1. Kinematics in One Dimension

2. Kinematics deals with the concepts that are needed to describe motion, without consideration of what causes the motion. Dynamics deals with the effect that forces have on motion. Analyzes the causes of motion. Together, kinematics and dynamics form the branch of physics known as Mechanics.

3. Mechanics Branch of Physics Kinematics  Study of motion Dynamics  What produces and effects motion

4. What is Motion? Motion Object is in motion when it is changing position with respect to the surroundings Example: Car driving down a road

5. Kinematics in One Dimension Concerned with the description of an object that moves along a straight-line path  Linear motion Convenient to specify motion along the x and y coordinate system Orient the straight line path along the coordinate axes  so motion is along one of the axes

6. Kinematics in One Dimension Important to specify not only speed but also the direction of motion (Up or down; North, South, East, or West) Coordinate (x and y) axes Objects to the right of the origin on the x axis = positive Objects to the left of the origin on the x axis = negative Objects above the origin on the y axis = positive Objects below the origin on the y axis = negative

7. Displacement versus Distance Displacement (x) Change in position of the object How far the object is from its starting point Displacement may not equal total distance traveled

8. Distance Distance Total path length traversed in moving from one location to another Example: Jimmy is driving to school but he forgets to pick up Johnny on the way…He now has to reverse his direction and drive back 2 miles Total Distance Traveled = 12 miles Total Displacement = 8 miles + 10 miles - 2 miles

9. Displacement versus Distance  Person walks 70m East then turns around and walks 30m West  Total distance traveled = 100m  Displacement = 40m, since the person is now only 40m from the starting point ** Displacement: Quantity with both magnitude and direction  Vector * Represented by arrows in diagrams * Displacement = Final position – Initial position

10. Displacement

14. Speed versus Velocity Speed and velocity Used synonymously in everyday conversation  But, the terms have different meanings in physics

15. Speed and Velocity Average speed Average speed is the total distance traveled divided by the time required to travel this distance. SI units for speed: meters per second (m/s)

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

17. Answer Distance traveled = (Average speed) (Elapsed Time) Distance= (2.22 m/s)(5400 s) = 12,000 m

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

19. Average Velocity The football player ran 50 m in 20 s. What is his average velocity? v = 50 m / 20 s = 2.5 m/s

20. Speed and Velocity Example: The World’s Fastest Jet-Engine Car Andy Green in the car ThrustSSC set a world record of 341.1 m/s (762.8 mph) 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.

21. Speed and Velocity (759 mph) (766.4 mph)

22. Speed versus Velocity Speed: Positive number, with units Does not take into account direction Speed is therefore a _ _ _ _ _ _ quantity? Velocity (v): Specifies both the magnitude (numerical value  how fast an object is moving) and also the direction in which an object is moving Velocity is therefore a _ _ _ _ _ _ quantity?

23. Instantaneous Velocity You drive a car 100 km along a straight road in one direction for 4 hours Average velocity = 25 km/hr…unlikely you were driving 25 km/hr at every instant Instantaneous velocity Average velocity at any instant of time Example: Speedometer reading Gives the speed of a car over a very short interval of time

24. Speed and Velocity instantaneous velocity The instantaneous velocity indicates how fast the car moves and the direction of motion at each instant of time. Animation of Average vs. Instantaneous Speed

25. Graphical Analysis - Uniform Linear Motion Given below is a strobe picture of a ball rolling across a table Strobe pictures reveal the position of the object at regular intervals of time, in this case, once each 0.1 seconds Notice that the ball covers an equal distance between flashes Assume distance between flashes = 20 cm Display the ball's behavior on an position versus time graph

26. Position versus Time Graph

27. Slope of a Straight Line Slope of a straight line Slope = On the position versus time graph Slope of the straight line = Average velocity Positive slope indicates that the objects displacement, away from the origin, increases with time  so, motion of the ball is in the positive x direction

28. Slope of a Position versus Time Graph

29. What does the Slope of a Position vs. Time Graph Tell Us? On the Position vs. Time Graph Slope of the straight line  20 cm/0.1 s = 200 cm/sec Represents the ball’s average velocity, as it moves across the table Ball’s velocity is positive, therefore it is moving in the positive direction –Ball's velocity  Vector quantity possessing both magnitude (200 cm/sec) and direction (positive)

30. Velocity versus Time Graph Slope of the straight line on the Position versus Time Graph  Average Velocity Velocity Time Average Velocity = 200 cm/s 200 cm/s

31. What would the Acceleration versus Time Graph look like? Hint: Slope of the straight line on the Velocity versus Time Graph  Average Acceleration Acceleration Time

32. Acceleration versus Time Graph * Slope of the straight line on the Velocity versus Time Graph = 0 ** Average Acceleration = 0 *** Ball is moving with constant velocity Acceleration Time

33. Acceleration Average Acceleration (a) Describes how rapidly the velocity of an object is changing Defined as the change in velocity divided by the time taken to make this change Car  velocity increases in magnitude from zero to 70 mph  acceleration

34. Constant Acceleration  Acceleration  Change in velocity per time interval  Velocity = m/s  Time = s  Acceleration formula leads to units of m/s 2 Animation of constant acceleration website

35. Acceleration acceleration Notion of acceleration emerges when a change in velocity is combined with the time during which the change occurs.

36. Acceleration DEFINITION OF AVERAGE ACCELERATION

37. Acceleration Acceleration and Increasing Velocity Example: Acceleration and Increasing Velocity Determine the average acceleration of the plane.

38. Acceleration

39. Example Acceleration and Decreasing Velocity

40. Acceleration

41. Acceleration Acceleration: Vector quantity  For one-dimensional motion: Utilize a (+) positive sign or (-) negative sign to indicate direction relative to a coordinate system (x or y axes) SI units for acceleration: meters/second 2  m/s 2 Acceleration Animation Website Animation - Direction of Acceleration and Velocity Website

42. Motion at Constant Acceleration Constant Acceleration: Instantaneous and average accelerations are equal Utilize a set of kinematic equations for constant acceleration

43. Equations of Kinematics for Constant Acceleration It is customary to dispense with the use of boldface symbols overdrawn with arrows for the displacement, velocity, and acceleration vectors. We will, however, continue to convey the directions with a plus or minus sign.

44. Equations of Kinematics for Constant Acceleration Let the object be at the origin when the clock starts.

45. Equations of Kinematics for Constant Acceleration

46. 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

47. Equations of Kinematics for Constant Acceleration

49. Example: Catapulting a Jet ** Find its displacement.

50. Equations of Kinematics for Constant Acceleration

52. Equations of Kinematics for Constant Acceleration

53. Applications of the Equations of Kinematics Solving Problems 1.Draw a diagram of the situation 2.Illustrate coordinate axes  decide which directions are to be called positive (+) and negative (-) 3.Write down the known quantities 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  the final velocity of one segment is the initial velocity for the next 6.Carry out the calculation, keeping track of the units 7.Is the result reasonable?

54. Applications of the Equations of Kinematics Example: 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

55. Applications of the Equations of Kinematics

56. xavvovo t +215000 m -10.0 m/s 2 ?+3250 m/s

57. Falling Objects  Galileo Galilei’s  Galileo Galilei’s contribution to our understanding of the motion of falling objects  ** In the absence of air resistance, at any given location on Earth, all objects fall with the same constant acceleration *** Acceleration due to gravity g = 9.8 m/s 2  Uniformly accelerated motion  object allowed to fall freely near the Earth’s surface

58. Acceleration due to Gravity Vector  Direction is toward the center of the Earth

59. 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. If the distance of the fall is small compared to the radius of the Earth, then the acceleration remains essentially constant throughout the descent. free-fall acceleration due to gravity. This idealized motion is called free-fall and the acceleration of a freely falling body is called the acceleration due to gravity.

60. Kinematic Equations: Constant Acceleration Kinematic Equations: Acceleration due to Gravity

61. Motion is Relative! Using the Earth as a frame of reference  most common Object with a positive velocity (+ v)  Moving to the right (Away from the reference point) Object with a negative velocity (- v)  Moving to the left (In the opposite direction with respect to the reference point) Hint: It is possible for an object to have a zero velocity (v = 0), while it changes direction, in the presence of an acceleration – * Just remember! It is impossible for acceleration to be zero and have a changing velocity

62. Things to Remember! Objects can either be at Rest Moving with a constant velocity or Accelerating If an object is at rest or moving with a constant velocity  Acceleration = 0! When present, we always assume accelerations are uniform, or constant

63. Freely Falling Bodies

64. Free-Fall Free-Fall Website:Elephant and Feather Website:Elephant and Feather  “Free Fall”  Different meaning to a physicist than it does to a skydiver  In physics  Free fall = One-dimensional motion of any object under the influence of gravity only  No air resistance or frictional effects of any kind  Whereas it is air resistance that makes skydiving a hobby rather than a crazy stunt!

65. Free Fall “Physics Free Fall” Since everything we observe falling is falling through the air, is “Physics Free Fall”, a useless idea in practice? No! Any falling object's motion is at least approximately free fall as long as: heavy –It is relatively heavy compared to its size »Dropping a ball or jumping off a chair = Free-fall motion »Dropping an unfolded piece of paper, or the motion of a dust particle floating in the air  Not Free Fall (If you crumble the paper into a "paper wad", however, its motion is approximately free fall)

66. Free Fall short time  It falls for a relatively short time  Jumping off a chair = Free fall  After you have jumped out of an airplane and fallen for several seconds  you are not in free fall, since air resistance is now a factor in your motion slowly  It is moving relatively slowly  Drop a ball or throw it down  Free fall  Shoot a ball of a cannon  Not free fall  * Note:  * Note: Object does not have to be falling to be in free fall  Example - Throwing a ball upward  Motion is still considered to be free fall, since it is moving under the influence of gravity

67. Freely Falling Bodies A Falling Stone Example: 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?

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

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

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

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

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

73. Freely Falling Bodies Acceleration Versus Velocity Conceptual Example: 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 a 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?

74. Note: As this occurs, a = 9.81 m/s 2 ! The object is changing direction at this point.

75. Freely Falling Bodies Conceptual Example: 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?

76. Analyzing Graphs Three types of graphs: Three states of Motion: Time Distance Velocity Acceleration Velocity = 0 (rest) Constant velocity Constant acceleration

77. Graphical Analysis of Velocity and Acceleration

78. Velocity versus Time Graph? Velocity Time (s) 4 m/s

79. Acceleration versus Time Graph Acceleration (m/s 2 ) Time

80. Graphical Analysis of Velocity and Acceleration

83. On velocity vs. time graphs, the slope of a straight line is the acceleration of the object, and the area under the curve equals the distance the object travels.

84. Websites One Dimensional Kinematics Review Lessons  1, 2, 3, 4, 5, 6 * Disclaimer: This powerpoint presentation is a compilation of various works.