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Topic #10: Special Relativity II

Topic #10: Special Relativity II. Relativistic momentum and energy p = g mu and E = g mc 2 E = mc 2 + K (simple) and E 2 = (mc 2 ) 2 + (pc) 2 (square) Conserved Quantities Total energy E and total momentum p are conserved in any reference frame.

anthony-maynard
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Topic #10: Special Relativity II

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  1. Topic #10: Special Relativity II • Relativistic momentum and energy • p = gmu and E = gmc2 • E = mc2 + K (simple) and E2 = (mc2)2 + (pc)2 (square) • Conserved Quantities • Totalenergy E and total momentum p are conserved in any reference frame. • Total rest mass mc2 is not conserved (cannot be obtained from adding up individual rest masses!). Note: Exception when all particles are at rest w/respect to each other. • Invariant Quantities • Totalrest mass mc2 and time-space interval Ds are invariant between different reference frames. • Lorentz Transform for E, p Relativity II

  2. Relativistic Momentum • Relativistic momentum must: • Be consistent with momentum conservation. • Reduce to classicalexpression for u << c. Relativity II

  3. Relativistic Energy: Definition &“Simple” Equation Total energy E =Kinetic energy K+Rest energy mc2 Relativity II

  4. Invariant “Square” Equations • Invariant quantities are equal in all inertial reference frames • All observers will measure the same value for the rest mass mc2and for thespace-time interval s. • Invariant quantities are called “4-vectors” • Rest mass mc2 is given by four quantities: E, px, py, pz • Space-time interval s is given by four quantities: t, x, y, z Rest Mass Eo: Space-Time Interval: Relativity II

  5. Invariant Equation: Classical vs. Relativistic Very Relativistic Limit Classical Limit E = g mc2 pc pc E = g mc2 mc2 mc2 E >> pc gives E  mc2 (g 1)  K  ½ mu2 or p2/2m E >> mc2 givesE  pc Accurate to 1% or better if E > 8mc2 If K/mc2  1%, then K approximation is accurate to 1.5%. Relativity II

  6. 6 Variables given by 5 Eqns. (3 def., simple, square) Requires mc2 Requires mc2 Requires mc2  Relativity II

  7. Problem: Find E, p, and K (given u, mc2 ) Find the total energy E, momentum p (MeV/c), and kinetic energy K for an electron with rest mass 0.511 MeV and speed u = 0.5c. u gives g Relativity II

  8. Problem: Find pc and u (given E, mc2 ) Find the momentum pc (MeV) and speed u of an electron with rest mass  0.511 MeV and total energy E = 10 MeV. Use Square Eqn. Relativity II

  9. Problem: Find mc2 and u (given p, E) Find the rest mass and speedu of a particle with momentum pc = 300 MeV and total energy E = 3500 MeV. Use Square Eqn. Relativity II

  10. Problem: Find E, pc and u(given K, mc2 ) A 2-GeVproton hits another 2-GeV proton in a head-on collision. Find the total energy E, momentum pc, and velocity u of each proton. Use Simple AND Square Eqns. EG , Baski Relativity II

  11. Problem: Find mc2 and K A S particle decays into a neutron (pc = 4702 MeV) and pion (pc = 169 MeV). Find the total rest mass and kinetic energy of the S particle. EG , Baski Relativity II

  12. Rest Mass: Basics • Internal energy of the system appears as an increase in rest mass of the system. • The mass of a bound system is less than that of the separate particles by Eb/c2, where Eb is the binding energy. • Example Rest Mass Values • Photon = 0 MeV, Electron = 0.511 MeV, Proton = 938.28 MeV • Total rest mass Eo of a system is NOT CONSERVED, i.e. does not equal sum of individual rest masses! • Total rest mass is INVARIANT, i.e. has same value in different reference frames. Relativity II

  13. Rest Mass: Example Problem per second /s Compute the rate at which the sun is losing mass, given that the mean radius R of the Earth's orbit is R = 1.50×108 km, and the intensity of solar radiation at the Earth (solar constant) is 1.36×103 W/m2Assuming that the sun radiates uniformly over a sphere of radius R, the total power P radiated by the sun is given by: Therefore, the sun is losing 4.3x109 kg of mass every second! At this rate, the sun would run out of fuel after ~ 1011 years. Relativity II

  14. Rest Mass: Homework Problem Two identical particles with rest mass 12 kg approach each other with equal but opposite velocities u1 = –u2 = 0.6c. Find the total rest mass of this system. Relativity II

  15. Rest Mass: Homework Problem An electron is moving left with velocity0.995cand a positron is moving right at velocity 0.9798c. Draw a diagram and find the total rest mass of this system. Relativity II

  16. Lorentz Transform of E,p Derivation from: Relativity II

  17. Lorentz Transform: Homework Problem Two electrons with each total energy E = 30 GeV approach each other with equal but opposite velocities. Find the momentum p of each electron before the collision in the lab frame. Also, find the energy E’ and momentum p’ of the left electron in the reference frameof the right electron. Note: Because u~c, use approximation E ~ pc. Relativity II

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