Representing Motion: Picturing, Describing, and Graphing Motion

1. Chapter 2: Representing Motion Click the mouse or press the spacebar to continue. Splash Screen

2. In this chapter you will: • 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 Intro

3. 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? Table of Contents

4. In this section you will: • Draw motion diagrams to describe motion. • Develop a particle model to represent a moving object. Section 2.1-1

5. 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 2.1-2

6. All Kinds of Motion 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 2.1-3

7. Movement Along a Straight Line 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 2.1-4

8. Movement Along a Straight Line In the figure below, the car has moved from point A to point B in a specific time period. Section 2.1-5

9. Motion Diagrams Click image to view movie. Section 2.1-6

10. Question 1 Explain how applying the particle model produces a simplified version of a motion diagram? Section 2.1-7

11. Answer 1 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 2.1-8

12. Question 2 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 2.1-9

13. Answer 2 Reason:A series of images showing the positions of a moving object at equal time intervals is called a motion diagram. Section 2.1-10

14. Question 3 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 2.1-11

15. Answer 3 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 2.1-12

16. End of Section 2.1

17. In this section you will: • 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. Section 2.2-1

18. 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 2.2-2

19. Coordinate Systems 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 2.2-3

20. Coordinate Systems 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 2.2-4

21. Coordinate Systems 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 2.2-5

22. Coordinate Systems The two arrows locate the runner’s position at two different times. Section 2.2-6

23. Coordinate Systems The length of how far an object is from the origin indicates its distance from the origin. Section 2.2-7

24. Coordinate Systems The arrow points from the origin to the location of the moving object at a particular time. Section 2.2-8

25. Coordinate Systems 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 2.2-9

26. Vectors and Scalars Quantities that have both size, also called magnitude, and direction, are called vectors, and can be represented by arrows. Quantities that are just numbers without any direction, such as distance, time, or temperature, are called scalars. Section 2.2-10

27. Vectors and Scalars 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. Section 2.2-11

28. Vectors and Scalars The vector that represents the sum of the other two vectors is called the resultant. The resultant always points from the tail of the first vector to the tip of the last vector. Section 2.2-12

29. Time Intervals and Displacement The difference between the initial and the final times is called the time interval. Section 2.2-13

30. Time Intervals and Displacement The common symbol for a time interval is ∆t, where the Greek letter delta, ∆, is used to represent a change in a quantity. Section 2.2-14

31. Time Intervals and Displacement 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. Section 2.2-15

32. Time Intervals and Displacement Also of importance is how the position changes. The symbol d may be used to represent position. 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. Section 2.2-16

33. Time Intervals and Displacement The figure below shows ∆d, an arrow drawn from the runner’s position at the tree to his position at the lamppost. Section 2.2-17

34. Time Intervals and Displacement The change in position during the time interval between ti and tf is called displacement. Section 2.2-18

35. Time Intervals and Displacement The length of the arrow represents the distance the runner moved, while the direction the arrow points indicates the direction of the displacement. Displacement is mathematically defined as follows: Displacement is equal to the final position minus the initial position. Section 2.2-19

36. Time Intervals and Displacement 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. Section 2.2-20

37. Time Intervals and Displacement To determine the length and direction of the displacement vector, ∆d = df− di, draw −di, which is di reversed. Then draw dfand copy −diwith its tail at df’stip. Add dfand −di. Section 2.2-21

38. Time Intervals and Displacement 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 2.2-22

39. Question 1 Differentiate between scalar and vector quantities. Section 2.2-23

40. Answer 1 Answer: 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 2.2-24

41. Question 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 2.2-25

42. Answer 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. Section 2.2-26

43. Question 3 Refer to the adjoining figure and calculate the time taken by the car to travel from one signal to another signal? A. 20 min B. 45 min C. 25 min D. 5 min Section 2.2-27

44. Answer 3 Reason: Time interval t = tf – ti Here tf = 01:45 and ti = 01:20 Therefore, t = 25 min Section 2.2-28

45. End of Section 2.2

46. In this section you will: • 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. Section 2.3-1

47. Position-Time Graphs Click image to view movie. Section 2.3-2

48. Using a Graph to Find Out Where and When 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 2.3-3

49. Using a Graph to Find Out Where and When 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 2.3-4

50. Using a Graph to Find Out Where and When 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 d represents the instantaneous position of the object—the position at a particular instant. Section 2.3-5