1. Unit 1 A Kinematics

2. Dynamics The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts Kinematics is a part of dynamics Kinematics is a part of dynamics In kinematics, you are interested in the description of motion In kinematics, you are interested in the description of motion Not concerned with the cause of the motion Not concerned with the cause of the motion

3. Brief History of Motion Sumaria and Egypt Sumaria and Egypt Mainly motion of heavenly bodies Mainly motion of heavenly bodies Greeks Greeks Also to understand the motion of heavenly bodies Also to understand the motion of heavenly bodies Systematic and detailed studies Systematic and detailed studies

4. “Modern” Ideas of Motion Galileo Galileo Made astronomical observations with a telescope Made astronomical observations with a telescope Experimental evidence for description of motion Experimental evidence for description of motion Quantitative study of motion Quantitative study of motion

5. Position Defined in terms of a frame of reference Defined in terms of a frame of reference One dimensional, so generally the x- or y- axis One dimensional, so generally the x- or y- axis

6. Vector Quantities Vector quantities need both magnitude (size) and direction to completely describe them Vector quantities need both magnitude (size) and direction to completely describe them Represented by an arrow, the length of the arrow is proportional to the magnitude of the vector Represented by an arrow, the length of the arrow is proportional to the magnitude of the vector Head of the arrow represents the direction Head of the arrow represents the direction Generally printed in bold face type Generally printed in bold face type

7. Scalar Quantities Scalar quantities are completely described by magnitude only Scalar quantities are completely described by magnitude only

8. Displacement Measures the change in position Measures the change in position Represented as  x (if horizontal) or  y (if vertical) Represented as  x (if horizontal) or  y (if vertical) Vector quantity Vector quantity + or - is generally sufficient to indicate direction for one-dimensional motion + or - is generally sufficient to indicate direction for one-dimensional motion Units are meters (m) in SI, centimeters (cm) in cgs or feet (ft) in US Customary Units are meters (m) in SI, centimeters (cm) in cgs or feet (ft) in US Customary

9. Displacements

10. Distance Distance may be, but is not necessarily, the magnitude of the displacement Blue line shows the distance Red line shows the displacement

11. Velocity It takes time for an object to undergo a displacement It takes time for an object to undergo a displacement The average velocity is rate at which the displacement occurs The average velocity is rate at which the displacement occurs generally use a time interval, so let t o =0 generally use a time interval, so let t o =0

12. Velocity continued Direction will be the same as the direction of the displacement (time interval is always positive) Direction will be the same as the direction of the displacement (time interval is always positive) + or - is sufficient + or - is sufficient Units of velocity are m/s (SI), cm/s (cgs) or ft/s (US Cust.) Units of velocity are m/s (SI), cm/s (cgs) or ft/s (US Cust.) Other units may be given in a problem, but generally will need to be converted to these Other units may be given in a problem, but generally will need to be converted to these

13. Speed Speed is a scalar quantity Speed is a scalar quantity same units as velocity same units as velocity total distance / total time total distance / total time May be, but is not necessarily, the magnitude of the velocity May be, but is not necessarily, the magnitude of the velocity

14. Instantaneous Velocity The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero The instantaneous velocity indicates what is happening at every point of time The instantaneous velocity indicates what is happening at every point of time

15. Uniform Velocity Uniform velocity is constant velocity Uniform velocity is constant velocity The instantaneous velocities are always the same The instantaneous velocities are always the same All the instantaneous velocities will also equal the average velocity All the instantaneous velocities will also equal the average velocity

16. Graphical Interpretation of Velocity Velocity can be determined from a position- time graph Velocity can be determined from a position- time graph Average velocity equals the slope of the line joining the initial and final positions Average velocity equals the slope of the line joining the initial and final positions Instantaneous velocity is the slope of the tangent to the curve at the time of interest Instantaneous velocity is the slope of the tangent to the curve at the time of interest The instantaneous speed is the magnitude of the instantaneous velocity The instantaneous speed is the magnitude of the instantaneous velocity

17. Average Velocity

18. Instantaneous Velocity

19. Acceleration Changing velocity (non-uniform) means an acceleration is present Changing velocity (non-uniform) means an acceleration is present Acceleration is the rate of change of the velocity Acceleration is the rate of change of the velocity Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust) Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust)

20. Average Acceleration Vector quantity Vector quantity When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are in the opposite directions, the speed is decreasing When the sign of the velocity and the acceleration are in the opposite directions, the speed is decreasing

21. Instantaneous and Uniform Acceleration The limit of the average acceleration as the time interval goes to zero The limit of the average acceleration as the time interval goes to zero When the instantaneous accelerations are always the same, the acceleration will be uniform When the instantaneous accelerations are always the same, the acceleration will be uniform The instantaneous accelerations will all be equal to the average acceleration The instantaneous accelerations will all be equal to the average acceleration

22. Graphical Interpretation of Acceleration Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graph Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graph

23. Average Acceleration

24. Relationship Between Acceleration and Velocity Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zero

25. Relationship Between Velocity and Acceleration Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer)

26. Relationship Between Velocity and Acceleration Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter)

27. Kinematic Equations Used in situations with uniform acceleration Used in situations with uniform acceleration

28. Notes on the equations Gives displacement as a function of velocity and time Gives displacement as a function of velocity and time

29. Notes on the equations Shows velocity as a function of acceleration and time Shows velocity as a function of acceleration and time

30. Graphical Interpretation of the Equation

31. Notes on the equations Gives displacement as a function of time, velocity and acceleration Gives displacement as a function of time, velocity and acceleration

32. Notes on the equations Gives velocity as a function of acceleration and displacement Gives velocity as a function of acceleration and displacement

33. Problem-Solving Hints Be sure all the units are consistent Be sure all the units are consistent Convert if necessary Convert if necessary Choose a coordinate system Choose a coordinate system Sketch the situation, labeling initial and final points, indicating a positive direction Sketch the situation, labeling initial and final points, indicating a positive direction Choose the appropriate kinematic equation Choose the appropriate kinematic equation Check your results Check your results

34. Free Fall All objects moving under the influence of only gravity are said to be in free fall All objects moving under the influence of only gravity are said to be in free fall All objects falling near the earth’s surface fall with a constant acceleration All objects falling near the earth’s surface fall with a constant acceleration Galileo originated our present ideas about free fall from his inclined planes Galileo originated our present ideas about free fall from his inclined planes The acceleration is called the acceleration due to gravity, and indicated by g The acceleration is called the acceleration due to gravity, and indicated by g

35. Acceleration due to Gravity Symbolized by g Symbolized by g g = 9.8 m/s² ≈ 10 m/s 2 g = 9.8 m/s² ≈ 10 m/s 2 g is always directed downward g is always directed downward toward the center of the earth toward the center of the earth

36. Free Fall -- an object dropped Initial velocity is zero Initial velocity is zero Let up be positive Let up be positive Use the kinematic equations Use the kinematic equations Generally use y instead of x since vertical Generally use y instead of x since vertical v o = 0 a = g

37. Free Fall -- an object thrown downward a = g a = g Initial velocity  0 Initial velocity  0 With upward being positive, initial velocity will be negative With upward being positive, initial velocity will be negative

38. Free Fall -- object thrown upward Initial velocity is upward, so positive Initial velocity is upward, so positive The instantaneous velocity at the maximum height is zero The instantaneous velocity at the maximum height is zero a = g everywhere in the motion a = g everywhere in the motion g is always downward, negative g is always downward, negative v = 0

39. Thrown upward, cont. The motion may be symmetrical The motion may be symmetrical then t up = t down then t up = t down then v f = -v o then v f = -v o The motion may not be symmetrical The motion may not be symmetrical Break the motion into various parts Break the motion into various parts generally up and down generally up and down

40. Non-symmetrical Free Fall Need to divide the motion into segments Need to divide the motion into segments Possibilities include Possibilities include Upward and downward portions Upward and downward portions The symmetrical portion back to the release point and then the non- symmetrical portion The symmetrical portion back to the release point and then the non- symmetrical portion

41. Combination Motions

42. 42 Chapter 3 Vectors and Two-Dimensional Motion

43. 43 Vector Notation When handwritten, use an arrow: When handwritten, use an arrow: When printed, will be in bold print: A When printed, will be in bold print: A When dealing with just the magnitude of a vector in print, an italic letter will be used: A When dealing with just the magnitude of a vector in print, an italic letter will be used: A

44. 44 Properties of Vectors Equality of Two Vectors Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction Two vectors are equal if they have the same magnitude and the same direction Movement of vectors in a diagram Movement of vectors in a diagram Any vector can be moved parallel to itself without being affected Any vector can be moved parallel to itself without being affected

45. 45 More Properties of Vectors Negative Vectors Negative Vectors Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions) Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions) A = -B A = -B Resultant Vector Resultant Vector The resultant vector is the sum of a given set of vectors The resultant vector is the sum of a given set of vectors

46. 46 Adding Vectors When adding vectors, their directions must be taken into account When adding vectors, their directions must be taken into account Units must be the same Units must be the same Graphical Methods Graphical Methods Use scale drawings Use scale drawings Algebraic Methods Algebraic Methods More convenient More convenient

47. 47 Adding Vectors Graphically (Triangle or Polygon Method) Choose a scale Choose a scale Draw the first vector with the appropriate length and in the direction specified, with respect to a coordinate system Draw the first vector with the appropriate length and in the direction specified, with respect to a coordinate system Draw the next vector with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of vector A and parallel to the coordinate system used for A Draw the next vector with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of vector A and parallel to the coordinate system used for A

48. 48 Graphically Adding Vectors, cont. Continue drawing the vectors “head-to-tail” Continue drawing the vectors “head-to-tail” The resultant is drawn from the origin of A to the end of the last vector The resultant is drawn from the origin of A to the end of the last vector Measure the length of R and its angle Measure the length of R and its angle Use the scale factor to convert length to actual magnitude Use the scale factor to convert length to actual magnitude

49. 49 Graphically Adding Vectors, cont. When you have many vectors, just keep repeating the process until all are included When you have many vectors, just keep repeating the process until all are included The resultant is still drawn from the origin of the first vector to the end of the last vector The resultant is still drawn from the origin of the first vector to the end of the last vector

50. 50 Alternative Graphical Method When you have only two vectors, you may use the Parallelogram Method When you have only two vectors, you may use the Parallelogram Method All vectors, including the resultant, are drawn from a common origin All vectors, including the resultant, are drawn from a common origin The remaining sides of the parallelogram are sketched to determine the diagonal, R The remaining sides of the parallelogram are sketched to determine the diagonal, R

51. 51 Notes about Vector Addition Vectors obey the Commutative Law of Addition Vectors obey the Commutative Law of Addition The order in which the vectors are added doesn’t affect the result The order in which the vectors are added doesn’t affect the result

52. 52 Vector Subtraction Special case of vector addition Special case of vector addition If A – B, then use A+(-B) If A – B, then use A+(-B) Continue with standard vector addition procedure Continue with standard vector addition procedure

53. 53 Multiplying or Dividing a Vector by a Scalar The result of the multiplication or division is a vector The result of the multiplication or division is a vector The magnitude of the vector is multiplied or divided by the scalar The magnitude of the vector is multiplied or divided by the scalar If the scalar is positive, the direction of the result is the same as of the original vector If the scalar is positive, the direction of the result is the same as of the original vector If the scalar is negative, the direction of the result is opposite that of the original vector If the scalar is negative, the direction of the result is opposite that of the original vector

54. 54 Components of a Vector A component is a part A component is a part It is useful to use rectangular components It is useful to use rectangular components These are the projections of the vector along the x- and y-axes These are the projections of the vector along the x- and y-axes

55. 55 Components of a Vector, cont. The x-component of a vector is the projection along the x-axis The x-component of a vector is the projection along the x-axis The y-component of a vector is the projection along the y-axis The y-component of a vector is the projection along the y-axis Then, Then,

56. 56 More About Components of a Vector The previous equations are valid only if θ is measured with respect to the x-axis The previous equations are valid only if θ is measured with respect to the x-axis The components can be positive or negative and will have the same units as the original vector The components can be positive or negative and will have the same units as the original vector The components are the legs of the right triangle whose hypotenuse is A The components are the legs of the right triangle whose hypotenuse is A May still have to find θ with respect to the positive x-axis May still have to find θ with respect to the positive x-axis

57. 57 Adding Vectors Algebraically Choose a coordinate system and sketch the vectors Choose a coordinate system and sketch the vectors Find the x- and y-components of all the vectors Find the x- and y-components of all the vectors Add all the x-components Add all the x-components This gives R x : This gives R x :

58. 58 Adding Vectors Algebraically, cont. Add all the y-components Add all the y-components This gives R y : This gives R y : Use the Pythagorean Theorem to find the magnitude of the Resultant: Use the Pythagorean Theorem to find the magnitude of the Resultant: Use the inverse tangent function to find the direction of R: Use the inverse tangent function to find the direction of R:

59. 59 Motion in Two Dimensions Using + or – signs is not always sufficient to fully describe motion in more than one dimension Using + or – signs is not always sufficient to fully describe motion in more than one dimension Vectors can be used to more fully describe motion Vectors can be used to more fully describe motion Still interested in displacement, velocity, and acceleration Still interested in displacement, velocity, and acceleration

60. 60 Displacement The position of an object is described by its position vector, r The position of an object is described by its position vector, r The displacement of the object is defined as the change in its position The displacement of the object is defined as the change in its position Δr = r f - r i Δr = r f - r i

61. 61 Velocity The average velocity is the ratio of the displacement to the time interval for the displacement The average velocity is the ratio of the displacement to the time interval for the displacement The instantaneous velocity is the limit of the average velocity as Δt approaches zero The instantaneous velocity is the limit of the average velocity as Δt approaches zero The direction of the instantaneous velocity is along a line that is tangent to the path of the particle and in the direction of motion The direction of the instantaneous velocity is along a line that is tangent to the path of the particle and in the direction of motion

62. 62 Acceleration The average acceleration is defined as the rate at which the velocity changes The average acceleration is defined as the rate at which the velocity changes The instantaneous acceleration is the limit of the average acceleration as Δt approaches zero The instantaneous acceleration is the limit of the average acceleration as Δt approaches zero

63. 63 Ways an Object Might Accelerate The magnitude of the velocity (the speed) can change The magnitude of the velocity (the speed) can change The direction of the velocity can change The direction of the velocity can change Even though the magnitude is constant Even though the magnitude is constant Both the magnitude and the direction can change Both the magnitude and the direction can change

64. 64 Projectile Motion An object may move in both the x and y directions simultaneously An object may move in both the x and y directions simultaneously It moves in two dimensions It moves in two dimensions The form of two dimensional motion we will deal with is called projectile motion The form of two dimensional motion we will deal with is called projectile motion

65. 65 Assumptions of Projectile Motion We may ignore air friction We may ignore air friction We may ignore the rotation of the earth We may ignore the rotation of the earth With these assumptions, an object in projectile motion will follow a parabolic path With these assumptions, an object in projectile motion will follow a parabolic path

66. 66 Rules of Projectile Motion The x- and y-directions of motion can be treated independently The x- and y-directions of motion can be treated independently The x-direction is uniform motion The x-direction is uniform motion a x = 0 a x = 0 The y-direction is free fall The y-direction is free fall a y = -g a y = -g The initial velocity can be broken down into its x- and y-components The initial velocity can be broken down into its x- and y-components

67. 67 Projectile Motion

68. 68 Some Details About the Rules x-direction x-direction a x = 0 a x = 0 x = v xo t x = v xo t This is the only operative equation in the x- direction since there is uniform velocity in that direction This is the only operative equation in the x- direction since there is uniform velocity in that direction

69. 69 More Details About the Rules y-direction y-direction free fall problem free fall problem a = -g a = -g take the positive direction as upward take the positive direction as upward uniformly accelerated motion, so the motion equations all hold uniformly accelerated motion, so the motion equations all hold

70. 70 Velocity of the Projectile The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point

71. 71 Some Variations of Projectile Motion An object may be fired horizontally An object may be fired horizontally The initial velocity is all in the x-direction The initial velocity is all in the x-direction v o = v x and v y = 0 v o = v x and v y = 0 All the general rules of projectile motion apply All the general rules of projectile motion apply

72. 72 Non-Symmetrical Projectile Motion Follow the general rules for projectile motion Follow the general rules for projectile motion Break the y-direction into parts Break the y-direction into parts up and down up and down symmetrical back to initial height and then the rest of the height symmetrical back to initial height and then the rest of the height

73. 73 Relative Velocity It may be useful to use a moving frame of reference instead of a stationary one It may be useful to use a moving frame of reference instead of a stationary one It is important to specify the frame of reference, since the motion may be different in different frames of reference It is important to specify the frame of reference, since the motion may be different in different frames of reference There are no specific equations to learn to solve relative velocity problems There are no specific equations to learn to solve relative velocity problems

74. 74 Solving Relative Velocity Problems The pattern of subscripts can be useful in solving relative velocity problems The pattern of subscripts can be useful in solving relative velocity problems Write an equation for the velocity of interest in terms of the velocities you know, matching the pattern of subscripts Write an equation for the velocity of interest in terms of the velocities you know, matching the pattern of subscripts