PHYSICS Principles and Problems Chapter 2: Representing Motion CHAPTER 2 Representing Motion BIG IDEA You can use displacement and velocity to describe an object’s motion. CHAPTER 2 Table Of Contents Section 2.1 Picturing Motion Section 2.2 Where and When? Section 2.3 Position-Time Graphs Section 2.4 How Fast? Click a hyperlink to view the corresponding slides. Exit SECTION Picturing Motion 2.1 MAIN IDEA You can use motion diagrams to show how an object’s position changes over time. Essential Questions • How do motion diagrams represent motion? • How can you use a particle model to represent a moving object? SECTION Picturing Motion 2.1 Review Vocabulary • Model a representation of an idea, event, structure or object to help people better understand it. New Vocabulary • Motion diagram • Particle model SECTION 2.1 Picturing Motion All Kinds of Motion • Perceiving motion is instinctive—your eyes pay more attention to moving objects than to stationary ones. Movement is all around you. • Movement travels in many directions, such as the straight-line path of a bowling ball in a lane’s gutter, the curved path of a tether ball, the spiral of a falling kite, and the swirls of water circling a drain. SECTION 2.1 Picturing Motion All Kinds of Motion (cont.) • When an object is in motion, its position changes. Its position can change along the path of a straight line, a circle, an arc, or a back-and-forth vibration. SECTION 2.1 Picturing Motion All Kinds of Motion (cont.) • Straight-line motion follows a path directly between two points without turning left or right. –Ex. Forward and backward, up and down, or north and south. • A description of motion relates to place and time. You must be able to answer the questions of where and when an object is positioned to describe its motion. SECTION 2.1 Picturing Motion Motion Diagrams Click image to view movie. SECTION 2.1 Section Check Explain how applying the particle model produces a simplified version of a motion diagram? SECTION 2.1 Section Check Answer Answer: Keeping track of the motion of the runner is easier if we disregard the movements of the arms and the legs, and instead concentrate on a single point at the center of the body. In effect, we can disregard the fact that the runner has some size and imagine that the runner is a very small object located precisely at that central point. A particle model is a simplified version of a motion diagram in which the object in motion is replaced by a series of single points. SECTION 2.1 Section Check Which statement describes best the motion diagram of an object in motion? A. a graph of the time data on a horizontal axis and the position on a vertical axis B. a series of images showing the positions of a moving object at equal time intervals C. a diagram in which the object in motion is replaced by a series of single points D. a diagram that tells us the location of the zero point of the object in motion and the direction in which the object is moving SECTION 2.1 Section Check Answer Reason: A series of images showing the positions of a moving object at equal time intervals is called a motion diagram. SECTION 2.1 Section Check What is the purpose of drawing a motion diagram or a particle model? A. to calculate the speed of the object in motion B. to calculate the distance covered by the object in a particular time C. to check whether an object is in motion D. to calculate the instantaneous velocity of the object in motion SECTION 2.1 Section Check Answer Reason: In a motion diagram or a particle model, we relate the motion of the object with the background, which indicates that relative to the background, only the object is in motion. SECTION Where and When? 2.2 MAIN IDEA A coordinate system is helpful when you are describing motion. Essential Questions • What is a coordinate system? • How does the chosen coordinate system affect the sign of objects’ positions? • How are time intervals measured? • What is displacement? • How are motion diagrams helpful in answering questions about an object’s position or displacement? SECTION Where and When? 2.2 Review Vocabulary • Dimension extension in a given direction;one dimension is along a straight line; three dimensions are height, width and length. New Vocabulary • Coordinate system • Vector • Origin • Scalar • Position • Time interval • Distance • Displacement • Magnitude • Resultant SECTION 2.2 Where and When? Coordinate Systems • A coordinate system tells you the location of the zero point of the variable you are studying and the direction in which the values of the variable increase. • The origin is the point at which both variables have the value zero. SECTION 2.2 Where and When? Coordinate Systems (cont.) • In the example of the runner, the origin, represented by the zero end of the measuring tape, could be placed 5 m to the left of the tree. SECTION 2.2 Where and When? Coordinate Systems (cont.) • The motion is in a straight line, thus, your measuring tape should lie along that straight line. The straight line is an axis of the coordinate system. SECTION 2.2 Where and When? Coordinate Systems (cont.) • You can indicate how far away an object is from the origin at a particular time on the simplified motion diagram by drawing an arrow from the origin to the point representing the object, as shown in the figure. SECTION 2.2 Where and When? Coordinate Systems (cont.) • The two arrows locate the runner’s position at two different times. • Because the motion in the figure below is in one direction, the arrow lengths represent distance. SECTION 2.2 Where and When? Coordinate Systems (cont.) • The length of how far an object is from the origin indicates its distance from the origin. SECTION 2.2 Where and When? Coordinate Systems (cont.) • A position 9 m to the left of the tree, 5 m left of the origin, would be a negative position, as shown in the figure below. SECTION 2.2 Where and When? Vectors and Scalars • Quantities that have both size, also called magnitude, and direction, are called vectors, and can be represented by arrows. – Vector quantities will be represented by boldface letters. • Quantities that are just numbers without any direction, such as distance, time, or temperature, are called scalars. – Scalars quantities will be represented by regular letters. SECTION 2.2 Where and When? Vectors and Scalars (cont.) • The difference between the initial and the final times is called the time interval. SECTION 2.2 Where and When? Vectors and Scalars (cont.) • The common symbol for a time interval is ∆t, where the Greek letter delta, ∆, is used to represent a change in a quantity. SECTION 2.2 Where and When? Vectors and Scalars (cont.) • The time interval is defined mathematically as follows: • Although i and f are used to represent the initial and final times, they can be initial and final times of any time interval you choose. • The time interval is a scalar because it has no direction. SECTION 2.2 Where and When? Vectors and Scalars (cont.) • The figure below shows the position of the runner at both the tree and the lamppost. • These arrows have magnitude and direction. • Position is a vector with the arrow’s tail at the origin and the arrow’s tip at the place. SECTION 2.2 Where and When? Vectors and Scalars (cont.) • The symbol x is used to represent position vectors mathematically. • Xi represents the position at the tree, xf represents the position at the lamppost and ∆x, represents the change in position, displacement, from the tree to the lamppost. SECTION 2.2 Where and When? Vectors and Scalars (cont.) • Displacement is defined mathematically as: ∆x = xf - xi • Remember that the initial and final positions are the start and end of any interval you choose, so a plus and minus sign might be used to indicate direction. SECTION 2.2 Where and When? Vectors and Scalars (cont.) • A vector that represents the sum o f two other vectors is called a resultant. • The figure to the right shows how to add and subtract vectors in one dimension. SECTION 2.2 Where and When? Vectors and Scalars (cont.) • To completely describe an object’s displacement, you must indicate the distance it traveled and the direction it moved. Thus, displacement, a vector, is not identical to distance, a scalar; it is distance and direction. • While the vectors drawn to represent each position change, the length and direction of the displacement vector does not. • The displacement vector is always drawn with its flat end, or tail, at the earlier position, and its point, or tip, at the later position. SECTION 2.2 Section Check Differentiate between scalar and vector quantities. SECTION 2.2 Section Check Answer Reason: Quantities that have both magnitude and direction are called vectors, and can be represented by arrows. Quantities that are just numbers without any direction, such as time, are called scalars. SECTION Section Check 2.2 What is displacement? A. the vector drawn from the initial position to the final position of the motion in a coordinate system B. the distance between the initial position and the final position of the motion in a coordinate system C. the amount by which the object is displaced from the initial position D. the amount by which the object moved from the initial position SECTION 2.2 Section Check Answer Reason: Options B, C, and D are all defining the distance of the motion and not the displacement. Displacement is a vector drawn from the starting position to the final position. SECTION 2.2 Section Check Refer to the adjoining figure and calculate the time taken by the car to travel from one signal to another signal? A. 20 min C. 25 min B. 45 min D. 5 min SECTION 2.2 Section Check Answer Reason: Time interval t = tf – ti Here tf = 01:45 and ti = 01:20 Therefore, t = 25 min SECTION Position-Time Graphs 2.3 MAIN IDEA You can use position-time graphs to determine an object’s position at a certain time. Essential Questions • What information do position-time graphs provide? • How can you use a position-time graph to interpret an object’s position or displacement? • What are the purposes of equivalent representations of an object’s motion? SECTION 2.3 Position-Time Graphs Review Vocabulary • Intersection a point where lines meet and cross. New Vocabulary • Position-time graph • Instantaneous position SECTION 2.3 Position-Time Graphs Finding Positions Click image to view movie. SECTION 2.3 Position-Time Graphs Finding Positions (cont.) • Graphs of an object’s position and time contain useful information about an object’s position at various times. It can be helpful in determining the displacement of an object during various time intervals. SECTION 2.3 Position-Time Graphs Finding Positions (cont.) • The data in the table can be presented by plotting the time data on a horizontal axis and the position data on a vertical axis, which is called a position-time graph. SECTION 2.3 Position-Time Graphs Finding Positions (cont.) • To draw the graph, plot the object’s recorded positions. Then, draw a line that best fits the recorded points. This line represents the most likely positions of the runner at the times between the recorded data points. • The symbol x represents the instantaneous position of the object—the position at a particular instant. SECTION 2.3 Position-Time Graphs Finding Positions (cont.) • Words, pictorial representations, motion diagrams, data tables, and position-time graphs are all representations that are equivalent. They all contain the same information about an object’s motion. • Depending on what you want to find out about an object’s motion, some of the representations will be more useful than others. SECTION 2.3 Position-Time Graphs Multiple Objects on a Position-Time Graph In the graph, when and where does runner B pass runner A? SECTION 2.3 Position-Time Graphs Multiple Objects on a Position-Time Graph (cont.) Step 1: Analyze the Problem Restate the questions. Question 1: At what time do A and B have the same position? Question 2: What is the position of runner A and runner B at this time? SECTION 2.3 Position-Time Graphs Multiple Objects on a Position-Time Graph (cont.) Step 2: Solve for the Unknown SECTION 2.3 Position-Time Graphs Multiple Objects on a Position-Time Graph (cont.) Question 1 In the figure, examine the graph to find the intersection of the line representing the motion of A with the line representing the motion of B. These lines intersect at 45 s. SECTION 2.3 Position-Time Graphs Multiple Objects on a Position-Time Graph (cont.) Question 2 In the figure, examine the graph to find the intersection of the line representing the motion of A with the line representing the motion of B. The position of both runners is about 190m from the origin. SECTION 2.3 Position-Time Graphs Multiple Objects on a Position-Time Graph (cont.) B passes A about 190 m beyond the origin, 45.0 s after A has passed the origin. SECTION 2.3 Position-Time Graphs Considering the Motion of Multiple Objects The steps covered were: Step 1: Analyze the Problem Restate the questions. Step 2: Solve for the Unknown SECTION 2.3 Section Check A position-time graph of an athlete winning the 100-m run is shown. Estimate the time taken by the athlete to reach 65 m. A. 6.0 s B. 6.5 s C. 5.5 s D. 7.0 s SECTION 2.3 Section Check Answer Reason: Draw a horizontal line from the position of 65 m to the line of best fit. Draw a vertical line to touch the time axis from the point of intersection of the horizontal line and line of best fit. Note the time where the vertical line crosses the time axis. This is the estimated time taken by the athlete to reach 65 m. SECTION 2.3 Section Check A position-time graph of an athlete winning the 100-m run is shown. What was the instantaneous position of the athlete at 2.5 s? A. 15 m B. 20 m C. 25 m D. 30 m SECTION 2.3 Section Check Answer Reason: Draw a vertical line from the position of 2.5 m to the line of best fit. Draw a horizontal line to touch the position axis from the point of intersection of the vertical line and line of best fit. Note the position where the horizontal line crosses the position axis. This is the instantaneous position of the athlete at 2.5 s. SECTION 2.3 Section Check From the following position-time graph of two brothers running a 100-m dash, at what time do both brothers have the same position? The smaller brother started the race from the 20-m mark. SECTION 2.3 Section Check Answer Reason: The two brothers meet at 6 s. In the figure, we find the intersection of lines representing the motion of one brother with the line representing the motion of other brother. These lines intersect at 6 s and at 60 m. SECTION How Fast? 2.4 MAIN IDEA An object’s velocity is the rate of change in its position. Essential Questions • What is velocity? • What is the difference between speed and velocity? • How can you determine an object’s average velocity from a position-time graph? • How can you represent motion with pictorial, physical, and mathematical models? SECTION How Fast? 2.4 Review Vocabulary • Absolute value magnitude of a number, regardless of sign. New Vocabulary • Average velocity • Average speed • Instantaneous velocity SECTION 2.4 How Fast? Velocity and Speed • Suppose you recorded two joggers in one motion diagram, as shown in the figure below. The position of the jogger wearing red changes more than the of the jogger wearing blue • For a fixed time, the magnitude of the displacement (∆x), is greater for the jogger in red. • If each jogger travels 100m, the time interval (∆t) would be smaller for the jogger in red. SECTION 2.4 How Fast? Velocity and Speed (cont.) • Recall from Chapter 1 that to find the slope, you first choose two points on the line. • Next, you subtract the vertical coordinate (x in this case) of the first point from the vertical coordinate of the second point to obtain the rise of the line. • After that, you subtract the horizontal coordinate (t in this case) of the first point from the horizontal coordinate of the second point to obtain the run. • Finally, you divide the rise by the run to obtain the slope. SECTION 2.4 How Fast? Velocity and Speed (cont.) • The slopes of the two lines are found as follows: • A greater slope, shows that the red jogger traveled faster. SECTION 2.4 How Fast? Velocity and Speed (cont.) • The unit of the slope is meters per second. In other words, the slope tells how many meters the runner moved in 1 s. • The slope is the change in position, divided by the time interval during which that change took place, or (xf - xi) / (tf - ti), or Δx/Δt. • When Δx gets larger, the slope gets larger; when Δt gets larger, the slope gets smaller. SECTION 2.4 How Fast? Velocity and Speed (cont.) • The slope of a position-time graph for an object is the object’s average velocity and is represented by the ratio of the change of position to the time interval during which the change occurred. Average Velocity ≡ Δx _______ Δt (xf - xi) = ________ (tf - ti) • The symbol ≡ means that the left-hand side of the equation is defined by the right-hand side. SECTION 2.4 How Fast? Velocity and Speed (cont.) • It is a common misconception to say that the slope of a positiontime graph gives the speed of the object. • The slope of the positiontime graph on the right is –5.0 m/s. It indicates the average velocity of the object and not its speed. SECTION 2.4 How Fast? Velocity and Speed (cont.) • The object moves in the negative direction at a rate of 5.0 m/s. SECTION 2.4 How Fast? Velocity and Speed (cont.) • The slope’s absolute value is the object’s average speed, 5.0m/s, which is the distance traveled divided by the time taken to travel that distance. SECTION 2.4 How Fast? Velocity and Speed (cont.) • If an object moves in the negative direction, then its displacement is negative. The object’s velocity will always have the same sign as the object’s displacement. SECTION 2.4 How Fast? Velocity and Speed (cont.) The graph describes the motion of a student riding his skateboard along a smooth, pedestrian-free sidewalk. What is his average velocity? What is his average speed? SECTION 2.4 How Fast? Velocity and Speed (cont.) Step 1: Analyze and Sketch the Problem Identify the coordinate system of the graph. SECTION 2.4 How Fast? Velocity and Speed (cont.) Step 2: Solve for the Unknown SECTION 2.4 How Fast? Velocity and Speed (cont.) Identify the unknown variables. Unknown: SECTION 2.4 How Fast? Velocity and Speed (cont.) Find the average velocity using two points on the line. Use magnitudes with signs indicating directions. Δx = _____ Δt (xf - xi) = ______ (tf - ti) SECTION 2.4 How Fast? Velocity and Speed (cont.) Substitute x2 = 12.0 m, x1 = 6.0 m, t2 = 8.0 s, t1 = 4.0 s: SECTION 2.4 How Fast? Velocity and Speed (cont.) Step 3: Evaluate the Answer SECTION 2.4 How Fast? Velocity and Speed (cont.) Are the units correct? m/s are the units for both velocity and speed. Do the signs make sense? The positive sign for the velocity agrees with the coordinate system. No direction is associated with speed. SECTION 2.4 How Fast? Velocity and Speed (cont.) The steps covered were: Step 1: Analyze and Sketch the Problem Identify the coordinate system of the graph. SECTION 2.4 How Fast? Velocity and Speed (cont.) The steps covered were: Step 2: Solve for the Unknown Find the average velocity using two points on the line. The average speed is the absolute value of the average velocity. Step 3: Evaluate the Answer SECTION 2.4 How Fast? Velocity and Speed (cont.) • A motion diagram shows the position of a moving object at the beginning and end of a time interval. During that time interval, the speed of the object could have remained the same, increased, or decreased. All that can be determined from the motion diagram is the average velocity. • The speed and direction of an object at a particular instant is called the instantaneous velocity. • The term velocity refers to instantaneous velocity and is represented by the symbol v. SECTION 2.4 How Fast? Velocity and Speed (cont.) • Although the average velocity is in the same direction as displacement, the two quantities are not measured in the same units. • Nevertheless, they are proportional—when displacement is greater during a given time interval, so is the average velocity. • A motion diagram is not a precise graph of average velocity, but you can indicate the direction and magnitude of the average velocity on it. SECTION 2.4 How Fast? Equation of Motion • Using the position-time graph used before with a slope of -5.0m/s, remember that you can represent any straight line with the equation, y = mx + b. • y is the quantity plotted on the vertical axis, m is the line’s slope, x is the quantity plotted on the horizontal axis and b is the line’s y-intercept. SECTION 2.4 How Fast? Equation of Motion (cont.) • Based on the information shown in the table, the equation y = mx + b becomes x = t + xi, or, by inserting the values of the constants, x = (–5.0 m/s)t + 20.0 m. • You cannot set two items with different units equal to each other in an equation. Comparison of Straight Lines with Position-Time Graphs General Variable Specific Motion Variable y x m Value in Graph -5.0m/s x t b xi 20.0m SECTION 2.4 How Fast? Equation of Motion (cont.) • An object’s position is equal to the average velocity multiplied by time plus the initial position. • This equation gives you another way to represent the motion of an object. SECTION 2.4 Section Check Which of the following statements defines the velocity of the object’s motion? A. the ratio of the distance covered by an object to the respective time interval B. the rate at which distance is covered C. the distance moved by a moving body in unit time D. the ratio of the displacement of an object to the respective time interval SECTION 2.4 Section Check Answer Reason: Options A, B, and C define the speed of the object’s motion. The velocity of a moving object is defined as the ratio of the displacement (x) to the time interval (t). SECTION 2.4 Section Check Which of the statements given below is correct? A. Average velocity cannot have a negative value. B. Average velocity is a scalar quantity. C. Average velocity is a vector quantity. D. Average velocity is the absolute value of the slope of a position-time graph. SECTION 2.4 Section Check Answer Reason: Average velocity is a vector quantity, whereas all other statements are true for scalar quantities. SECTION 2.4 Section Check The position-time graph of a car moving on a street is given here. What is the average velocity of the car? A. 2.5 m/s B. 5 m/s C. 2 m/s D. 10 m/s SECTION 2.4 Section Check Answer Reason: The average velocity of an object is the slope of a position-time graph. CHAPTER Representing Motion 2 Resources Physics Online Study Guide Chapter Assessment Questions Standardized Test Practice SECTION Picturing Motion 2.1 Study Guide • A motion diagram shows the position of an object at successive equal time intervals. • In the particle model motion diagram, an object’s position at successive times is represented by a series of dots. The spacing between dots indicates whether the object is moving faster or slower. SECTION Where and When? 2.2 Study Guide • A coordinate system gives the location of the zero point of the variable you are studying and the direction in which the values of the variable increase. • A vector drawn from the origin of a coordinate system to an object indicates the object’s position in that coordinate system. The directions chosen as positive and negative on the coordinate system. SECTION Where and When? 2.2 Study Guide • A time interval is the difference between two times. • Change in position is displacement, which has both magnitude and direction. SECTION Where and When? 2.2 Study Guide • On a motion diagram, the displacement vector’s length represents how far the object was displaced. The vector points in the direction of the displacement, from xi to xf. SECTION Position-Time Graphs 2.3 Study Guide • Position-time graphs provide information about the motion of objects. They also might indicate where and when two objects meet. • The line on a position-time graph describes an object’s position at each time. • Motion can be described using words, motion diagrams, data tables or graphs. SECTION How Fast? 2.4 Study Guide • An object’s velocity tells how fast it is moving and in what direction it is moving. • Speed is the magnitude of velocity. • Slope on a position-time graph described the average velocity of the object. SECTION How Fast? 2.4 Study Guide • You can represent motion with pictures and physical models. A simple equation relates an object’s initial position (xi), its constant average velocity, its position (x) and the time (t) since the object was at its initial position. CHAPTER 2 Representing Motion Chapter Assessment What should be true about the motion of an object in order for you to treat that object as if it were a particle? A. The object should be no smaller than your fist. B. The object should be small compared to its motion. C. The object should be no larger than you can lift. D. The object should not be moving faster than the speed of sound. CHAPTER 2 Representing Motion Chapter Assessment Reason: you can treat even planets and stars as particles as long as those objects are small compared to the motion you are studying. CHAPTER 2 Representing Motion Chapter Assessment Which is the distance and direction from one point to another? A. Displacement B. Magnitude of distance C. Position D. Velocity CHAPTER 2 Representing Motion Chapter Assessment Reason: Velocity is speed and direction. CHAPTER 2 Representing Motion Chapter Assessment On a position-time graph, how would you indicate that object A has a greater velocity than object B? A. Make the slope for object A less than the slope for object B. B. Make the slope for object A greater than the slope for object B. C. Make the y-intercept for object A less then the y-intercept for object B. D. Make the y-intercept for object A greater than the yintercept for object B. CHAPTER 2 Representing Motion Chapter Assessment Answer: The slope of a line on a position-time graph indicates the object’s velocity. CHAPTER 2 Representing Motion Chapter Assessment A car is moving at a constant speed of 25 m/s. How far does this car move in 0.2 s, the approximate reaction time for an average person? A. 5 m B. 10 m C. 25 m D. 50 m CHAPTER 2 Representing Motion Chapter Assessment Reason: (25m/s)(0.2s) = 5m CHAPTER 2 Representing Motion Chapter Assessment Which is a measurement of velocity? A. 20 m B. 33 km/s C. 300 km west D. 7800 m/s north CHAPTER 2 Representing Motion Chapter Assessment Reason: Velocity measures both speed and direction. CHAPTER Representing Motion 2 Standardized Test Practice Which statement about velocity vectors is true? A. All velocity vectors are positive. B. Velocity vectors have magnitude but no direction. C. Velocity vectors and displacement vectors are the same thing. D. A velocity vector’s length should be proportional to the object’s speed. CHAPTER 2 Representing Motion Standardized Test Practice What is the average speed of a sprinter who completes a 55-m dash in 6.2 s? A. 6.2 m/s B. 7.1 m/s C. 8.9 m/s D. 11 m/s CHAPTER 2 Representing Motion Standardized Test Practice Car A is moving faster than Car B on the highway. Which statement describes the particle model motion diagrams for Car A and Car B? A. The does for Car A are farther apart than the dots for Car B. B. The dots for Car A are closer together than the dots for Car B. C. The slope of the motion diagram is greater for Car A than for Car B. D. The slope of the motion diagram is less for Car A than for Car B. CHAPTER 2 Representing Motion Standardized Test Practice An athlete runs four complete laps around a 200-m track. What is the athlete’s displacement? A. 0 m B. 200 m C. 400 m D. 800 m CHAPTER 2 Representing Motion Standardized Test Practice Which correctly describes a relationship between an object’s particle model motion diagram and that object’s graph of position v. time? A. If the dots on the motion diagram are closer together, then the slope of the graph is greater. B. If the dots on the motion diagram are farther apart, then the slope of the graph is greater. C. If the dots on the motion diagram are closer together, then the y-intercept of the graph is less. D. If the dots on the motion diagram are farther apart, then the yintercept of the graph is less. CHAPTER 2 Representing Motion Standardized Test Practice Test-Taking Tip Stock up on Supplies Bring all your test-taking tools: number two pencils, black and blue pens, erasers, correction fluid, a sharpener, a ruler, a calculator, and a protractor. CHAPTER 2 Representing Motion Chapter Resources Coordinate Systems CHAPTER 2 Representing Motion Chapter Resources Coordinate Systems Showing Position CHAPTER 2 Representing Motion Chapter Resources Motion Diagram Showing Negative Position CHAPTER 2 Representing Motion Chapter Resources Position-Time Graph for the Runner End of Custom Shows