Linear motion of a particle

This PPT covers linear motion of an object in a very systematic and lucid manner. I hope this PPT will be helpful for instructor's as well as students.

1. Linear motion

2. Linear motion The motion of an object along a straight path is called a linear motion.

3. A train running on a straight railway track

4. Vehicles moving on a straight highway

5. Sprinters running on a straight athletic track

6. Motion of a bowling ball

7. Motion diagrams Frame-1

8. Motion diagrams Frame-2

9. Motion diagrams Frame-3

10. Motion diagrams Frame-4

11. Motion diagram of a car moving towards a tree Motion diagrams

12. Motion diagrams A racing car running on a track Images that are equally spaced indicate an object moving with constant speed.

13. Motion diagrams A cyclist starting a race An increasing distance between the images s hows that the object is speeding up.

14. Motion diagrams A car stopping for a red light A decreasing distance between the images s hows that the object is slowing down.

15. Particle model 1 Motion diagram of a racing car moving with constant speed 2 3 4 Same motion d iagram using the particle model

16. Position The location of an object at a particular instant is called its position.

17. Position Your school Your position E W (reference point) (direction) 4 km (distance)

18. Coordinate system The school defines the origin of the coordinate system 1 -1 2 3 4 Your position is at +4 km (km)

19. Case-I: Position is zero The car is at origin   -20 -10 10 2 30 4 50 6 (m) x

20. Case-II: Position is positive The car is at positive side of origin   -20 -10 10 2 30 4 50 6   (m) x

21. Case-III: Position is negative   The car is at negative side of origin -50 -40 -30 -20 -10 10 2 30   (m) x

22. The displacement of an object is the change in the position of an object in a particular direction.   Displacement

23. Case-I: Displacement is positive     If object is moving towards right, the displacement is positive because   -20 -10 10 2 30 4 50 6       A B (m) x

24. Case-II: Displacement is negative -20 -10 10 2 30 4 50 6     B A     If object is moving towards left, the displacement is negative because   (m) x  

25. Case-III: Displacement is zero     If object come to its initial position, the displacement is zero because   -20 -10 10 2 30 4 50 6     A B (m) x

26. Distance The actual path covered by an object is called a distance.  

27. Distance                 -20 -10 10 2 30 4 50 6 A B C (m) x  

28. The rate of change of position of an object in a given direction is called its velocity. V elocity   Vector quantity (magnitude & direction). SI unit is metre /second (m/s). Dimensional formula is [ ].  

29. Case-I: V elocity is positive Velocity is positive because change in position is positive. 1 20 30 4 5 6 7 8 (m) x s         1 s 2 s 3 s 4 s    

30. Case-II: V elocity is negative 1 20 30 4 5 6 7 8 (m) x s 1 s 2 s 3 s 4 s         Velocity is negative because change in position is negative.    

31. Case-III: V elocity is zero 1 20 30 4 5 6 7 8 (m) x 0 s, 1 s, 2 s, 3 s, 4s   Velocity is zero because change in position is zero.  

32. Speed The rate at which an object covers a distance is called a speed.   Scalar quantity (only magnitude). SI unit is metre /second (m/s). Dimensional formula is [ ].  

33. 1 20 30 4 5 6 7 8 (m) x 0 s   Case-I: Speed is positive 1 s 2 s 3 s 4 s   If an object moves it cover some distance and h ence the speed is positive.      

34. Case-II: Speed is zero 1 20 30 4 5 6 7 8 (m) x 0 s, 1 s, 2 s, 3 s, 4s   If an object is at rest the distance covered is zero and hence the speed is zero.  

35. Average velocity 1 20 30 4 5 6 7 8 (m) x A           2 s 4 s 3 s B C D E s 1 s D      

36. Average velocity For an object moving with variable velocity, average velocity is defined as the ratio of its total displacement to the total time interval in which that displacement occurs.  

37. Average velocity     1 20 30 4 5 6 7 8 (m) x A   2 s 4 s 3 s B C D E s 1 s D      

38. Average velocity If and are the positions of an object at times and , then the average velocity from time to is given by    

39. Average velocity     1 20 30 4 5 6 7 8 (m) x A   2 s 4 s 3 s B C D E s 1 s D      

40. Average speed         1 20 30 4 5 6 7 8 (m) x   A s B 1 s 2 s C D 3 s D 4 s E      

41. For an object moving with variable speed, the average speed is the total distance travelled by the object divided by the total time taken to cover that distance.   Average speed

42. Average speed     1 20 30 4 5 6 7 8 (m) x   A s B 1 s 2 s C D 3 s D 4 s E      

43. Instantaneous velocity The velocity of an object at a particular instant of time or at a particular point of its path is called its instantaneous velocity. 1 20 30 4 5 6 7 8 (m) x 0 s 1 s         A B 2 s C D 3 s

44. Instantaneous velocity x             x                    

45. Instantaneous velocity x               x        

46. Instantaneous velocity Instantaneous velocity is equal to the limiting value of the average velocity of the object in a small time interval taken around that instant, when time interval approaches zero.  

47. Instantaneous speed The magnitude of instantaneous velocity is called instantaneous speed. 1 20 30 4 5 6 7 8 (m) x 0 s         1 s 2 s 3 s A B C D

48. Acceleration The rate of change of velocity of an object with time is called its acceleration.   Vector quantity (magnitude & direction). SI unit is metre /second/second ( m/ ). Dimensional formula is [ ].  

49. Case-I: Velocity is constant   Acceleration is zero because change in velocity is zero.         0 s 1 s 2 s 3 s x            

50.   Case-II: Velocity is increasing towards right Acceleration is positive because change in velocity is positive.           0 s 1 s 2 s 3 s   x        

51. Case-III: Velocity is decreasing towards right   Acceleration is negative because change in velocity is negative.         0 s 1 s 2 s 3 s x            

52.   Case-IV: Velocity is increasing towards left Acceleration is negative because change in velocity is negative.         0 s 3 s 2 s 1 s x            

53.   Case-V: Velocity is increasing towards left Acceleration is positive because change in velocity is positive.         3 s s 1 s 2 s x            

54. Important note If the signs of the velocity and acceleration of an object are the same, the speed of the object increases. If the signs are opposite, the speed decreases.

55. Average acceleration                 0 s 8 s 2 s x       4 s 6 s     A B C D E

56. Average acceleration For an object moving with variable velocity, the average acceleration is defined as the ratio of the total change in velocity of the object to the total time interval taken.  

57. Average acceleration             0 s 8 s 2 s x       4 s 6 s     A B C D E

58. If and are the velocities of an object at times and , then the average acceleration from time to is given by     Average acceleration

59. Average acceleration             0 s 8 s 2 s x       4 s 6 s     A B C D E

60. Instantaneous acceleration The acceleration of an object at a particular instant of time or at a particular point of its path is called its instantaneous acceleration. 1 20 30 4 5 6 7 8 (m) x 0 s 1 s 2 s 3 s         A B C D

61. Instantaneous acceleration t             t                    

62. Instantaneous acceleration t               t        

63. Instantaneous acceleration Instantaneous acceleration is equal to the limiting value of the average acceleration of the object in a small time interval taken around that instant, when time interval approaches zero.  

64. Uniformly accelerated motion The motion in which the velocity of an object changes with uniform rate or constant rate is called uniformly accelerated motion.

65. Uniformly accelerated motion A car slowing down after a red signal A car speeding up after a green signal

66. Kinematical equations of motion         x     Velocity after a certain time:   By definition,       This is the first kinematical equation.

67. Kinematical equations of motion Displacement in a certain time: By definition,     But,           T his is the second kinematical equation.  

68. Kinematical equations of motion Velocity after certain displacement: We know that,       Also,       multiplying equations (1) & (2), we get       T his is the third kinematical equation.    

69. Equations of motion by calculus method First equation of motion: By definition,       When time = 0, velocity =   When time = t, velocity =     Integrating equation (1) within the above limits of time & velocity, we get          

70. Equations of motion by calculus method S econd equation of motion: By definition,       When time = 0, position =   When time = t, position =     Integrating equation (1) within the above limits of time & position , we get            

71. Equations of motion by calculus method S econd equation of motion: By definition,       When time = 0, velocity = , position =   Integrating equation (1) within the above limits of velocity & position, we get       When time = t, velocity = , position =          

72. Graphical r epresentation o f motion

73. Position-time graph Time (s) Position (m) 30 5 55 10 40 15 20 -35 25 -55 5 10 15 20 25 30 t -40 -20 2 4 6 -60 x

74. Average velocity from position-time graph 5 10 15 20 25 30 t -40 -20 2 4 6 -60 x     Slope of line AB gives average velocity between points A & B.         A B (5,55) (25,-55)  

75. Instantaneous velocity from position-time graph 5 10 15 20 25 30 t -40 -20 2 4 6 -60     A B (5,5) (25,-55)   Slope of tangent at point B gives instantaneous velocity at point B.        

76. Analyzing nature of motion through position–time graph t x A B         The particle is at rest on positive side of origin.

77. t x       Analyzing nature of motion through position–time graph A B C       The particle is at negative side of origin and moving with constant velocity towards origin. For portion AB: For portion BC:     The particle is at origin and moving with constant velocity towards right side of origin.    

78. Analyzing nature of motion through position–time graph t x             The particle is at positive side of origin and moving with constant velocity towards origin. For portion AB: For portion BC:     The particle is at origin and moving with constant velocity towards left side of origin. A B C    

79. Analyzing nature of motion through position–time graph t x           The particle is at negative side of origin and moving with increasing velocity towards origin. For portion AB: For portion BC:     The particle is at origin and moving with increasing velocity towards right side of origin.       A B C    

80. Analyzing nature of motion through position–time graph t x       A B C     The particle is at negative side of origin and moving with decreasing velocity towards origin. For portion AB: For portion BC:     The particle is at origin and moving with decreasing velocity towards right side of origin.          

81. Analyzing nature of motion through position–time graph t x           The particle is at positive side of origin and moving with increasing velocity towards origin. For portion AB: For portion BC:     The particle is at origin and moving with increasing velocity towards left side of origin.     A B C      

82. Analyzing nature of motion through position–time graph t x           The particle is at positive side of origin and moving with decreasing velocity towards origin. For portion AB: For portion BC:     The particle is at origin and moving with decreasing velocity towards left side of origin.     A B C      

83. Velocity-time graph Time (s) velocity (m/s) -40 5 10 20 15 10 20 -20 25 -40 5 10 15 20 25 30 t -40 -20 2 4 6 -60 v

84. 5 10 15 20 25 30 t -40 -20 2 4 6 -60 Average acceleration from velocity-time graph     Slope of line AB gives average acceleration between points A & B.     v     B (25,-40) A (5,0)  

85. 5 10 15 20 25 30 t -40 -20 2 4 6 -60 Instantaneous acceleration from velocity-time graph     Slope of line AB gives instantaneous acceleration between points A & B.     v     B (25,-40) A (5,7)  

86. Analyzing nature of motion through velocity–time graph t v A B       The particle is moving with constant velocity towards right.    

87. Analyzing nature of motion through velocity–time graph t v       A B C     The particle is moving towards left and its velocity is uniformly decreasing. For portion AB: For portion BC:             The particle is moving towards right and its velocity is uniformly increasing.              

88. Analyzing nature of motion through velocity–time graph t v         A B C   The particle is moving towards right and its velocity is uniformly decreasing. For portion AB: For portion BC:             The particle is moving towards left and its velocity is uniformly increasing.              

89. t v         A B C Analyzing nature of motion through velocity–time graph   The particle is moving towards left and its velocity is decreasing slowly at non-uniform rate. For portion AB: For portion BC:                           The particle is moving towards right and its velocity is increasing rapidly at non-uniform rate.

90. Analyzing nature of motion through velocity–time graph t v       A B C     For portion AB: For portion BC:                           The particle is moving towards left and its velocity is decreasing rapidly at non-uniform rate. The particle is moving towards right and its velocity is increasing slowly at non-uniform rate.

91. Analyzing nature of motion through position–time graph t v       A B C     For portion AB: For portion BC:                           The particle is moving towards right and its velocity is decreasing slowly at non-uniform rate. The particle is moving towards left and its velocity is increasing rapidly at non-uniform rate.

92. t v       A B C Analyzing nature of motion through position–time graph   For portion AB: For portion BC:                           The particle is moving towards right and its velocity is decreasing rapidly at non-uniform rate. The particle is moving towards left and its velocity is increasing slowly at non-uniform rate.

93. Acceleration–time graph Time (s) acceleration (m/ ) 20 5 50 10 30 15 20 -30 25 -40 Time (s) 20 5 50 10 30 15 20 -30 25 -40 5 10 15 20 25 30 t -40 -20 2 4 6 -60 a

94. Analyzing nature of motion through acceleration–time graph t a A B     The particle is moving with constant acceleration.    

95. Analyzing nature of motion through acceleration–time graph t a       A B C     The acceleration of particle is uniformly decreasing towards left. For portion AB: For portion BC:         The acceleration of particle is uniformly increasing towards right.      

96. Analyzing nature of motion through acceleration–time graph t a         A B C   The acceleration of particle is uniformly decreasing towards right. For portion AB: For portion BC:         The acceleration of particle is uniformly increasing towards left.      

97. Graphical i ntegration in motion analysis

98. t a                      

99. t v                      

100. Thank you