Class 5: Energy and Momentum In this class we will study how the basic concepts of mechanics, energy and momentum, must be modified to be consistent with the postulates of relativity
Class5:EnergyandMomentum
Inthisclasswewillstudyhowthebasicconceptsofmechanics,energyandmomentum,
mustbemodifiedtobeconsistentwiththepostulatesofrelativity
Class5:EnergyandMomentum
Attheendofthissessionyoushouldbeableto…
• … recallhowthedefinitionsofmomentum andenergy aremodifiedinrelativity
• … applyconservationlawsofthesequantities,tostudyrelativisticparticlecollisions
• … befamiliarwiththeconceptoftherestmassofaparticle,andhowitrepresentsanequivalentenergy
• ...understandhowtocalculatetheenergyandmomentumofphotons,particlestravellingatthespeedoflight
Relativisticmechanics
• Everydayintheworld’smostenergeticparticlecolliders,billionsofparticlessmashtogetheratrelativisticspeeds
• ProtonsintheLargeHadronCollideraremovingwithspeed0.99999999𝑐 !!
• Dothenormallawsofmechanicsapplyatthesespeeds?
https://www.pinterest.com.au/pin/542120873871190260/
• Themostimportantaspectofmechanicsisconservationlaws
• Theyallowustoanalysethemostcomplexinteractionsintermsofsimpleprinciples
https://www.flexptnj.com/the-ride-of-your-life/ http://www.alomabowlingcenters.com/boardwalk/bowl-a-strike-at-boardwalk-bowl/
Relativisticmechanics
ModifyingNewtonianmechanics
• Topreserveconservationlawsathighspeedsrequiresustomodifythedefinitionsofenergyandmomentum
• Toseewhy,considerasimplecollisionintwoframes:
Before:
After:
Frame𝑆′
𝑢 𝑢𝑚𝑚
2𝑚
Frame𝑆 [sittingwith2nd particle](classical)
2𝑢𝑚𝑚
2𝑚
𝑢
Momentumconserved Momentumconserved
𝑢
ModifyingNewtonianmechanics
Before:
After:
Frame𝑆′
𝑢 𝑢𝑚𝑚
2𝑚
Frame𝑆(relativistic)
2𝑢1 + 𝑢-/𝑐-
𝑚𝑚
2𝑚
Momentumconserved Momentumnotconserved
• Topreserveconservationlawsathighspeedsrequiresustomodifythedefinitionsofenergyandmomentum
• Toseewhy,considerasimplecollisionintwoframes:
ModifyingNewtonianmechanics
• Thisissueissolvedbymodifyingthedefinitionofthemassofaparticlesuchthatitincreaseswithspeed𝒖
• Particlemass𝑚 𝑢 = 𝛾(𝑢)𝑚4 =56
789:/;:�
• 𝛾(𝑢) = 7789:/;:� isthenormal“Lorentzfactor”intermsof
thespeedoftheparticle,𝑢 [notaframetransformation]
• Themassatzerospeed,𝑚4,iscalledtherestmassoftheparticle andisaninvariantinallreferenceframes
Relativisticmomentum
• Therelativisticmomentumofaparticleisthendefinedintheusualway,asmass× velocity[NoteintheWorkbook]:
• Doesthisdefinitionmakesense?Showthatinthelow-𝑢limit(𝑢/𝑐 ≪ 1)werecovertheclassicaldefinitionofmomentum,𝑝 = 𝑚4𝑢 [mathshint: 1 + 𝑥 A ≈ 1 + 𝑛𝑥]
• Thetotalrelativisticmomentumisconservedincollisions[wewillseesomeexamplessoon]
Relativisticmomentum𝑝 𝑢 = 𝑚𝑢 = 𝛾𝑚4𝑢 =𝑚4𝑢
1 − 𝑢-/𝑐-�
Relativisticenergy
• Wecanusetheexpressionforrelativisticmomentumtocalculatethekineticenergygainedbyaparticleasitaccelerates
Kineticenergy𝑇 = W𝐹𝑑𝑥�
�
= W𝑑𝑝𝑑𝑡
�
�
𝑑𝑥 = W𝑑𝑥𝑑𝑡 𝑑𝑝
�
�
= W𝑣𝑑𝑝𝑑𝑣 𝑑𝑣
�
�
Integrationbyparts: 𝑇 = W 𝑣𝑑𝑝𝑑𝑣 𝑑𝑣
`a9
`a4= 𝑣𝑝 `a4
`a9 − W 𝑝𝑑𝑣`a9
`a4
𝑇 =𝑚4𝑣-
1 − 𝑣-/𝑐-�`a4
`a9
− 𝑚4𝑐- 1 − 𝑣-/𝑐-�
`a4
`a9
Kineticenergy𝑇 =𝑚4𝑐-
1 − 𝑢-/𝑐-� − 𝑚4𝑐- = 𝛾 𝑢 − 1 𝑚4𝑐-
Relativisticenergy
• Therelativistickineticenergyofaparticleisdefinedby
• Doesthisdefinitionmakesense?Showthatinthelow-𝑢limit(𝑢/𝑐 ≪ 1)werecovertheclassicaldefinitionofkineticenergy,𝑇 = 7
-𝑚4𝑢- [mathshint: 1 + 𝑥 A ≈ 1 + 𝑛𝑥]
Kineticenergy𝑇 𝑢 = 𝛾 𝑢 − 1 𝑚4𝑐- =𝑚4𝑐-
1 − 𝑢-/𝑐-� − 𝑚4𝑐-
Relativisticenergy
• Inrelativity,thekineticenergyofaparticle→ ∞ astheparticleapproachesthespeedoflight,𝑣 → 𝑐
Relativity
Classical
https://courses.lumenlearning.com/physics/chapter/28-6-relativistic-energy/
Anacceleratedparticlerequiresinfinite
energytoreachthespeedoflight!
• Theexpressionforkineticenergy,𝑇 = 𝛾 𝑢 − 1 𝑚4𝑐-,canbewrittenintermsofatotalenergy𝐸
• Weconcludethataparticlehasanintrinsicrestenergy,𝐸 0 = 𝑚4𝑐-,suchthat
Relativisticenergy
Totalenergy𝐸 𝑢 = 𝛾 𝑢 𝑚4𝑐- =𝑚4𝑐-
1 − 𝑢-/𝑐-�
Kineticenergy𝑇 = 𝐸 𝑢 − 𝐸(0)
Totalenergy = restenergy + kineticenergy
Rest-massenergy
• Whatistherest-massenergyofahuman?
• Isthisalotofenergy,ornot?
http://explorecuriocity.org/Explore/ArticleId/1606/e-mc-squared-einsteins-relativity-1606.aspx
Rest-massenergy
• Theideathatrest-massisaformofenergyisoneofthemostimportantaspectsofrelativity,andwewilldiscussitsapplicationsfurtherinthenextclass
https://www.space.com/19321-sun-formation.htmlhttps://www.elp.com/articles/2015/09/more-nuclear-power-plant-retirements-forecast.htmlhttps://physics.stackexchange.com/questions/56296/why-does-a-photon-colliding-with-an-atomic-nucleus-cause-pair-production
• Wehaveseenthatinrelativity,
• Combiningtheseequations,wefindtwousefulrelations:
Twousefulformulae
Momentum𝑝 𝑢 = 𝛾(𝑢)𝑚4𝑢 =𝑚4𝑢
1 − 𝑢-/𝑐-�
Energy𝐸 𝑢 = 𝛾(𝑢)𝑚4𝑐- =𝑚4𝑐-
1 − 𝑢-/𝑐-�
𝐸- = 𝑚4𝑐- - + 𝑝𝑐 - 𝑢 =𝑝𝑐-
𝐸
Energy– restmass– momentum Speed – momentum– energy
Photons
• 𝐸 = 56;:
789:/;:� :noparticlecantravelatthespeedoflight
• However,quantummechanicstellsusthatlightitselfcanbehaveasparticlesknownasphotons withenergy𝐸 = ℎ𝜈
• Thiscontradictionisresolvedifphotonshavezerorest-mass,𝑚4 = 0
• Using𝐸- = 𝑚4𝑐- - + 𝑝𝑐 -,wefindforphotons,𝐸 = 𝑝𝑐
• Momentum𝑝 = k;= lm
;
Relativisticcollisionexample
• Amassof1𝑘𝑔,movingatspeed𝑢 = 0.8𝑐,isabsorbedbyastationarymassof2𝑘𝑔.Atwhatspeed𝑈 doesthecombinedmassrecoil?Whatisthecombinedmass𝑀?
• Writeanequationfortheconservationofmomentum
• Writeanequationfortheconservationofenergy
• Eliminatevariablesbetweenthesetwoequationstosolveforthetwounknowns𝑈 and𝑀
𝑢 = 0.8𝑐𝑢 = 𝑈𝑚4 = 1
𝑚4 = 2𝑚4 = 𝑀
Relativisticcollisionexample
• Checkingyouranswers:𝑈 = s77𝑐 = 0.36𝑐,𝑀 = 3.42𝑘𝑔
• Asthisexampleshows,massisnotconservedincollisions!!(rest-massbefore= 1 + 2 = 3,rest-massafter= 3.42).Rather,energyandmomentumareconserved
• Question:wheredoestheextramasscomefrom?
Class5:EnergyandMomentum
Attheendofthissessionyoushouldbeableto…
• … recallhowthedefinitionsofmomentum andenergy aremodifiedinrelativity
• … applyconservationlawsofthesequantities,tostudyrelativisticparticlecollisions
• … befamiliarwiththeconceptoftherestmassofaparticle,andhowitrepresentsanequivalentenergy
• ...understandhowtocalculatetheenergyandmomentumofphotons,particlestravellingatthespeedoflight