – Graph Hopscotching”

  • M.

    Activity #37, “Worksheet – Going Up and Coming Down”

  • N.

    Activities #38, #39, #40, and #41 use a liquid level accelerometer to investigate the “Classification of Motion.” This series of four activities, appropriate for middle and high school students, is designed to develop an understanding of common characteristics of different types of acceleration. The basic types of acceleration include:

    • Speeding Up

    • Slowing Down

    • Changing Direction

    Students often believe that the direction of the acceleration and velocity are the same. However acceleration (i.e., speeding up) and deceleration (i.e., slowing down) can be positive or negative when the velocity is positive. Acceleration (i.e., speeding up) and deceleration (i.e., slowing down) can be negative or positive when the velocity is negative. Here is another way to think of this. If you are speeding up, the velocity and the acceleration must have the same direction. An example is dropping a ball. The velocity and the acceleration are in the same direction so the ball speeds up. If you are slowing down, the velocity and the acceleration must have the opposite direction. An example is throwing a ball up into the air. The velocity and the acceleration are in the opposite direction, so the ball slows down. The two activities of Simple Harmonic Motion are designed to help students confront and overcome this misconception.¹¹

    • –

      Demonstrate and discuss the liquid level accelerometer on a cart with springs.

    • –

      Demonstrate and discuss the liquid level accelerometer as a pendulum.

    • –

      Demonstrate the liquid level accelerometer on a record turntable or lazy susan.

    • –

      Discuss the observation that we cannot feel speed, but we can feel acceleration. Newton’s first law (aka, the law of inertia) is a statement of what we cannot feel.

  • O.

    Activity #42, “Kinematics of a Student– Acceleration” This activity provides a tactile experience of constant acceleration. It is appropriate for high school students.

  • P.

    Activity #43, “Differentiating Among Position, Velocity & Acceleration” This is a worksheet appropriate for high school students to review the signs for position, velocity, and acceleration.

  • Q.

    Activity #44, “Signs of Our Times (A Line Dance)” This song gives students kinesthetic examples for the sign of position, velocity, and acceleration, thereby reinforcing the concepts from Activity #43.

  • R.

    Activity #45, “Worksheet on Graphs with Direction”

  • S.

    Activity #46, “An ‘Acceleration Song’” This is a fun song for all ages about acceleration net force and types of acceleration.

    At this point you can do a demonstration of speed, velocity, and direction of force for a toy car (e.g., Stomper Car) traveling in a circle. Set up a tunnel that is along a tangent to the circle and cut or burn the string so the toy car will be able to “Drive through the Tunnel.”

  • T.

    Activity #47, “The Racetrack Game” This game illustrates the difference between speed and acceleration as well as the difference among acceleration that is speeding up, slowing down, and changing direction. This activity is appropriate for middle school and high school students.

  • U.

    Activity #48, “Yellow Light Problem” This activity provides a real-life application of kinematics concepts and equations. The goal of the laboratory activity is to investigate the effect of the time of a yellow light on the safety of a traffic intersection.

  • V.

    Activity #49, “Constant Acceleration Problem” This activity reviews both graphic and algebraic results of previous activities and is appropriate for high school students. Use whiteboarding technique.