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l5 - Review Conservation of momentum Photon mass Invariants...

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Unformatted text preview: Review Conservation of momentum Photon mass Invariants Collisions LHC Review: momentum and forces • We found that relativistic momentum goes like p 1 = γ m v 2 • For perpendicular forces ( vector F ⊥ vector v ) we found vector F = γ m vector a • The centripetal force for uniform circular motion is F = γ mv 2 / r . • For parallel forces ( vector F bardbl vector v ) we found vector F = γ 3 m vector a Review Conservation of momentum Photon mass Invariants Collisions LHC Review: relativistic energy • Relativistic kinetic energy is E k = ( γ − 1 ) m c 2 . • Classical limit reduces to E k = 1 2 mv 2 • Suggests total energy of E tot = E + E k = m c 2 + ( γ − 1 ) m c 2 = γ m c 2 (1) • Common unit for energies in modern physics: electron · Volt, where 1 eV=1 . 602 × 10- 19 Joules. Chemical bonds are typically 3–6 eV. • Can describe particle masses in energy units. • Electron: m e = 511 × 10 3 eV/ c 2 or just 511 keV. • Proton: m p = 939 × 10 6 eV/ c 2 or just 939 MeV. Review Conservation of momentum Photon mass Invariants Collisions LHC Relativistic conservation of momentum • Classically, kinetic energy is p 2 / 2 m . Consider p 2 in relativity: ( pc ) 2 = ( γ m vc ) 2 = ( γβ m c 2 ) 2 . (2) • If we then use E = m c 2 and β 2 = 1 − 1 γ 2 , we obtain p 2 c 2 = γ 2 ( 1 − 1 γ 2 ) E 2 = ( γ 2 − 1 ) E 2 = γ 2 E 2 − E 2 . (3) However, since we found before that the total energy is E = γ E , we have p 2 c 2 = E 2 − E 2 or E 2 = E 2 + p 2 c 2 . (4) • Therefore if E k ≫ E we have E k ,relativistic ≃ pc . Review...
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l5 - Review Conservation of momentum Photon mass Invariants...

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