1. Position and displacement
In this initial lesson students learn about distance, position, and displacement through the use of an interactive simulation The presentation contains introductory information to help students understand how these variables appear in both graphical formats and in word problems. This lesson considers only one dimensional situations. The next lesson extends these ideas to 2 dimensional systems.

2. Objectives
Describe motion in 1D using position, distance, and displacement.
Analyze motion in 1D using position, distance, and displacement.
Correctly use and interpret positive and negative values of position and displacement.
The lesson objectives clarify what students should know and be able to do.

3. Assessment
A robot car moves 200 m east, turns around and moves 600 m west. Describe this motion using position, displacement, and distance.
A woman starts at x = -10 m and has three successive displacements of -5 m. What is her final position?
What displacement moves an object from a position of 25 meters to a position of -15 meters?
These formative assessments will appear again at the end of the lesson, along with the answers.

4. Physics terms origin position displacement
New vocabulary will be defined and reinforced throughout the lesson.

5. Equations
The final position equals the initial position plus any displacements.

6. Describing motion
Before you can predict an object’s motion, you need to be able to describe it.
How do we describe an object’s position?
If the object moves, what is the difference between its distance and displacement?

7. Exploring the ideas Click on this interactive on page 74.
This interactive simulation allows students to use an active inquiry process to come to an understanding of the distinctions between position, distance, and displacement.

8. Engaging with the concept
Initial position or
final position can be entered here.
These next several slides show students how to use the interactive.

9. Engaging with the concept5
Enter a displacement of 5 meters
Click [Enter]

10. Engaging with the concept5 5 5
Final position
Distance moved

11. Engaging with the concept5 5
-10 5
Add a displacement of -10 meters
Point out that the red figure acts out the displacements, and makes the initial and final positions very clear.
Click [Enter]

12. Engaging with the concept
Final position 5 15
-10 -5
Is the distance moved equal to the sum of the displacements? How are these variables related?
Distance moved

13. Engaging with the concept
What is the difference between distance and position? 5 15
-10 -5

14. Engaging with the concept
What is the difference between distance and position?
Distance is always positive. It doesn’t tell you where you are because it has no direction information. 5 15
-10 -5
Position, on the other hand, DOES tell you where you are relative to a defined origin.

15. Engaging with the concept
What is the difference between distance and displacement? 5 15
-10 -5
Give students a moment to formulate an answer.

16. Engaging with the concept
What is the difference between distance and displacement?
Displacement is a change in position that includes direction. 5 15
-10 -5
The total distance traveled equals the sum of the magnitudes (absolute values) of the displacements.

17. The position equation
The final position equals the initial position plus any displacements.
How is position described?
The next six slides are focused on helping students understand what this equation is saying. the next two slides point out the importance of defining an origin when giving a position.

18. Importance of the origin
In order to describe where something is, always begin by deciding on an origin.
All positions can then be described by comparing them to the origin.

19. Importance of the origin
GPS systems use 0° latitude and longitude as the origin.
Students might be interested in knowing their own latitude and longitude. In solving typical physics problems, we do not usually need a global coordinate system.

20. Position Position tells you where you are relative to an origin.
Positive and negative values tell you whether you are in front or behind, or to the left or right of the origin.
A position value has no meaning if the origin is not clearly understood.

21. The position equation
The final position equals the initial position plus any displacements.
How is displacement described?

22. Coordinate systems + - + -
To describe displacement, you must create a coordinate system and choose which direction is positive and which is negative.
This is a choice! It may change for different problems. +
Left - Right - + Up
Down
North
South
West - East -

23. Displacement Displacement is a change in position.
Positive and negative values indicate direction.
You could also note that d = xf – xi . Displacement equals the change in position.

24. Displacement is a vector
Displacement is a vector because it contains direction information.
For motion along a line, direction is positive or negative.

25. Distance is a scalar Distance is a scalar quantity.
It does not include direction information.

26. Adding displacements
An ant starts at 2 m, and crawls forward 7.1 m. Then it turns around and crawls back 5.5 m. What is the ant’s final position?
The next two slides show graphical vector addition, and numerical vector addition of 1-D displacement vectors.

27. Adding displacements
An ant starts at 2 m, and crawls forward 7.1 m. Then it turns around and crawls back 5.5 m. What is the ant’s final position?
Add displacements graphically by drawing vectors to scale:
The first vector starts at the initial position.
The second vector starts at the end of the first vector.
Reinforce the fact that the second vector starts at the tip (or head) of the first vector.

28. Adding displacements
An ant starts at 2 m, and crawls forward 7.1 m. Then it turns around and crawls back 5.5 m. What is the ant’s final position?
You can also use numerical addition. This is faster and more accurate.
Reinforce the fact that the second vector starts at the tip (or head) of the first vector.

29. Problem solving
How do you recognize the initial position and displacements?
What do you choose as the initial position?
Example: “A boat sails 50 km north then 800 km south. What is the sailboat’s final position?”
This set of slides helps students interpret word problems and extract needed variable information from them.

30. Problem solving
How do you recognize the initial position and displacements?
What do you choose as the initial position?
Example: “A boat sails 50 km north then 800 km south. What is the sailboat’s final position?”
Let the initial position be zero km:
When nothing is said to establish a particular “start” you may assume the initial position is zero.

31. Problem solving
How do you recognize the initial position and displacements?
What are the displacements?
Example: “A boat sails 50 km north then 800 km south. What is the sailboat’s final position?”

32. Problem solving
How do you recognize the initial position and displacements?
What are the displacements?
Example: “A boat sails 50 km north then 800 km south. What is the sailboat’s final position?”
50 km north is a displacement of +50 km:
800 km south is a displacement of -800 km:
Displacements are movements, so words such as “move”, “sail”, “run”, and “travel” are clues.

33. Problem solving
How do you recognize the initial position and displacements?
What is the final position?
Example: “A boat sails 50 km north then 800 km south. What is the sailboat’s final position?”
You might also want to ask them for the distance traveled.

34. Assessment
A robot car moves 200 m east, turns around and moves 600 m west. Describe this motion using position, displacement, and distance.
The first assessment addresses the first objective: “Describe motion in 1D using position, distance, and displacement.” The answer appears on the next slide.

35. Assessment
A robot car moves 200 m east, turns around and moves 600 m west. Describe this motion using position, displacement, and distance.
The robot car starts at the origin with a position of 0 m.
It has a displacement of +200 m east, and then a displacement of 600 m west.
Its final position is 400 meters west of the origin.
The total distance traveled is 800 m.

36. Assessment
A woman starts at x = -10 m and has three successive displacements of -5 m. What is her final position?
The second assessment addresses the second objective: “Analyze motion in 1D using position, distance, and displacement.”

37. Assessment The final position is -25 m.
A woman starts at x = -10 m and has three successive displacements of -5 m. What is her final position?
The final position is -25 m.

38. Assessment
What displacement moves an object from a position of 25 meters to a position of -15 meters?
Asked: displacement Given: initial and final positions
Relationships:
Solution:
Answer:
The third assessment addresses the third objective: “Correctly use and interpret positive and negative values of position and displacement.”

39. Assessment
What displacement moves an object from a position of 25 meters to a position of -15 meters?
Asked: displacement Given: initial and final positions
Relationships: xf = xi + d Solution: d = xf – xi d = -15 m – 25 m = -40 m
Answer: The displacement is -40 meters

40. Speed and velocity
Interactive calculators are used to build qualitative and quantitative understanding of the speed and velocity equations, including the concepts of positive and negative velocity and the comparison with positive and negative position. The lesson builds a conceptual understanding for the units of m/s, and then presents the speed and velocity equations. Students act out positive and negative positions and velocities to build comprehension.

41. Objectives
Describe one dimensional motion using equations for speed and velocity.
Analyze one dimensional motion using equations for speed and velocity.
Define and identify positive and negative velocities.
The lesson objectives clarify what students should know and be able to do.

42. Assessment
A swallow moves 80 meters in 5.0 seconds. What is its speed? What is its velocity? How can the two be different?
Write an English sentence that means the same as this equation:
Give an example of an object with positive position and negative velocity.
These formative assessments will appear again at the end of the lesson, along with the answers.

43. Physics terms speed velocity
New vocabulary will be defined and reinforced throughout the lesson.

44. Equations
The speed is the distance traveled divided by the time taken.
The velocity is the change in position divided by the change in time.

45. Speed versus velocity
Are speed and velocity just two different words for the same thing?
In everyday life you probably use the words speed and velocity interchangeably.
In physics class, speed and velocity are related, but not exactly the same.
Ask students to use the words speed or velocity in a sentence.

46. Exploring the ideas
Click the interactive calculator on page 80 on speed.
This interactive simulation helps students use the equation for speed three different ways.

47. The speed equation
A car travels 30 meters in a trip that lasts 2.0 seconds.
What is the car’s speed? 30
2.0
Speed
This slide provides an example of solving this equation for speed. To enter data, students need to highlight an entry, click [Clear], enter the new value using the keyboard, then click [Enter].

48. The speed equation
A car travels 30 meters in a trip that lasts 2.0 seconds.
What is the car’s speed?
15 m/s 30 15
2.0
Speed
Click [Run] to see the car act out the meaning of the equation.
Point out that the units in the equation shown work also.

49. The speed equation
If you go a distance of 45 meters at a speed of 16 m/s, how long does this trip last?
What variable are you solving for? 45 16
Now students solve the speed equation for time.

50. The speed equation
If you go a distance of 45 meters at a speed of 16 m/s, how long does this trip last?
2.8 seconds 45 16
2.81
Time
Point out that the units in the equation shown work also.

51. The speed equation A fox runs at a speed of 9.7 m/s for 12 seconds.
How far does the fox run?
9.7 12
Now students solve the speed equation for distance.

52. The speed equation A fox runs at a speed of 9.7 m/s for 12 seconds.
How far does the fox run?
116 meters
116.4
9.7 12
Distance
Point out that the units in the equation shown work also.

53. How fast is fast?
See if you can come up with an example of when an actual object might move at each speed.
0.1 m/s
1 m/s
10 m/s
100 m/s
1000 m/s
Because students are more familiar with units of mile per hour, they need to work at developing a feel for values like 1 m/s or 10 m/s so they will know if data or solutions containing speed information is reasonable.

54. How fast is fast?
See if you can come up with an example of when an actual object might move at each speed.
0.1 m/s about 0.22 mph (10 cm/s), the tip of the second hand on your clock
1 m/s
10 m/s
100 m/s
1000 m/s

55. This is an excellent benchmark to remember!
How fast is fast?
See if you can come up with an example of when an actual object might move at each speed.
0.1 m/s about 0.22 mph (10 cm/s), the tip of the second hand on your clock
1 m/s mph, a slow walk
10 m/s
100 m/s
1000 m/s
This is an excellent benchmark to remember!

56. How fast is fast?
See if you can come up with an example of when an actual object might move at each speed.
0.1 m/s about 0.22 mph (10 cm/s), the tip of the second hand on your clock
1 m/s mph, a slow walk
10 m/s 22 mph, a brisk bike riding speed
100 m/s
1000 m/s
Point out that as a rough estimate, the value in m/s is roughly half of the value in mph.

57. How fast is fast?
See if you can come up with an example of when an actual object might move at each speed.
0.1 m/s about 0.22 mph (10 cm/s), the tip of the second hand on your clock
1 m/s mph, a slow walk
10 m/s 22 mph, a brisk bike riding speed
100 m/s 220 mph, a supercar’s top driving speed
1000 m/s

58. How fast is fast?
See if you can come up with an example of when an actual object might move at each speed.
0.1 m/s about 0.22 mph (10 cm/s), the tip of the second hand on your clock
1 m/s mph, a slow walk
10 m/s 22 mph, a brisk bike riding speed
100 m/s 220 mph, a supercar’s top driving speed
1000 m/s ,200 mph, about the F14 fighter jet’s top speed

59. Speed speed Distance is always positive. Time is always positive.
Therefore, speed is always positive!
So how do we tell the difference between moving backward and forward?

60. Speed speed Distance is always positive. Time is always positive.
Therefore, speed is always positive!
So how do we tell the difference between moving backward and forward?
We need a new variable: velocity!

61. Velocity
velocity
The velocity is defined as the change in position divided by the change in time.
Velocity has direction as well as magnitude.

62. Velocity velocity The symbol Δ translates to “the change in.”
The velocity is defined as the change in position divided by the change in time.
The symbol Δ translates to “the change in.”
If x = position then Δx means “the change in position.”

63. Velocity velocity What does Δt mean?
The velocity is defined as the change in position divided by the change in time.
What does Δt mean?

64. Velocity velocity What does Δt mean?
The velocity is defined as the change in position divided by the change in time.
What does Δt mean?
If t = time then Δt means “the change in time.”

65. Velocity
velocity
The velocity is defined as the change in position divided by the change in time.
A change in position, Δx, can be positive or negative.
That means that velocity can be positive or negative.
The sign of the velocity comes from the sign of the change in x.

66. Velocity
velocity
The velocity is defined as the change in position divided by the change in time.
A change in position, Δx, can be positive or negative.
That means that velocity can be positive or negative.
Moving forward is a positive velocity.
Moving backward is a negative velocity.

67. Velocity is a vector Velocity can have negative or positive values.
The sign of the velocity tells you the direction of motion.
Velocity is a vector. A vector is a type of variable which includes directional information in a mathematically useful way.

68. Exploring the ideas
Click the interactive calculator on page 80 on velocity.

69. The velocity equation
A car starts at 30 m and finishes at 10 m in a trip that takes 2.0 seconds.
Notice: The change in position is -20 m. 10 30
-20
What is the car’s velocity? 2 2
Velocity
Enter your initial and final values here.
Point out how to enter value.

70. The velocity equation
A car starts at 30 m and finishes at 10 m in a trip that takes 2.0 seconds.
Notice: The change in position is -20 m. 10 30
-20
-10
What is the car’s velocity? -10 m/s
Click [Run] and see the car drive backwards. 2 2
Velocity
The car in the interactive calculator shows negative velocity by driving backwards, to the left. Negative velocity means “headed in the negative direction” regardless of position.

71. The velocity equation
A car travels at 15 m/s. Four seconds after starting out, it is at a position of 40 m.
What is the car’s change in position?
Where did the car start from? 40 15 4 4
Change in position

72. The velocity equation
A car travels at 15 m/s. Four seconds after starting out, it is at a position of 40 m.
What is the car’s change in position? +60 m
Where did the car start from? -20 m 40
-20 60 15 4 4
Change in position

73. The change in position What is the change in position?
The variable x stands for position.
Subscript “i” means “initial” and subscript “f” means “final”.
These next slides elaborate on the definition of velocity.

74. The change in position What is the change in position?
Δx = xf – xi = (8 m) – (2 m) = 6 meters

75. Velocity
What is the velocity if the change in time (Δt) is 2 seconds?

76. Velocity The velocity is +3 m/s.
What is the velocity if the change in time (Δt) is 2 seconds?
The velocity is +3 m/s.

77. What if the person STARTS at 8 m and moves to the left?
What is the velocity if the change in time (Δt) is 2 seconds?

78. What if the person STARTS at 8 m and moves to the left?
What is the velocity if the change in time (Δt) is 2 seconds?
The man is heading in the negative direction so he has a negative velocity.

79. - + Positive and negative positions and velocities
Can there be positive position and negative velocity?
I need a volunteer to show me how. - +
The next group of slides go carefully through all the possible sign combinations for position and velocity. Students are often confused by the meaning of negative velocity and need to work this through in detail. If possible, draw the origin and the positive and negative directions on the side board so a volunteer can act out the motion for the class.

80. - + Positive and negative positions and velocities Positive position
Can there be positive position and negative velocity?
Positive
position - +

81. - + Positive and negative positions and velocities Positive position
Can there be positive position and negative velocity?
Positive
position
Negative
velocity - +
Having students act out these four cases can make a difficult concept clear and enjoyable. Kinesthetic learners will appreciate the chance to put the idea into action.

82. - + Positive and negative positions and velocities
Can there be negative position and positive velocity?
I need a volunteer to show me how. - +

83. - + Positive and negative positions and velocities Negative position
Can there be negative position and positive velocity?
Negative
position - +

84. - + Positive and negative positions and velocities Negative position
Can there be negative position and positive velocity?
Negative
position
Positive
velocity - +

85. - + Positive and negative positions and velocities
Can there be negative position and negative velocity?
I need a volunteer to show me how. - +

86. - + Positive and negative positions and velocities Negative position
Can there be negative position and negative velocity?
Negative
position
Negative
velocity - +

87. - + Positive and negative positions and velocities
Can there be positive position and positive velocity?
I need a volunteer to show me how. - +

88. - + Positive and negative positions and velocities Positive Positive
Can there be positive position and positive velocity?
Positive
position
Positive
velocity - +

89. The terminology of motion
Tell a story that illustrates how each picture relates to the meaning of each word.

90. The terminology of motion
Tell a story that illustrates how each picture relates to the meaning of each word.
For example: A hiker needs to go southwest on a map and this vector represents the hiker’s movement.
English language learners benefit from a chance to stop and verbally assimilate and employ new vocabulary.

91. Assessment
A swallow moves 80 meters in 5 seconds. What is its speed? What is its velocity? How can the two be different?
The answer to question 1 appears on the next slide. Give students time to formulate an answer.

92. Assessment
A swallow moves 80 meters in 5 seconds. What is its speed? What is its velocity? How can the two be different?
Write an English sentence that means the same as this equation:
The speed is 16 m/s. The velocity could be either +16 m/s or -16 m/s depending on the bird’s direction.
The answer to question 2 appears on the next slide. Ask a volunteer to read their sentence.

93. Assessment
A swallow moves 80 meters in 5 seconds. What is its speed? What is its velocity? How can the two be different?
Write an English sentence that means the same as this equation:
The speed is 16 m/s. The velocity could be either +16 m/s or -16 m/s depending on the bird’s direction.
The velocity is the change in position divided by the change in time.

94. Assessment
Give an example of an object with positive position and negative velocity.
This question addresses the third objective: “Define and identify positive and negative velocities.” The answer appears on the next slide.

95. Assessment - + Negative velocity Positive position
Give an example of an object with positive position and negative velocity.
A person to the right of the origin but walking to the left would have a positive position and negative velocity.
Negative
velocity
Positive
position - +