1Nearing the End• All homework solutions are now available• Check CULearn for all scores (HW, Clickers, Exams)– Any requested changes due by this Friday• Final exam is Tuesday, December 15, 2009 4:30 pm – 7:00 pm in G125 (this room)• Additional review help (in the Physics Help Room), Thursday 10:00 am – noonFriday 1:30 – 4:00 pm• I have to travel to Brookhaven National Lab next week and so the exam will be proctored.Final Exam• There are 15 multiple choice questions– scored as right or wrong• There are 5 long answer questions– each has multiple parts– must show all work to receive full credit• Formula sheet for final exam is posted• Final exam is comprehensive• BRING A CALCULATOR!!!Special RelativityPostulates of relativity:1. All physical laws are the same in all inertial reference frames2. The speed of light is the same in all inertial reference framesProper time Δt0between two events is the time measured in the frame in which both events occur at the same location.Time dilation (moving clocks run slower): Δt=γΔt0Length contraction (moving objects are shorter in the direction they are traveling): L = L0/γRelativity of simultaneity: events which are simultaneous in oneframe may not be in another; even the order may be switched.Proper length L0of an object is the length measured in the rest frame of the object.Clicker question Jimmy sees two lightning flashes and determines that one happens 10 ns before the other. Which statement is true:A. Another inertial observer must find the order of events, the location of events, and the time difference to be the same.B. Another inertial observer must find the order of events to be the same but the location and time difference may be differentC. Another inertial observer must find the order of events and the locations to be the same but the time difference may be different.D. Any other inertial observer might find the order of events, the locations, and the time difference to be different.E. Say what?In general, length, time, and order can be different for different inertial observers. However, due to causality and finite speed of light, if the events are far enough apart it is impossible for the order to be reversed.Special RelativityLorentz transformations are used to transform coordinates (position and time) of an event from a rest frame S to a moving frame S′.)/(' ' ' )(2cvxttzzyyvtxx −===−=′γγVelocity addition formula is also different than the Galilean velocity addition formula:2/1 cuvvuu−−=′Spacetime diagrams can be used to understand where & when events occur and how they are related in the space-time continuum.The inverse Lorentz transformations go the other way (S′ to S) and can be obtained by swapping primes and signs.xctFuture(forward light cone)ElsewherePast (backward light cone)Solving for v:To getClicker questionTwo events take place 90 m apart with an intervening time interval of 0.60 μs in reference frame S. What is the speed of the reference frame S′ which measures the proper time between the two events?A. 0B. 0.25cC. 0.50cD. 0.90cE. cProper time Δt0between two events is the time measured in the frame in which both events occur at the same location.So in the S′ frame, both events must occur at point 0. That is, x′ = 0 when x = 90 m and t = 0.60 μs.)()(2xcvttzzyyvtxx−=′=′=′−=′γγ0)(=−=′vtxxγrequires x – vt = 0, i.e. x = vt.ctxv 5.0sm/ 150s .6m 90====μμ2Clicker questionA person on Earth sees two spaceships heading toward each other.The earthling sees rocket A moving at 0.8c. In the Earth frame, how fast are the two spaceships moving toward each other?A. 0B. 0.6cC. 0.8cD. 0.98cE. 1.6c2/1 cvuvuu′++′=2/1 cuvvuu−−=′The person on Earth sees that they are closing at 1.6c. This does not mean that they see anything moving faster than the speed of light. It is just that they are approaching each other at that speed.Clicker questionA person on Earth sees two spaceships heading toward each other.The earthling measures ship A moving at 0.6c and ship B at 0.8c. In the reference frame of ship A, how fast is ship B moving?A. 0B. 0.6cC. 0.8cD. 0.95cE. 1.6c2/1 cuvvuu−−=′2/1 cvuvuu′++′=u is the speed of something in the S frame.u′ is the speed of the same thing in the S′ frame.v is the relative speed between the S and S′ frames.We know u: the speed of rocket B in the Earth frame (S frame)We know v: the relative speed between the S frame (Earth) and the S′ frame (rocket A)So we need u′: the speed of rocket B in the S′ framecccvuvuu 95.048.014.1/12=+=′++′=Energy-momentum relationship:Define kinetic energy:Special Relativity – Chapter 2Conservation of momentum and energy continues to hold for isolated systems if we redefine momentum and energy as:mupuγ=2mcEuγ=for mass m with velocity u and 22/1/1 cuu−=γ222rest)1( mcmcmcEEKEuu−=−=−=γγThis gives a rest energy of2restmcE =2222)()( mcpcE +=For a massless particle like a photon, this reduces topcE =One other equation isEpc /=βThis is it for equations in Chapter 2. Usually use conservationof energy and conservation of momentum to solve problems.Clicker question A Λ particle with mass 1116 MeV/c2decays at rest to a proton with mass 938 MeV/c2and a pion with mass 140 MeV/c2. What can we say about the proton and pion momentum and energy?A. The proton and pion have the same magnitude momentum and energy.B. The proton and pion have the same energy but the proton has more momentum.C. The proton and pion have the same momentum but the proton has more energy.D. The proton and pion have equal and opposite momenta and the proton has more energy.E. Need more informationConservation of momentum requires the proton and pionmomentum add to 0 so must be equal and opposite. Since E2= (pc)2+ (mc2)2, the proton has more energy due to larger mass.2222)()( mcpcE +=2restmcE =mupuγ=2mcEuγ=2rest)1( mcEEKEu−=−=γEpc /=βPhotoelectric effectElectronsTest metalObservations didn’t match theory:1. Minimum frequency needed to get current no matter the intensity2. Current depended on frequency as well as intensity.1905: Einstein’s solution: photons have energy hf and only one photon can interact with electron at a time. Minimum frequency leads to minimum energy to eject the electron from the metal (overcomework function). Energy above minimum goes into electron KE.Conservation of energy: Ephoton= E to escape metal + electron KEφ−= hfKEmaxA. 230 nmB. 400 nmC. 430