Position-time graphs

1. Notes on Position-time graphs

2. 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

3. Position-time graph Where is the object at the 3 second mark? 9 second mark?

4. 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

5. Position Time graphs

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

7. Moving very fast

8. Moving very slow

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

10. Moving forward, backward at constant speed

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

12. Speeding up

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

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

17. Slowing down

18. Slowing down moving forward

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

21. 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

22. 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

23. 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

24. 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

25. 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

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

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

28. Drawing and analyzing Position time graphs

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

30. 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?

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

32. 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

33. Start position

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

35. 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

36. Given the graph, describe the motion

37. Given description of motion, produce corresponding graph

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

39. 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

40. 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

41. 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

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

43. Determine the velocity from a position time graph

44. Position –time graph with number values

45. 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?

46. 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

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

48. 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

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

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