1. Relativistic Mechanics Relativistic Mass and Momentum

2. Relativistic Mechanics In classical mechanics, momentum is always conserved.

3. Relativistic Mechanics In classical mechanics, momentum is always conserved. However this is not true under a Lorentz transform.

4. Relativistic Mechanics If momentum is conserved for an observer in a reference frame

5. Relativistic Mechanics If momentum is conserved for an observer in a reference frame, it is not conserved for another observer in another inertial frame.

6. Relativistic Mass After V=0 Stationary Frame Before v 1 v 2

7. Relativistic Mass Moving Frame of mass 1 Before 1 v′xv′x 2 After V′V′

8. Relativistic Mass In the stationary frame, However in the moving frame, P final > P initial. P i = mv – mv = 0 (assume the masses are equal

9. Relativistic Momentum For momentum to be conserved the expression for the momentum must be rewritten as follows: (Relativistic Momentum)

10. Relativistic Mass Where is the rest mass.

11. Relativistic Mass The expression is produced by assuming that mass is not absolute. Instead the mass of an object varies with velocity i.e..

12. Relativistic Mass The expression for the relativistic mass is (Relativistic mass)

13. Relativistic Mass From the expression for the momentum, the relativistic force is defined as,

14. Stationary Frame P i = mv – mv = 0 (assume the masses are equal) P f = (m1+m2)x0 = 0 P i = P f. Therefore Momentum is conserved. After V=0 Before v 1 v 2

15. Moving Frame of mass1 Before 1 v′xv′x 2 After V′V′

16. Moving Frame of mass1 Before 1 v′xv′x 2 After V′V′

17. Moving Frame of mass1 Before 1 v′xv′x 2 After V′V′

18. Moving Frame of mass1 Before 1 v′xv′x 2 After V′V′

19. Moving Frame of mass1 Before 1 v′xv′x 2 After V′V′

20. Moving Frame of mass1 Before 1 v′xv′x 2 After V′V′

21. Moving Frame of mass1 Hence momentum is not conserved.

22. Moving Frame of mass1 Now consider what happens when the modification to the mass is introduced.

23. Moving Frame of mass1 Recall the equations for the final and initial momentum. where

24. Moving Frame of mass1 Consider the initial momentum wherereally corresponds to the mass NB:Since it is at rest in the moving frame

25. Moving Frame of mass1 Consider the initial momentum (Relativistic mass)

26. Moving Frame of mass1 where

27. Moving Frame of mass1 where

28. Moving Frame of mass1 where

29. Moving Frame of mass1 where simplify

30. Moving Frame of mass1 where simplify

31. Moving Frame of mass1 Therefore the initial momentum

32. Moving Frame of mass1 Therefore the initial momentum

33. Moving Frame of mass1 Therefore the initial momentum

34. Moving Frame of mass1 Now consider the final momentum

35. Relativistic Momentum Hence momentum is conserved.

36. Relativistic Momentum Consider the difference in the relativistic momentum and classical momentum.

37. Relativistic Momentum

38. Relativistic Energy

39. To derive an expression for the relativistic energy we start with the work-energy theorem

40. Relativistic Energy To derive an expression for the relativistic energy we start with the work-energy theorem (Work done)

41. Relativistic Energy Using the chain rule repeatedly this can rewritten as

42. Relativistic Energy Using the chain rule repeatedly this can rewritten as

43. Relativistic Energy Using the chain rule repeatedly this can rewritten as (Work done accelerating an object from rest some velocity)

44. Relativistic Energy

46. Integrating by substitution:

47. Relativistic Energy Integrating by substitution:

48. Relativistic Energy Integrating by substitution:

49. Relativistic Energy

54. Therefore

55. Relativistic Energy Therefore However this equal to

56. Relativistic Energy Therefore However this equal to therefore the Relativistic Kinetic energy is

57. Relativistic Energy The velocity independent term ( ) is the rest energy – the energy an object contains when it is at rest.

58. Relativistic Energy (Correspondence Principle) At low speeds we can write the kinetic energy as

59. Relativistic Energy (Correspondence Principle) At low speeds we can write the kinetic energy as Using a Taylor expansion we get,

60. Relativistic Energy (Correspondence Principle) At low speeds we can write the kinetic energy as Using a Taylor expansion we get,

61. Relativistic Energy (Correspondence Principle) Taking the first 2 terms of the expansion

62. Relativistic Energy (Correspondence Principle) Taking the first 2 terms of the expansion

63. Relativistic Energy (Correspondence Principle) Taking the first 2 terms of the expansion So that (classical expression for the Kinetic energy)

64. Relativistic Energy Note that even an infinite amount of energy is not enough to achieve c.

65. Relativistic Energy The expression for the relativistic kinetic energy is often written as

66. Relativistic Energy The expression for the relativistic kinetic energy is often written as or equivalent as

67. Relativistic Energy The expression for the relativistic kinetic energy is often written as or equivalent as whereis the total relativistic energy

68. Relativistic Energy So that

69. Relativistic Energy So that If the object also has potential energy it can be shown that

70. Relativistic Energy The relationship is just Einstein’s mass-energy equivalence equation, which shows that mass is a form of energy.

71. Relativistic Energy  An important fact of this relationship is that the relativistic mass is a direct measure of the total energy content of an object.

72. Relativistic Energy  An important fact of this relationship is that the relativistic mass is a direct measure of the total energy content of an object.  It shows that a small mass corresponds to an enormous amount of energy.

73. Relativistic Energy  An important fact of this relationship is that the relativistic mass is a direct measure of the total energy content of an object.  It shows that a small mass corresponds to an enormous amount of energy. This is the foundation of nuclear physics.

74. Relativistic Energy Anything that increases the energy in an object will increase its relativistic mass.

75. Relativistic Energy In certain cases the momentum or energy is known instead of the speed. The relationship involving the momentum is derived as follows:

76. Relativistic Energy In certain cases the momentum or energy is known instead of the speed.

77. Relativistic Energy Note that

78. Relativistic Energy Note that

79. Relativistic Energy Note that

80. Relativistic Energy Note that Since

81. Relativistic Energy Note that Since

82. Relativistic Energy When the object is at rest, so that

83. Relativistic Energy When the object is at rest, so that For particles having zero mass (photons) we see that the expected relationship relating energy and momentum for photons and neutrinos which travel at the speed of light.