2. Representing Motion Chapter 2

3. Representing Motion Represent motion through the use of words, motion diagrams, and graphs. Use the terms position, distance, displacement, and time interval in a scientific manner to describe motion. Chapter 2 In this chapter you will:

4. Table of Contents Chapter 2: Representing Motion Section 2.1: Picturing Motion Section 2.2: Where and When? Section 2.3: Position-Time Graphs Section 2.4: How Fast? Chapter 2

5. Picturing Motion Draw motion diagrams to describe motion. Develop a particle model to represent a moving object. In this section you will: Section 2.1

6. Picturing Motion Pages 31 - 33 Reading assignment for section one: Section 2.1

7. Picturing 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. 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. 1.) When an object is in motion its position changes. Section 2.1

8. Picturing Motion 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. In the figure below, the car has moved from point A to point B in a specific time period. 2.) What two things does a description of motion relate? Place and time Section 2.1

9. 3.)What is a motion diagram ? A series of images showing the positions of a moving object at equal time intervals.

10. 4. What is the particle model ? A simplified version of a motion diagram in which the object in motion is replaced by a series of single points.

11. Picturing Motion Section 2.1 Motion Diagrams Click image to view movie.

12. Section Check Explain how applying the particle model produces a simplified version of a motion diagram? Question 1 Section 2.1

13. Section Check Answer 1 Section 2.1 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.

14. Section Check Which statement describes best the motion diagram of an object in motion? Question 2 Section 2.1 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. Diagram in which the object in motion is replaced by a series of single point. D. A diagram that tells us the location of zero point of the object in motion and the direction in which the object is moving.

15. Section Check Answer: B Answer 2 Section 2.1 Reason: A series of images showing the positions of a moving object at equal time intervals is called a motion diagram.

16. Section Check What is the purpose of drawing a motion diagram or a particle model? Question 3 Section 2.1 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.

17. Section Check Answer: C Answer 3 Section 2.1 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.

18. Where and When? Define coordinate systems for motion problems. Recognize that the chosen coordinate system affects the sign of objects’ positions. Define displacement. Determine a time interval. Use a motion diagram to answer questions about an object’s position or displacement. In this section you will: Section 2.2

19. Where and When? Pages 34 - 37 Section 2 Reading assignment Section 2.2

20. Where and When? 5.) What does a coordinate system tell you? A coordinate system tells you the location of the zero point of the variable you are studying and the direction in which the value of the variable increases. Coordinate Systems Section 2.2

21. Where and When? 6.) What is the origin? The point at which all variables have the value zero. Coordinate Systems Section 2.2

22. Where and When? 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. 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. Coordinate Systems Section 2.2

23. Where and When? 7) What can we use to indicate how far an object is from the origin at a particular time? You can indicate the displacement by drawing an arrow from the origin to the position at a given time. Coordinate Systems Section 2.2

24. 8.) The objects motion is represented by the arrow. Where and When? Coordinate Systems Section 2.2

25. 9.) What does the length of the arrow represent? The distance the object is from the origin. Where and When? Coordinate Systems Section 2.2 The arrow points from the origin to the location of the moving object at a particular time.

26. Where and When? 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. 10.) Is it possible to have a negative position? Yes. If the object is on the opposite side of what is considered to be the positive direction. Section 2.2

27. Where and When? 11.) What is a vector? A quantity that has both size ( often called magnitude) and direction. A vector can be represented by arrows. Velocity is speed in a given direction Displacement is distance in a particular direction Force the size of the force and the direction of the force is important. Vectors and Scalars Section 2.2

28. Where and When? 12.) What is a scalar? Vectors and Scalars Section 2.2 Quantities that are just numbers without any direction, such as distance, time, or temperature, are called scalars.

29. Where and When? 13.) What is important when vectors are added graphically? Vectors and Scalars Section 2.2 To add vectors graphically, the length of a vector should be proportional to the magnitude of the quantity being represented. So it is important to decide on the scale of your drawings. The important thing is to choose a scale that produces a diagram of reasonable size with a vector that is about 5–10 cm long.

30. Where and When? 14. The vector that represents the sum of the other two vectors is called the resultant. Vectors and Scalars Section 2.2

31. Where and When? 15.) What is the direction of the resultant vector? Vectors and Scalars Section 2.2 The resultant always points from the tail of the first vector to the tip of the last vector.

32. Where and When? 16.) How are vectors added? Vectors and Scalars Section 2.2 To add two vectors place the vectors tip to tail. The resultant points from the tail of the first vector to the tip of the last vector.

33. Where and When? 17.) The difference between the initial and the final times is called the time interval. Time Intervals and Displacement Section 2.2

34. Where and When? 18.) What is the common symbol for a time interval? Time Intervals and Displacement Section 2.2 The common symbol for a time interval is ∆t, where the Greek letter delta, ∆, is used to represent a change in a quantity.

35. Where and When? 19.) How is the time interval defined mathematically? The time interval is defined mathematically as follows: Time Intervals and Displacement Section 2.2

36. Where and When? 20.) In physics, how do we represent position? Time Intervals and Displacement Section 2.2 In physics, a position is a vector with its tail at the origin of a coordinate system and its tip at the place where the object is located at that time.

37. Where and When? The figure below shows ∆d, an arrow drawn from the runner’s position at the tree to his position at the lamppost. Section 2.2 21.) What is displacement? The change in position during the time interval between t i and t f is called displacement.

38. Where and When? The length of the arrow represents the distance the runner moved, while the direction the arrow points indicates the direction of the displacement. 22.How is displacement defined mathematically? Displacement is mathematically defined as follows: Time Intervals and Displacement Section 2.2 Displacement is equal to the final position minus the initial position.

39. Where and When? To subtract vectors, reverse the subtracted vector and then add the two vectors. This is because A – B = A + (–B). The figure a below shows two vectors, A, 4 cm long pointing east, and B, 1 cm long also pointing east. Figure b shows –B, which is 1 cm long pointing west. The resultant of A and –B is 3 cm long pointing east. 23.) How are vectors subtracted? Section 2.2

40. Where and When? To determine the length and direction of the displacement vector, ∆d = d f − d i, draw −d i, which is d i reversed. Then draw d f and copy −d i with its tail at d f ’s tip. Add d f and −d i. 24.) Displacement is a vector because it describes the distance the object traveled and the direction it moved. Section 2.2

41. Where and When? Time Intervals and Displacement Section 2.2

42. Where and When? 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

43. Section Check Differentiate between scalar and vector quantities? Question 1 Section 2.2

44. Section Check Answer 1 Section 2.2 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.

45. Section Check What is displacement? Question 2 Section 2.2 A. The vector drawn from the initial position to the final position of the motion in a coordinate system. B. The length of 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.

46. Section Check Answer: A Answer 2 Section 2.2 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.

47. Refer the adjoining figure and calculate the time taken by the car to travel from one signal to another signal? Question 3 Section 2.2 Section Check A. 20 min B. 45 min C. 25 min D. 5 min

48. Section Check Answer: C Answer 3 Section 2.2 Reason: Time interval  t = t f - t i Here t f = 01:45 and t i = 01:20 Therefore,  t = 25 min

49. Position-Time Graphs Develop position-time graphs for moving objects. Use a position-time graph to interpret an object’s position or displacement. Make motion diagrams, pictorial representations, and position-time graphs that are equivalent representations describing an object’s motion. In this section you will: Section 2.3

50. Position-Time Graphs Pages 38 - 42 Reading assignment Section 3 Section 2.3

51. Position-Time Graphs Position Time Graphs Section 2.3 Click image to view movie.

52. Graphs of an object’s position and time contain useful information about an object’s position at various times and can be helpful in determining the displacement of an object during various time intervals. Position-Time Graphs Using a Graph to Find Out Where and When Section 2.3 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.

53. A graph created by plotting the time data on the horizontal axis and the position data on the vertical axis. Position-Time Graphs 25.) What is a position-time graph? Section 2.3

54. 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. Position-Time Graphs Using a Graph to Find Out Where and When Section 2.3 The symbol d represents the instantaneous position of the object—the position at a particular instant.

55. This line represents the most likely positions of the runner at the times between the recorded data points. Position-Time Graphs 26.) What does the best fit line represent on a position-time graph? Section 2.3

56. The instantaneous position of the object at the particular instant. Position-Time Graphs 27.) What does d represent? Section 2.3

57. Position-Time Graphs 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. Equivalent Representations Section 2.3

58. Position-Time Graphs Considering the Motion of Multiple Objects Section 2.3 In the graph, when and where does runner B pass runner A?

59. 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. Position-Time Graphs Section 2.3 Considering the Motion of Multiple Objects

60. These lines intersect at 45.0 s and at about 190 m. Position-Time Graphs Section 2.3 Considering the Motion of Multiple Objects

61. B passes A about 190 m beyond the origin, 45.0 s after A has passed the origin. Position-Time Graphs Section 2.3 Considering the Motion of Multiple Objects

62. 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. Question 1 Section 2.3 Section Check A. 6.0 s B. 6.5 s C. 5.5 s D. 7.0 s

63. Section Check Answer : B Answer 1 Section 2.3 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.

64. 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? Question 2 Section 2.3 Section Check A. 15 m B. 20 m C. 25 m D. 30 m

65. Section Check Answer: C Answer 2 Section 2.3 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.

66. From the following position- time graph of two brothers running a 100-m run, analyze at what time do both brothers have the same position. The smaller brother started the race from the 20- m mark. Question 3 Section 2.3 Section Check

67. Answer 3 Section 2.3 The two brothers meet at 6 s. In the figure, we find the intersection of line 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.

68. Practice Problems 14 – 18 page 41. eSection 3 problems 19 – 23 page 42.

69. How Fast? Define velocity. Differentiate between speed and velocity. Create pictorial, physical, and mathematical models of motion problems. In this section you will: Section 2.4

70. How Fast? Pages 43 - 47. Section 4 reading assignment: Section 2.4

71. How Fast? Suppose you recorded two joggers on one motion diagram, as shown in the below figure. From one frame to the next, you can see that the position of the jogger in red shorts changes more than that of the one wearing blue. Velocity Section 2.4

72. In other words, for a fixed time interval, the displacement, ∆d, is greater for the jogger in red because she is moving faster. She covers a larger distance than the jogger in blue does in the same amount of time. Now, suppose that each jogger travels 100 m. The time interval, ∆t, would be smaller for the jogger in red than for the one in blue. How Fast? Velocity Section 2.4

73. How Fast? The slopes of the two lines are found as follows: Average Velocity Section 2.4

74. How Fast? The unit of the slope is meter 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 (d f - d i ) / (t f - t i ), or Δd/Δt. When Δd gets larger, the slope gets larger; when Δt gets larger, the slope gets smaller. Average Velocity Section 2.4

75. How Fast? 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. 29.) What is average velocity? Section 2.4

76. How Fast? 30.) How is average velocity expressed mathematically? Section 2.4 Average velocity is defined as the change in position, divided by the time during which the change occurred. The symbol ≡ means that the left-hand side of the equation is defined by the right-hand side.

77. How Fast? It is a common misconception to say that the slope of a position-time graph gives the speed of the object. The slope of the position-time graph on the right is –5.0 m/s. It indicates the average velocity of the object and not its speed. The object moves in the negative direction at a rate of 5.0 m/s. Average Velocity Section 2.4

78. How Fast? 31.) The absolute value of the slope of a position-time graph tells you the average speed of the object, that is, how fast the object is moving. Section 2.4

79. How Fast? Section 2.4 32.) What does the sign of the slope tell you? The sign of the slope tells you in what direction the object is moving. The combination of an object’s average speed,, and the direction in which it is moving is the average velocity.

80. How Fast? Section 2.4 33.) How will the sign of the displacement and the velocity compare? 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.

81. How Fast? Average Speed 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

82. Average Speed Identify the coordinate system of the graph. How Fast? Section 2.4

83. Average Speed Find the average velocity using two points on the line. How Fast? Section 2.4 Use magnitudes with signs indicating directions.

84. Example Problem Substitute d 2 = 12.0 m, d 1 = 6.0 m, t 2 = 8.0 s, t 1 = 4.0 s: How Fast? Section 2.4

85. How Fast? 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. Average Speed Section 2.4

86. How Fast? 34.) What is instantaneous velocity? The speed and direction of an object at a particular instant is called the instantaneous velocity. Section 2.4 Instantaneous Velocity.

87. How Fast? Although the average velocity is in the same direction as displacement, the two quantities are not measured in the same units. However, they are proportional—when displacement is greater during a given time interval, so is the average velocity. 35.) How do displacement and velocity compare? Section 2.4

88. 36.) What is the equation for a straight line? y = m x + b How Fast? Using Equations Section 2.4 Based on the information shown in the table, the equation y = mx + b becomes d = t + d i, or, by inserting the values of the constants, d = (–5.0 m/s)t + 20.0 m. You cannot set two items with different units equal to each other in an equation.

89. 36.) What is the equation for a straight line? y = m x + b How Fast? Using Equations Section 2.4

90. 37.) Using a position time graph, what is the equation for the line? How Fast? Using Equations Section 2.4 Based on the information shown in the table, the equation y = mx + b becomes d = t + d i, or, by inserting the values of the constants, d = (–5.0 m/s)t + 20.0 m.

91. How Fast? An object’s position is equal to the average velocity multiplied by time plus the initial position. Equation of Motion for Average Velocity 38.) What is the above equation in words? Section 2.4

92. Section Check Which of the following statement defines the velocity of the object’s motion? Question 1 Section 2.4 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.

93. Section Check Answer: D Answer 1 Section 2.4 Reason: Options A, B, and C define the speed of the object’s motion. Velocity of a moving object is defined as the ratio of the displacement (  d) to the time interval (  t).

94. Section Check Which of the statements given below is correct? Question 2 Section 2.4 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.

95. Section Check Answer: C Answer 2 Section 2.4 Reason: Average velocity is a vector quantity, whereas all other statements are true for scalar quantities.

96. The position-time graph of a car moving on a street is as given here. What is the average velocity of the car? Question 3 Section 2.4 Section Check A. 2.5 m/s B. 5 m/s C. 2 m/s D. 10 m/s

97. Section Check Answer: C Answer 3 Section 2.4 Reason: Average velocity of an object is the slope of the position-time graph.

98. Position-Time Graphs Considering the Motion of Multiple Objects In the graph, when and where does runner B pass runner A? Section 2.3 Click the Back button to return to original slide.

99. End of Chapter Representing Motion Chapter 2