1. Homework Day 1 Questions 1-10 Problems 1-12 odd only Day 2 Problems 13-29 odd only Day 3 Review Answers 1-29 on line Day 4 Problems 34, 38, 40, 44, 47,49, 59 Day 5 Quiz

2. Kinematics deals with the concepts that are needed to describe motion. Dynamics deals with the effect that forces have on motion. Together, kinematics and dynamics form the branch of physics known as Mechanics.

3. Defining the important variables Kinematics is a way of describing the motion of objects without describing the causes. You can describe an object’s motion: In wordsMathematicallyPictoriallyGraphically No matter HOW we describe the motion, there are several KEY VARIABLES that we use. SymbolVariableUnits tTimes aAccelerationm/s/s x or yDisplacementm vovo Initial velocitym/s vFinal velocitym/s g or a g Acceleration due to gravity m/s/s

4. A Scalar vs a Vector  Scalar – physical quantity that is specified in terms of a single real number, or magnitude Ex. Length, temperature, mass, speed  Vector – physical quantity that is specified by both magnitude and direction Ex. Force, velocity, displacement, acceleration  We represent vectors graphically or quantitatively :

5. More about Vectors A vector is represented on paper by an arrow 1. the length represents magnitude 2. the arrow faces the direction of motion 3. a vector can be “picked up” and moved on the paper as long as the length and direction its pointing does not change

6. Characteristics of a Vector Quantity Has magnitude & direction Requires 3 things: 1. A value 2. Appropriate units 3. A direction! Ex. Acceleration: 9.8 m/s 2 down Velocity: 25 mph West

7. Understanding Vector Directions To accurately draw a given vector, start at the second direction and move the given degrees to the first direction. N S EW 30° N of E Start on the East origin and turn 30° to the North

8. Graphical Representation Practice 5.0 m/s East (suggested scale: 1 cm = 1 m/s) 300 Newtons 60° South of East (suggested scale: 1 cm = 100 N) 0.40 m 25° East of North (suggested scale: 5 cm = 0.1 m)

9. Displacement Displacement (x or y) "Change in position" It is not necessarily the total distance traveled. In fact, displacement and distance are entirely different concepts. Displacement is relative to an axis. o "x" displacement means you are moving horizontally either right or left. o "y" displacement means you are moving vertically either up or down. o The word change is expressed using the Greek letter DELTA ( Δ ). o To find the change you ALWAYS subtract your FINAL - INITIAL position o It is therefore expressed as either Δx = x f - x i or Δy = y f - y i Distance - How far you travel regardless of direction.

11. Displacement

13. Example Suppose a person moves in a straight line from the lockers( at a position x = 1.0 m) toward the physics lab(at a position x = 9.0 m), as shown below The answer is positive so the person must have been traveling horizontally to the right.

14. Example Suppose the person turns around! The answer is negative so the person must have been traveling horizontally to the left What is the DISPLACEMENT for the entire trip? What is the total DISTANCE for the entire trip?

15. 2.1 Displacement

19. Displacement

20. Average Velocity Velocity is defined as: “The RATE at which DISPLACEMENT changes”. Rate = ANY quantity divided by TIME. Average SPEED is simply the “RATE at which DISTANCE changes”.

21. 2.2 Speed and Velocity Average speed is the distance traveled divided by the time required to cover the distance. SI units for speed: meters per second (m/s)

22. 2.2 Speed and Velocity Average speed is the distance traveled divided by the time required to cover the distance. SI units for speed: meters per second (m/s)

23. 2.2 Speed and Velocity Example 1 Distance Run by a Jogger How far does a jogger run in 1.5 hours (5400 s) if his average speed is 2.22 m/s?

24. 2.2 Speed and Velocity Average velocity is the displacement divided by the elapsed time.

25. Example A quarterback throws a pass to a defender on the other team who intercepts the football. Assume the defender had to run 50 m away from the quarterback to catch the ball, then 15 m towards the quarterback before he is tackled. The entire play took 8 seconds. Let's look at the defender's average velocity: Let's look at the defender's speed: “m/s” is the derived unit for both speed and velocity.

26. ARE SPEED AND VELOCITY ALWAYS THE SAME?

27. Average Velocity

28. 2.2 Speed and Velocity Example 2 The World’s Fastest Jet-Engine Car Andy Green in the car ThrustSSC set a world record of 341.1 m/s in 1997. To establish such a record, the driver makes two runs through the course, one in each direction, to nullify wind effects. From the data, determine the average velocity for each run.

29. 2.2 Speed and Velocity

30. 2.3 Acceleration The notion of acceleration emerges when a change in velocity is combined with the time during which the change occurs.

31. Average Acceleration Acceleration is the RATE at which VELOCITY changes. A truck accelerates from 10 m/s to 30 m/s in 2.0 seconds. What is the acceleration? Suppose the same truck then slows down to 5 m/s in 4 seconds. What is the acceleration? “m/s/s” or “m/s 2” is the derived unit for acceleration.

33. What do the “signs”( + or -) mean? QuantityPositiveNegative DisplacementYou are traveling north, east, right, or in the +x or +y direction. You are traveling south, west, left, or in the –x or –y direction. VelocityThe rate you are traveling north, east, right, or in the +x or +y direction. The rate you are traveling south, west, left, or in the –x or –y direction. AccelerationYour velocity(speed) is increasing in a positive direction or your speed is decreasing in a negative direction. Your velocity(speed) is decreasing in a positive direction or your speed is increasing in a negative direction.

34. Beware – the signs can confuse! Suppose a ball is thrown straight upwards at 40 m/s. It takes 4 seconds to reach its maximum height, then another 4 seconds back down to the point where it was thrown. Assume it is caught with the same speed it was thrown. Calculate the acceleration upwards and downwards. This negative sign came from using the DELTA This negative sign came from the DIRECTION of the velocity. It is no surprise you get a negative answer both ways as gravity acts DOWNWARDS no matter if the ball goes up or down. It is GRAVITY which changes the ball’s velocity.

35. 2.3 Acceleration DEFINITION OF AVERAGE ACCELERATION

36. Acceleration

38. What does the sign indicate ?

39. 2.3 Acceleration Example 3 Acceleration and Increasing Velocity Determine the average acceleration of the plane.

40. 2.3 Acceleration

41. Example 3 Acceleration and Decreasing Velocity

42. 2.4 Equations of Kinematics for Constant Acceleration Five kinematic variables: 1. displacement, x 2. acceleration (constant), a 3. final velocity (at time t), v 4. initial velocity, v o 5. elapsed time, t

43. The 3 Kinematic equations There are 3 major kinematic equations than can be used to describe the motion in DETAIL. All are used when the acceleration is CONSTANT. REMEMBER! Any equation with x (horizontal motion) can be replaced with y (vertical motion) Go to Kinematic PowerPoint

44. 2.5 Applications of the Equations of Kinematics Reasoning Strategy 1. Make a drawing. 2. Decide which directions are to be called positive (+) and negative (-). 3. Write down the values that are given for any of the five kinematic variables. 4. Verify that the information contains values for at least three of the five kinematic variables. Select the appropriate equation. 5. When the motion is divided into segments, remember that the final velocity of one segment is the initial velocity for the next. 6. Keep in mind that there may be two possible answers to a kinematics problem.

45. 2.5 Applications of the Equations of Kinematics Example 8 An Accelerating Spacecraft A spacecraft is traveling with a velocity of +3250 m/s. Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s 2. What is the velocity of the spacecraft when the displacement of the craft is +215 km, relative to the point where the retrorockets began firing? xavvovo t +215000 m-10.0 m/s 2 ?+3250 m/s

46. 2.4 Equations of Kinematics for Constant Acceleration

47. Example 6 Catapulting a Jet Find its displacement.

48. 2.4 Equations of Kinematics for Constant Acceleration

49. 2.5 Applications of the Equations of Kinematics xavvovo t +215000 m-10.0 m/s 2 ?+3250 m/s

50. 2.6 Freely Falling Bodies In the absence of air resistance, it is found that all bodies at the same location above the Earth fall vertically with the same acceleration. This idealized motion is called free-fall and the acceleration of a freely falling body is called the acceleration due to gravity.

51. 2.6 Freely Falling Bodies

53. Example 10 A Falling Stone A stone is dropped from the top of a tall building. After 3.00s of free fall, what is the displacement y of the stone?

54. 2.6 Freely Falling Bodies yavvovo t ?-9.80 m/s 2 0 m/s3.00 s

55. 2.6 Freely Falling Bodies yavvovo t ?-9.80 m/s 2 0 m/s3.00 s

56. 2.6 Freely Falling Bodies Example 12 How High Does it Go? The referee tosses the coin up with an initial speed of 5.00m/s. In the absence if air resistance, how high does the coin go above its point of release?

57. 2.6 Freely Falling Bodies yavvovo t ?-9.80 m/s 2 0 m/s+5.00 m/s

58. 2.6 Freely Falling Bodies yavvovo t ?-9.80 m/s 2 0 m/s+5.00 m/s

59. 2.6 Freely Falling Bodies Conceptual Example 14 Acceleration Versus Velocity There are three parts to the motion of the coin. On the way up, the coin has a vector velocity that is directed upward and has decreasing magnitude. At the top of its path, the coin momentarily has zero velocity. On the way down, the coin has downward-pointing velocity with an increasing magnitude. In the absence of air resistance, does the acceleration of the coin, like the velocity, change from one part to another?

60. 2.6 Freely Falling Bodies Conceptual Example 15 Taking Advantage of Symmetry Does the pellet in part b strike the ground beneath the cliff with a smaller, greater, or the same speed as the pellet in part a?

61. Interpret this graph…

62. Graphing Motion Tell me a story.

63. Describing in Words

64. Describe the motion of the object. When is the object moving in the positive direction? Negative direction. When is the object stopped? When is the object moving the fastest? The slowest?

66. http://www.physicsclassroom.com/mmedia/kinema/avd. cfmwww.physicsclassroom.com/mmedia/kinema/avd. cfm

67. Average Velocity

68. Two Stage Rocket Between which time does the rocket have the greatest acceleration? At which point does the velocity of the rocket change.

69. There are 3 types of MOTION graphs Displacement(position or X) vs. Time Velocity vs. Time Acceleration vs. Time There are 2 basic graph models Slope Area

70. Slope – A basic graph model A basic model for understanding graphs in physics is SLOPE. Using the model - Look at the formula for velocity. Who gets to play the role of the slope? Who gets to play the role of the y-axis or the rise? Who get to play the role of the x-axis or the run? What does all the mean? It means that if your are given a Displacement vs. Time graph, to find the velocity of an object during specific time intervals simply find the slope. Velocity Displacement Time

71. Velocity Velocity changes when an object… –Speeds Up –Slows Down –Change direction

72. Instantaneous Velocity

73. Position-Time Graphs We can use a postion-time graph to illustrate the motion of an object. Postion is on the y- axis Time is on the x-axis

74. Plotting a Distance-Time Graph Axis –Distance (position) on y- axis (vertical) –Time on x-axis (horizontal) Slope is the velocity –Steeper slope = faster –No slope (horizontal line) = staying still

75. Slope – A basic graph model Let’s look at another model Who gets to play the role of the slope? Who gets to play the role of the y-axis or the rise? Who get to play the role of the x-axis or the run? What does all the mean? It means that if your are given a Velocity vs. Time graph. To find the acceleration of an object during specific time intervals simply find the slope. Acceleration Velocity Time

76. Where and When We can use a position time graph to tell us where an object is at any moment in time. Where was the car at 4 s? 30 m How long did it take the car to travel 20 m? 3.2 s

77. Speeding Up and Slowing Down The graphs on the left represent an object speeding up. The graphs on the right represent an object that is slowing down.

78. Displacement vs. Time graph What is the velocity of the object from 0 seconds to 3 seconds? The velocity is the slope!

79. Displacement vs. Time graph What is the velocity of the object from 7 seconds to 8 seconds? Once again...find the slope! A velocity of 0 m/s. What does this mean? It is simple....the object has simply stopped moving for 1 second.

80. It is very important that you are able to look at a graph and explain it's motion in great detail. These graphs can be very conceptual. Look at the time interval t = 0 to t = 9 seconds. What does the slope do? It increases, the velocity is increasing Look at the time interval t = 9 to t = 11 seconds. What does the slope do? No slope. The velocity is ZERO. Look at the time interval t = 11 to t = 15 seconds. What does the slope do? The slope is constant and positive. The object is moving forwards at a constant velocity. Look at the time interval t = 15 to t = 17 seconds. What does the slope do? The slope is constant and negative. The object is moving backwards at a constant velocity.

81. Constant Velocity (ac) Objects with a constant velocity have no acceleration This is graphed as a flat line on a velocity time graph.

82. Accelerated Motion In a velocity time graph a line with no slope means constant velocity and no acceleration. In a velocity time graph a sloping line means a changing velocity and the object is accelerating.

83. Changing Velocity (Ac) Objects with a changing velocity are undergoing acceleration. Acceleration is represented on a velocity time graph as a sloped line.

84. Accelerated Motion In a position/displacement time graph a straight line denotes constant velocity. In a position/displacement time graph a curved line denotes changing velocity (acceleration). The instantaneous velocity is a line tangent to the curve.

85. Positive and Negative Velocity The first set of graphs show an object traveling in a positive direction. The second set of graphs show an object traveling in a negative direction.

86. Velocity-Time Graphs (ac) Velocity is placed on the vertical or y-axis. Time is place on the horizontal or x-axis. We can interpret the motion of an object using a velocity-time graph.

87. Velocity vs. Time Graph What is the acceleration from 0 to 6s? What is the acceleration from 6 to 9s? You could say one of two things here: The object has a ZERO acceleration The object has a CONSTANT velocity What is the acceleration from 14 to 15s? A negative acceleration is sometimes called DECELERATION. In other words, the object is slowing down. An object can also have a negative acceleration if it is falling. In that case the object is speeding up. CONFUSING? Be careful and make sure you understand WHY the negative sign is there.

88. Velocity vs. Time Graph Conceptually speaking, what is the object doing during the time interval t = 9 to t = 13 seconds? Does the steepness or slope increase or decrease? The slope INCREASES! According to the graph the slope gets steeper or increases, but in a negative direction. What this means is that the velocity slows down with a greater change each second. The deceleration, in this case, get larger even though the velocity decreases. The velocity goes from 60 to 55 ( a change of 5), then from 55 to 45 ( a change of 10), then from 45 to 30 ( a change of 15), then from 30 to 10 ( a change of 20). Do you see how the change gets LARGER as the velocity gets SMALLER?

89. Area – the “other” basic graph model Another basic model for understanding graphs in physics is AREA. Let's try to algebraically make our formulas look like the one above. We'll start with our formula for velocity. Who gets to play the role of the base? Time What kind of graph is this? Who gets to play the role of the Area ? A Velocity vs. Time graph ( velocity = y- axis & time = x-axis) Who gets to play the role of the height? Velocity Displacement

90. What is the displacement during the time interval t = 0 to t = 5 seconds? That happens to be the area! What is the displacement during the time interval t = 8 to t = 12 seconds? Once again...we have to find the area. During this time period we have a triangle AND a square. We must find the area of each section then ADD them together.

91. Area – the “other” basic graph model Let's use our new model again, but for our equation for acceleration. What does this mean? Who gets to play the role of the base? Who gets to play the role of the height? What kind of graph is this? Who gets to play the role of the Area? Time Acceleration An Acceleration vs. Time graph ( acceleration = y-axis & time = x-axis) The Velocity

92. Acceleration vs. Time Graph What is the velocity during the time interval t = 3 and t = 6 seconds? Find the Area!

93. Summary There are 3 types of MOTION graphs Displacement(position or X) vs. Time Velocity vs. Time Acceleration vs. Time There are 2 basic graph models Slope Area

94. You must know this! t (s) x (m) v (m/s)a (m/s/s) slope = v slope = a area = x area = v

95. You must know this !

96. What can this graph tell us? Two groups (6 min brainstorm)

100. QUALITATIVE GRAPHING What is happening? Represents something not moving Represents objects moving at constant velocity.

101. QUALITATIVE GRAPHING What is happening? Objects speeding up in the positive direction. Objects moving at constant velocity in the negative direction

102. Qualitative Graphing What is happening? Object slowing down in the positive direction Object slowing down in the negative direction Objects slowing down in the positive direction

103. Describe the Motion in Words. Hint: Look at the axes!

104. Velocity vs Acceleration Graph

105. Velocity and Acceleration

106. What if this was a position (x) vs time graph?

107. Comparing and Sketching graphs One of the more difficult applications of graphs in physics is when given a certain type of graph and asked to draw a different type of graph t (s) x (m) slope = v t (s) v (m/s) List 2 adjectives to describe the SLOPE or VELOCITY 1. 2. The slope is CONSTANT The slope is POSITIVE How could you translate what the SLOPE is doing on the graph ABOVE to the Y axis on the graph to the right?

108. Example t (s) x (m) t (s) v (m/s) 1 st line 2 nd line 3 rd line The slope is constant The slope is “+” The slope is “-” The slope is “0”

109. Example – Graph Matching t (s) v (m/s) t (s) a (m/s/s) t (s) a (m/s/s) t (s) a (m/s/s) What is the SLOPE(a) doing? The slope is increasing

110. Displacement from a Velocity-Time Graph The shaded region under a velocity time graph represents the displacement of the object. The method used to find the area under a line on a velocity-time graph depends on whether the section bounded by the line and the axes is a rectangle, a triangle

111. 2.7 Graphical Analysis of Velocity and Acceleration

114. Wrap Up and Review One Dimensional Motion