1. Position-Time graphs

2. WHY GRAPH?

3. With accuracy Can be understood by everybody in the world
What is the best way to describe motion of an object to somebody that did not witness it?
With accuracy
Can be understood by everybody in the world

4. Write a description of my motion on a piece of paper
Describe my motion
Write a description of my motion on a piece of paper

5. A better way Represent the motion in the universal language: Math
Draw a graph

6. Reasons Universal language Convey more information visually
Less time to draw than write

7. GRAPH SET UP

8. Axis
What do the numbers mean

9. Direction
Is direction of motion indicated on the graph?

10. POSITION-TIME GRPAHS

11. Position-time graphs
Position value is recorded as the vertical (y) component
Time value is recorded as the horizontal
The point (4,-8) means you are at -8 units from the central reference point at the 4 second mark 11

12. Position-time graph Where is the object at the 3 second mark?12

13. What information does the graph tell you about motion
Shape of the line
Straight-vs-curved
Tilt of the line
Flat-vs-slanted
Tilted upward-vs-tilted downward
Vertical lines
Placement of the line
Start position
Postive-vs negative territory 13

14. Position Time graphs 14

15. Position Time graphs Graph indicates Positive motion Constant velocity
Between fast and slow rate of motion 15

16. Moving very fast

17. Moving very slow

18. At rest
How does a graph indicate that the object does not move

19. Moving forward, backward at constant speed

20. Which line shows no motion? Which line shows fastest rate of motion?

21. Speeding up

22. 22

23. Graph indicates
Non-constant velocity
Positive motion
Getting faster

24. 24

25. Graph indicates
Negative motion
Non-constant velocity
Getting faster

26. Slowing down

27. Slowing down moving forward

28. 28

29. Graph indicates
Negative motion
Non-constant velocity
Slowing down

30. Forward, backward, or stopped
If the graph is horizontal, then no motion has occurred.
The position (vertical) value did not change over time
If the final position is more positive than the initial position , it moved forward
If the final position is less positive than initial position, it moved backward 30

31. Is the object moving at a constant rate?
Constant velocity means the rate of motion does not change over time
Graphs show constant velocity by creating a straight line.
Angle (tilt) of line does not matter 31

32. How fast is it going? Constant velocity can occur in any direction
Being stopped gives you a constant velocity of zero (0).
The tilt of a straight line will indicate forward, or backward motion
Slope of the line is a measure of the object’s velocity
The amount of tilt will indicate how fast th object goes 32

33. What if it is not a straight line?
Then the velocity is not constant
If the velocity changes over time the car accelerates
Any change in velocity indicates acceleration
On a P-T graph, acceleration is indicated by a curved line 33

34. It is getting faster or slower?
Tangent lines
Pick 2 point along section of graph
Draw tangent lines
If the slope of line increases, then object is getting faster
Divide section up into equal 2 blocks of time.
Compare the displacement in each
If amount of displacement increases, it is getting faster 34

35. Tangent line
Line that touches a graph at only one point 35

36. Mathematical method to determine slope of tangent lines
Determine the rate of change
Derivatives in calculus 36

37. DRAWING AND ANALYZING POSITION TIME GRAPHS

38. Draw the graph Draw a graph that would represent the following motion:
Positive Motion
Non-Constant Velocity
Slowing Down

39. Information from graphs
For each section with the same type of motion, you should be able to determine:
Is the object moving forward, backward, or stopped
Is the motion constant or not?
If constant, is the rate of motion fast or slow?
If not constant, is the object getting faster or slower? 39

40. Position-Time 4 2 5 3 6 1 2

41. Position time graph
Each point on the graph indicates the position of the object at a certain time
Shows both distance and displacement
Y-axis indicates position
X-axis indicates time

42. Start position

43. Most graphs are made from a combination of different types of motions

44. What you should be able to tell me about the graph
Whether object is moving or not
Which direction it moves
Whether motion is constant or not
Whether object speeds up or slows

45. GIVEN THE GRAPH, DESCRIBE THE MOTION

46. GIVEN DESCRIPTION OF MOTION, PRODUCE CORRESPONDING GRAPH

47. Creating a graph from written information
Draw a copy on a piece of paper 47

48. In the next graph…
You will create a graph that represents the following motion
Section 1- starts at the -2 meter position and moves with slow positive constant velocity
Section 2- moves with fast negative constant velocity 48

49. Information for next graph…
Section 3- moves with positive non-constant velocity and is getting faster
Section 4- moves with a constant velocity of zero
Section 5 – moves with a negative non-constant velocity and is slowing down 49

50. Example #1 Starting from a position of (-3).
Object speeds up, moving forward to the origin
Object maintains constant velocity moving forward, reaches (4)
Object slows down, moving forward, reaches (6)
Object Stops for several seconds
Object speeds up moving backwards

51. Assumption
If the question does not specify times, assume that the displacement of interest is over the entire graph

52. DETERMINE THE VELOCITY FROM A POSITION TIME GRAPH

53. Position –time graph with number values53

54. What is the velocity of the car during the first 1.5 second?
Is it constant
Is it relatively fast or slow?
How do you find its actual value? 54

55. To answer the initial problem
(3 – 0)(m) / (1.5 – 0)(s) = 3 m/s
Find the rest of the constant velocity values shown on the graph 55

56. Calculation of a constant velocity
Slope of the line = steepness
To determine slope, find the rise over run
Rise = change in the y values between initial and final points Run change in the x values
V = (y2 –y1) / (x2 – x1) 56

57. Slope of the line
Constant velocity is demonstrated by a slanted straight line on a P-T graph
The steepness indicates how fast the object moves
To measure the steepness of a line, calculate the slope

58. How to calculate the slope
Rise= change in the position
Run= change in the time
Slope = Rise / Run

59. Calculating the velocity
Use (y2-y1) / (x2 – x1) to calculate the slope (velocity)
Organization of information
Starts with identification of x and y values

60. Calculate velocity on each section

61. DETERMINE INSTANTANEOUS VELOCITY FROM POSITION-TIME GRAPH

62. Calculating the instantaneous velocity value
For any section on a graph with constant velocity, all points in time within that section have the same velocity value.

63. DETERMINE DISTANCE AND DISPLACEMENT FROM POSITION TIME GRAPH

64. Overall Displacement
Look at the difference between final position and start position
Look at nothing else!!!
What is the overall displacement of the above graph? 64

65. Determine overall displacement
On a p-t graph:
Displacement = the difference between the start position and final position for the portion of the graph you are interested in
D = y2 – y1

66. Example

67. Find the overall displacement for:
The entire graph
the first 20 seconds
The last 20 seconds

68. Overall Distance
Measure the change in position between each hill and valley of the graph
Then add up the changes
There are no subtractions!!
What is the total distance of the given graphs? 68