1. EPPT M2 INTRODUCTION TO RELATIVITY K Young, Physics Department, CUHK  The Chinese University of Hong Kong

2. CHAPTER 6 VELOCITY, MOMENTUM and ENERGY

3. Displacement Velocity Momentum, Conservation Force Newton's second law

4. Objectives  Momentum  Collisions

5. Momentum

6.  Newtonian momentum is wrong  Should transform as 4-vector  Form of p and E

7. Four-velocity

8.  Coordinates = ( time, space )  Displacement = change of postion

9. Example  P travels to a star 5 ly away  At a speed 0.5c

10.  Four velocity  Displacement per unit proper time

14. Example Particle is travelling at 300 m s -1

15. Example Particle is travelling at 0.6c

16. Case of low velocities  is just ordinary velocity   carries no information

17. Time component carries no extra information  True in general

18. Three spatial components etc.

19. Four-Momentum

20.  Momentum = mass velocity  Now is more convenient

21. Explicit expression

22.  If, = ordinary expression  Recover Newtonian physics  If  as

23. Spatial component v = c p = m v v pxpx Do not call this effective mass M!

24. Time component

25. Assuming mass does not change

26. Apart from additive constant, which does not matter

28.  Provided m  0, takes E =  to reach v = c  Therefore can never attain v = c v E v = c E 0 = m c 2

29. T here was a young fellow named Bright Who travelled much faster than light. He set off one day, in a relative way And come back the previous night! Faster than light?

30. Kinetic energy

31. Application to collisions "Classical" collisions / Elastic collisions

32. Nuclei / Elementary particles Mass is "converted" to energy

33. Analogy

34. Relation between E and p Newtonian Relativistic

35. System of units E :eV MeV GeV pc:eVMeV GeV p:eV/cMeV/cGeV/c mc 2 :eVMeVGeV m:eV/c 2 MeV/c 2 GeV/c 2

36. ParticleMass (MeV/ c 2 ) electron0.5110 muon105.7 proton938.3 neutron939.6

37. Conservation of four -momentum

38. The four-momentum  Recall  Contains energy + momentum

39. Conservation law For an isolated system, the total 4 – momentum is conserved.

40. Collisions

41. Example 1 1/3 2 0 2 v 1 u

42. Better to analyze in terms of p  p, E directly measured and quoted  v = 0.999… inconvenient  formulas apply to massless particles (photons)

43. known

44. Example Production of p at threshold

45. Example P = 150 GeV M = 90 GeV  Q Q Z Z  e+e+ ee

46. Energy in the CM frame

47. I n a collision, much of the energy of the projectile is used to carry the whole system forward; only a small fraction is used to produce new particles

48. Example ME Both of mass M E * in CM = ? Fixed target experiments are inefficient Colliding beams much better

49. Principle of Relativity

50. Conservation

51. Linear transformation Principle of Relativity

52. Objectives  Momentum  Collisions

53. Acknowledgment  I thank Miss HY Shik and Mr HT Fung for design