1Phy107, Lect 20 1Fri. Mar. 9, 2006Hour Exam 2: Wednesday, March 14• In-class, covering waves, electromagnetism, and relativity• Twenty multiple-choice questions• Will cover: Chap. 8, 9.1-9.3, 10 (NOT chap. 11)All lecture material• You should bring– 1 page notes, written double sided– #2 Pencil and a Calculator– Review Monday March 12– Review questions online under “Review Quizzes” linkHW#6 at WileyPlus due Tues midnite: covers relativityHomework & ExamPhy107, Lect 20 2Fri. Mar. 9, 2006From last time… Relativity!Proper time: tpTime interval measured in frame where events occur atsame spatial locationTime dilation: t= γtpTime intervals in other frames longer by a factor γ (gamma always bigger than 1)Length contraction: L= Lp/γTime intervals in other frames longer by a factor γ (gamma always bigger than 1)Conclusion: separate time and space separations not best wayto think about the world we live in.Phy107, Lect 20 3Fri. Mar. 9, 2006An analogy• Imagine you knowonly two dimensions.• You see a 3D objectfor the first time• Distance betweencorners (length ofred line drawn on theflat page) seems tobe differentdepending on howyou look at it.• Clearly just because we don’t understand full three-dimensional distance between points.• The 3D distance does not change with viewpoint.Phy107, Lect 20 4Fri. Mar. 9, 2006The real ‘distance’ between events• Relativity has an analog of this• It is a ‘four-dimensional’ distance• Separation in both space and time.• In 4D: 3 space + 1 time• The same or ‘invariant’ in any inertial frame! x2" c2t2# separation( )2" c2time interval( )2! x2+ y2+ z2" c2t2Phy107, Lect 20 5Fri. Mar. 9, 2006Time interval = 1.413 yrsTime interval = 4.526 yrsEvent separation = 0 LYEvent separation = 4.3 LYShip FrameEarth FrameA relativistic invariant quantity ! separation( )2" c2time interval( )2= 0 " c 1.413yrs( )( )2= "2.0 LY2 ! separation( )2" c2time interval( )2= 4.3( )2" c 4.526yrs( )( )2= "2.0 LY2• The quantity (separation)2-c2(time interval)2 isthe same for all observers• It mixes the space and time coordinatesPhy107, Lect 20 6Fri. Mar. 9, 2006Geometrical space-time diagrams• Minkowski developed the space-time diagram:– A geometrical way to understand reference frames andrelativity• Think of events, such as the lightening strikes, aspositions and times: (x, t)• Use event ‘coordinates’ of (x, ct)– Units of ct = (m/s)(s) = meters: x and ct in meters• Represent relativistic events graphically2Phy107, Lect 20 7Fri. Mar. 9, 2006The space-time continuum• An event is indicated bya point in this graph.xct• The time-dependentmotion of a particlewould be a string ofthese points.–This is called theparticle’s ‘world-line’x1ct1Phy107, Lect 20 8Fri. Mar. 9, 2006Constant velocity motion• Worldline of an objectmoving at constantvelocity is a linexctWorld line low ofvelocity objectWorld line of highervelocity objectCoordinate axesPhy107, Lect 20 9Fri. Mar. 9, 2006QuestionWhich events have a space-time separation of zero?A. A & BB. A & CC. C & DD. None of themxctWorldline oflight beamABCDPhy107, Lect 20 10Fri. Mar. 9, 2006Universal space-time distance(space separation)2-c2(time interval)2 = (space-time distance)2Think of all the different observers measuring different spatialseparations, different time intervals.Suppose the universal (space-time distance)2 < 0Minimum time interval occurs forspace separation is zero -> events occur at same spatial location.This was our condition for measuring the proper time.In another frame, spatial separation not zeromeasured time interval must be longerTime dilation!Phy107, Lect 20 11Fri. Mar. 9, 2006Newton again• Fundamental relations of Newtonian physics– acceleration = (change in velocity)/(change in time)– acceleration = Force / mass– Work = Force x distance– Kinetic Energy = (1/2) (mass) x (velocity)2– Change in Kinetic Energy = net work done• Newton predicts that a constant force gives– Constant acceleration– Velocity proportional to time– Kinetic energy proportional to (velocity)2Phy107, Lect 20 12Fri. Mar. 9, 2006Forces, Work, and Energy in RelativityWhat about Newton’s laws?• Relativity dramatically altered our perspective ofspace and time– But clearly objects still move,spaceships are accelerated by thrust,work is done,energy is converted.• How do these things work in relativity?3Phy107, Lect 20 13Fri. Mar. 9, 2006Applying a constant force• Particle initially at rest,then subject to a constant force starting at t=0, Δmomentum =momentum = (Force) x (time)• Using momentum = (mass) x (velocity),Velocity increases without bound as time increasesRelativity says no. The effect of the force gets smaller and smaller as velocity approaches speed of lightPhy107, Lect 20 14Fri. Mar. 9, 200600.20.40.60.810 1 2 3 4 5SPEED / SPEED OF LIGHTTIMENewtonEinstein! vc=t / tot / to( )2+ 1, to=FmocRelativistic speed of particlesubject to constant force• At small velocities(short times) themotion is describedby Newtonian physics• At higher velocities,big deviations!• The velocity neverexceeds the speed oflightPhy107, Lect 20 15Fri. Mar. 9, 2006Momentum in Relativity• The relationship between momentum andforce is very simple and fundamental ! change in momentumchange in time= ForceMomentum is constant for zero forceandThis relationship is preserved in relativityPhy107, Lect 20 16Fri. Mar. 9, 2006Relativistic momentum• Relativity concludes that the Newtoniandefinition of momentum(pNewton=mv=mass x velocity)is accurate at low velocities,but not at high velocitiesThe relativistic momentum is: ! prelativistic="mv"=11# (v /c)2RestmassvelocityRelativistic gammaPhy107, Lect 20 17Fri. Mar. 9, 2006Was Newton wrong?• Relativity requires a different concept ofmomentum• But not really so different!• For small velocities << light speedγ≈1, and so prelativistic ≈ mv• This is Newton’s momentum• Differences only occur at velocities that are a substantialfraction of the speed of light! prelativistic="mv"=11# (v /c)2Phy107, Lect 20 18Fri. Mar. 9, 2006Relativistic Momentum• Momentum can be increasedarbitrarily, but velocity neverexceeds c• We still use• For constant force we still havemomentum = Force x time,but the velocity never exceeds c• Momentum has been redefined! prelativistic="mv =mv1# (v /c)200.20.40.60.810 1 2 3 4 5SPEED / SPEED OF LIGHTRELATIVISTIC MOMENTUM! vc=p / pop / po( )2+ 1, po= moc !
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