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# Kinematics 2012

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• Note that a motion described as a constant, positive velocity results in a line of constant and positive slope when plotted as a position-time graph.
• Note that a motion described as a changing, positive velocity results in a line of changing and positive slope when plotted as a position-time graph.
• Note that a motion described as a constant, positive velocity results in a line of constant and positive slope when plotted as a position-time graph.
• Note that a motion described as a constant, positive velocity results in a line of constant and positive slope when plotted as a position-time graph.
• The velocity vs. time graphs for the two types of motion - constant velocity and changing velocity (acceleration) - can be summarized as follows.
• The velocity vs. time graphs for the two types of motion - constant velocity and changing velocity (acceleration) - can be summarized as follows.
• ### Kinematics 2012

1. 1. MECHANICSMechanics is a branch of physics which dealswith the study of the forces which act on abody and keep it in equilibrium or in motion. Itis one of the largest subjects in science andtechnology. The bodies whose motion westudy in "Mechanics" are macroscopic bodiesi.e. bodies that we can easily see.However, we may study motion with respect tosolids, liquids and gases, where we may bedealing with the motion of large molecules.
2. 2. MECHANICSMechanics is classified into three divisionsas:Statics • Statics is a branch of physics (mechanics) which concerned with equilibrium state of bodies under the action of forces. When a system of bodies is in static equilibrium, the system is either at rest, or moving at constant velocity through its center of mass. It can also be understood as the study of the forces affecting non- moving objects
3. 3. MECHANICSDynamics • The branch of physics (mechanics) which deals with the effect of forces on the motion of bodies. It can also be understood as the study of the forces affecting moving objects.Kinematics • It is the branch of physics (mechanics) concerned with the motions of objects without being concerned with the forces that cause the motion.
4. 4. KINEMATICSIn this unit, we study mechanics, which is thestudy of the motion of objects and the relatedconcepts of force and energy. This section isdevoted to the study of kinematics, adescription of how objects move. We seemany examples of moving objects in our livesfrom automobiles to ball motion in sports, tothe motion of the moon and the sun andsatellites.
5. 5. SCALAR AND VECTORQUANTITIESPhysics is a mathematical science - that is, theunderlying concepts and principles have amathematical basis. Throughout the course of ourstudy of physics, we will encounter a variety ofconcepts which have a mathematical basisassociated with them.The motion of objects can be described by words -words such as distance, displacement, position,speed, velocity, and acceleration. Thesemathematical quantities which are used todescribe the motion of objects can be divided intotwo categories.
6. 6. Scalar and VectorQuantitiesThe quantity is either a vector or a scalar.These two categories can be distinguishedfrom one another by their distinct definitions: • Scalars are quantities that are fully described by a magnitude alone. • Vectors are quantities that are fully described by both a magnitude and a direction.
7. 7. Scalar and VectorVideo
8. 8. Scalar and VectorQuantitiesCheck Your UnderstandingCategorize each quantity as being either a vector or a scalar. a. 5 m b. 30 m/sec, East c. 5 mi., 5o d. 20 degrees Celsius e. 256 bytes f. 4000 Calorie
9. 9. POSITION, DISPLACEMENT, &DISTANCEDistance and displacement are twoquantities which may seem to mean thesame thing, yet they have distinctlydifferent meanings and definitions.Distance is a scalar quantity which refers to"how much ground an object has covered"during its motion.Displacement is a vector quantity whichrefers to "how far out of place an object is";it is the objects change in position.
10. 10. POSITION, DISPLACEMENT, &DISTANCEThe position of an object is a description of itslocation. In order to describe the position of anobject, one must know the distance and directionfrom a known origin.Example • A physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North
11. 11. POSITION, DISPLACEMENT, &DISTANCEExample • Use the diagram to determine the distance traveled by the skier and the resulting displacement during these three minutes.
12. 12. POSITION, DISPLACEMENT, &DISTANCEQuestion:What are the position, distance and thedisplacement of the racecar drivers in the Indy500?
13. 13. VECTORS & DIRECTION• Examples of vector include displacement, velocity, acceleration, and force. Each of these quantities are unique in that a full description of the quantity demands that both a magnitude and a direction are listed.• Vector quantities are not fully described unless both magnitude and direction are described• Vector quantities are often represented by scaled diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction.
14. 14. VECTORS & DIRECTIONObserve that there are severalcharacteristics of this diagram whichmake it an appropriately drawn vectordiagram. • a scale is clearly listed • an arrow (with arrowhead) is drawn in a specified direction; thus, the vector has a head and a tail. • the magnitude and direction of the vector is clearly labeled; in this case, the diagram shows magnitude is 20m and the direction is (30 degrees West of North).
15. 15. VECTORS & DIRECTIONThe direction of a vector is often expressed as ancounterclockwise angle of rotation of the vectorabout its "tail" from due East. Using this convention,a vector with a direction of 30 degrees is a vectorthat has been rotated 30 degrees in acounterclockwise direction relative to due east. Avector with a direction of 160 degrees is a vectorthat has been rotated degrees in acounterclockwise direction relative to due east. Avector with a direction of 270 degrees is a vectorthat has been rotated 270 degrees in acounterclockwise direction relative to due east.
16. 16. VECTORS & DIRECTION
17. 17. VECTORS & DIRECTIONThe magnitude of a vector in a scaled vectordiagram is depicted by the length of the arrow. Thearrow is drawn a precise length in accordance witha chosen scale. For example, if a diagram shows avector with a magnitude of 15 km and the scaleused constructing the diagram is 1 cm = 5 km, thevector arrow is drawn with a length of 3 cm.
18. 18. COMPOSITION (ADDING) OFVECTORS• The composition of vectors is the process of combining vectors to find a single vector that has the same effect as the combination of single vectors.• Each separate vector is called a component and the single vector that produces the same result as the combined components is called the resultant.• The resultant vector is a vector that replaces two or more vectors that have been added together.
19. 19. COMPOSITION OF VECTORSThere are a variety of methods fordetermining the magnitude and directionof the result of adding two or morevectors. These methods include: • the Pythagorean theorem and trigonometric methods (Component Method) • the head-to-tail method using a scaled vector diagram (Graphical Method)
20. 20. ADDING LINEAR VECTORS• Two vectors, going in the same direction can be added together by simply placing them “head-to- tail”. The resultant will have a new magnitude that is equal to the sum of the two individual vector magnitudes.• Vectors can also be subtracted. The direction of the resultant vector is the same as the direction of the largest vector.• To determine the magnitude of the resultant vector you place them “tail-to-tail”. The longer one then cancels out the shorter one, leaving the resultant.
22. 22. ADDING LINEARVECTORSPractice Problems: 1. A jet fighter plane traveling south at 490 km/h fires a missile forward at 150 km/h. What is the velocity of themissile? 2. A child pulls east on a rope with a force of 30 newtons. Another child pulls with a force of 25 newtons in the oppositedirection. What is the resultant force on therope?
23. 23. ADDING PERPENDICULARVECTORSThe Pythagorean theorem is a useful method fordetermining the result of adding two (and only two)vectors that make a right angle to each other. Themethod is not applicable for adding more than twovectors or for adding vectors which are not at 90-degrees to each other.
24. 24. Adding PerpendicularVectorsExample • A hiker leaves camp and hikes 11 km, north and then hikes 11 km east. Determine the resulting displacement of the hiker.
25. 25. Adding PerpendicularVectorsThe direction of vector R in the diagram above can bedetermined by use of trigonometric functions. Recallthe meaning of the useful mnemonic - SOH CAHTOA.
26. 26. Adding PerpendicularVectorsThe three equations below summarize these threefunctions in equation form.
27. 27. Adding PerpendicularVectorsExample • A hiker leaves camp and hikes 11 km, north and then hikes 11 km east. Determine the resulting displacement of the hiker.
29. 29. GRAPHICAL METHOD• The magnitude and direction of the sum of two or more vectors can also be determined by use of an accurately drawn scaled vector diagram. Using a scaled diagram, the head-to-tail method is to determine the resultant.• The head-to-tail method involves drawing a vector to scale on a sheet of paper beginning at a designated starting position; where the head of this vector ends the tail of the next vector begins (thus, head-to-tail method).
30. 30. GRAPHICAL METHOD• The process is repeated for all vectors that are added. Once all vectors have been added head- to-tail, the resultant is drawn from the tail of the first vector to the head of the last vector; i.e., from start to finish. Once the resultant is drawn, its length can be measured and converted to real units using the given scale.• The direction of the resultant can be determined by using a protractor and measuring its counterclockwise angle of rotation from due East.
31. 31. A VECTOR JOURNEYYour assignment is to find the displacementof the following journey. You must draw ascaled vector Diagram, showing thedisplacement.Starting Point – X marks the spot (Mr.Coulter’s Room)End Point – X marks the spot (in front of thewater fountain in the north hallway)
32. 32. VECTOR COMPONENTS• Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts.• Each part of a two-dimensional vector is known as a component. The components of a vector depict the influence of that vector in a given direction.• The combined influence of the two components is equivalent to the influence of the single two- dimensional vector. The single two-dimensional vector could be replaced by the two components.
33. 33. VECTOR COMPONENTS
34. 34. VECTOR RESOLUTIONThere are two basic methods for determining themagnitudes of the components of a vector directedin two dimensions. The process of determining themagnitude of a vector is known as vectorresolution. The two methods of vector resolutionthat we will examine are: • The Parallelogram Method • Trigonometric Method
35. 35. THE PARALLELOGRAMMETHODThe parallelogram method of vector resolution involvesusing an accurately drawn, scaled vector diagram todetermine the components of the vector. The methodinvolves drawing the vector to scale in the indicateddirection, sketching a parallelogram around the vector.
36. 36. THE TRIGONOMETRICMETHODThe trigonometric method of vector resolution involves usingtrigonometric functions to determine components of thevector.
37. 37. Adding Non-Perpendicular or Non-Linear VectorsUsing the components of vectors, it is now easy toadd vectors that are not linear or perpendicular toone another using the component method.
38. 38. Adding Non-Perpendicular or Non-Linear VectorsExample:Find the resultant vector of A and B given in thegraph below.
39. 39. Adding Non-Perpendicular or Non-Linear Vectors We use trigonometric equations first and find the components of the vectors then, make addition and subtraction between the vectors sharing same direction.
40. 40. Adding Non-Perpendicular or Non-Linear Vectors Vector X Component Y Component A B Result- ant
41. 41. INSTANT & INTERVAL OF TIME• The notion of time is important in physics. A common misconception is that an instant in time is a very short interval of time. However, an instant is considered to be a single clock reading (t). If time were plotted on an axis, an instant is just a single coordinate along that axis.• An interval is a duration in time, i.e. the interval separating two instants on the time axis (∆t). The combination of a clock reading and instantaneous position is called an event and later becomes useful introducing relativity.
42. 42. INSTANT & INTERVAL OFTIMENote: ∆, pronounced “delta”, is used to represent the phrase: “change in”, and is calculated as “final – initial”. Ex. ∆ t is pronounced as: the change in time (∆ t = tfinal – tinitial ) ∆ v is pronounced as: the change in velocity (∆ v = vfinal - vinitial )
43. 43. INSTANT & INTERVAL OF TIMENote: ∆, pronounced “delta”, is used to represent the phrase: “change in”, and is calculated as “final – initial”. Ex. ∆ t is pronounced as: the change in time (∆ t = tfinal – tinitial ) ∆ v is pronounced as: the change in velocity (∆ v = vfinal - vinitial )
44. 44. Describing Motion 15 10Position 5 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Time
45. 45. POSITION VS.TIMEGRAPHS
46. 46. Position vs. Time GraphsThe specific features of the motion of objects are demonstratedby the shape and the slope of the lines on a position vs. timegraph.The first part involves a study of the relationship between theshape of a p-t graph and the motion of the object.
47. 47. Position vs. Time GraphsConsider a car moving with a constant,rightward (+) velocity - say of +10 m/s.
48. 48. Position vs. Time GraphsConsider a car moving with a constant,rightward (+) velocity - say of +10 m/s.
49. 49. Position vs. Time GraphsConsider a car moving with a rightward (+),changing velocity - that is, a car that is movingrightward but speeding up or accelerating.
50. 50. Position vs. Time GraphsConsider a car moving with a rightward (+),changing velocity - that is, a car that is movingrightward but speeding up or accelerating.
51. 51. Position vs. Time GraphsThe slope of the line on a position-time graphreveals useful information about the velocity ofthe object.If the velocity is constant, then the slope isconstant (i.e., a straight line).If the velocity is changing, then the slope ischanging (i.e., a curved line).If the velocity is positive, then the slope ispositive (i.e., moving upwards and to the right).
52. 52. Position vs. Time GraphsSlow, Rightward (+), Fast, Rightward (+),Constant Velocity Constant Velocity
53. 53. THE MEANING OF SLOPE FOR A P-T GRAPHThe slope of a position vs. time graph revealspertinent information about an objects velocity. • a small slope means a small velocity; • a negative slope means a negative velocity; • a constant slope (straight line) means a constant velocity; • a changing slope (curved line) means a changing velocity.
54. 54. The Meaning of Slope for a p-t GraphConsider a car moving at a constant velocityof +5 m/s for 5 seconds, abruptly stopping,and then remaining at rest (v = 0 m/s) for 5seconds.
55. 55. The Meaning of Slope for a p-t GraphConsider a car moving at a constant velocity of+5 m/s for 5 seconds, abruptly stopping, andthen remaining at rest (v = 0 m/s) for 5seconds.
56. 56. Determining the Slope for a p-t Graph The slope of the line on a position vs. time graph is equal to the velocity of the object.
57. 57. The Meaning of Slope for a p-t GraphDetermine the velocity (i.e., slope) of the objectas portrayed by the graph below.
58. 58. VELOCITY VSTIMEGRAPHS
59. 59. Velocity vs. Time GraphsVelocity is a vector quantity that refers to "the rate at which anobject changes its position." As such, velocity is "direction-aware." When evaluating the velocity of an object, one mustkeep track of direction.To describe the motion of an object, velocity vs. time graphs canbe used.
60. 60. Velocity vs. Time GraphsConsider a car moving with a constant, rightward(+) velocity - say of +10 m/s. As learned earlier, acar moving with a constant velocity is a car withzero acceleration.
61. 61. Velocity vs. Time GraphsNote that a motion described as a constant,positive velocity results in a line of zero slope (ahorizontal line has zero slope) when plotted asa velocity-time graph.
62. 62. Velocity vs. Time GraphsConsider a car moving with a rightward (+),changing velocity - that is, a car that is movingrightward but speeding up or accelerating. Sincethe car is moving in the positive direction andspeeding up, the car is said to have a positiveacceleration.
63. 63. Velocity vs. Time GraphsNote that a motion described as a changing,positive velocity results in a sloped line whenplotted as a velocity-time graph. The slope of theline is positive, corresponding to the positiveacceleration.
64. 64. Velocity vs. Time GraphsPositive Velocity Positive VelocityPositive Acceleration Zero Acceleration
65. 65. Velocity vs. Time Graphs
66. 66. KINEMATIC EQUATIONSThese arethe basicequationsthat youwill beworkingwith. Keepin mindthat theseequationscan berearrangedorcombinedto makedifferentequations
67. 67. KINEMATIC EQUATIONSIllustrative Example #1An object moving at 3.0 m/s accelerates for 4.0 swith a uniform acceleration of 2.0 m/s2. Find thedisplacement of the object.
68. 68. KINEMATIC EQUATIONSIllustrative Example #2A car traveling at 10.0 m/s accelerates at a rate of3.0 m/s2 to a speed of 25.0 m/s. What is thedisplacement of the car during the acceleration?
69. 69. KINEMATIC EQUATIONSIllustrative Example #3If an object is accelerated at 5.00 m/s2 and starts from rest, what isits velocity after 20.0 m?