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EPPT M2 INTRODUCTION TO RELATIVITY

EPPT M2 INTRODUCTION TO RELATIVITY. K Young , Physics Department, CUHK  The Chinese University of Hong Kong. CHAPTER 6 VELOCITY, MOMENTUM and ENERGY. Displacement. Velocity. Momentum, Conservation. Force Newton's second law. Objectives. Momentum Collisions. Momentum. Momentum.

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EPPT M2 INTRODUCTION TO RELATIVITY

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  1. EPPT M2INTRODUCTION TO RELATIVITY K Young, Physics Department, CUHK The Chinese University of Hong Kong

  2. CHAPTER 6VELOCITY, MOMENTUM and ENERGY

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

  4. Objectives • Momentum • Collisions

  5. Momentum

  6. Momentum • 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

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

  12. Example Particle is travelling at 0.6c

  13. is just ordinary velocity • carries no information Case of low velocities

  14. Time component carries no extra information • True in general

  15. etc. Three spatial components

  16. Four-Momentum

  17. Momentum = mass velocity • Now is more convenient

  18. Explicit expression

  19. If , = ordinary expression • If • as • Recover Newtonian physics

  20. px p = mv v v = c Do not call this effective mass M! Spatial component

  21. Time component

  22. Assuming mass does not change

  23. Apart from additive constant, which does not matter

  24. E E0 = m c2 v v = c • Provided m 0, takes E = to reach v = c • Therefore can never attain v = c

  25. Faster than light? There 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!

  26. Kinetic energy

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

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

  29. Analogy

  30. Relation between E and p Newtonian Relativistic

  31. System of units E :eV MeV GeV pc : eV MeV GeV p : eV/c MeV/c GeV/c mc2 : eV MeV GeV m : eV/c2 MeV/c2 GeV/c2

  32. Particle Mass (MeV/ c2) electron 0.5110 muon 105.7 proton 938.3 neutron 939.6

  33. Conservation of four -momentum

  34. The four-momentum • Recall • Contains energy + momentum

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

  36. Collisions

  37. 1/3 0 u v 2 2 1 1 Example

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

  39. known

  40. known Example Production of p at threshold

  41. e Q q Z P = 150 GeV M = 90 GeV Q e+ Example

  42. Energy in the CM frame

  43. In 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

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

  45. Principle of Relativity

  46. Conservation

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