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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

In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

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Page 1: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

Class5:EnergyandMomentum

Inthisclasswewillstudyhowthebasicconceptsofmechanics,energyandmomentum,

mustbemodifiedtobeconsistentwiththepostulatesofrelativity

Page 2: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

Class5:EnergyandMomentum

Attheendofthissessionyoushouldbeableto…

• … recallhowthedefinitionsofmomentum andenergy aremodifiedinrelativity

• … applyconservationlawsofthesequantities,tostudyrelativisticparticlecollisions

• … befamiliarwiththeconceptoftherestmassofaparticle,andhowitrepresentsanequivalentenergy

• ...understandhowtocalculatetheenergyandmomentumofphotons,particlestravellingatthespeedoflight

Page 3: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

Relativisticmechanics

• Everydayintheworld’smostenergeticparticlecolliders,billionsofparticlessmashtogetheratrelativisticspeeds

• ProtonsintheLargeHadronCollideraremovingwithspeed0.99999999𝑐 !!

• Dothenormallawsofmechanicsapplyatthesespeeds?

https://www.pinterest.com.au/pin/542120873871190260/

Page 4: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

• Themostimportantaspectofmechanicsisconservationlaws

• Theyallowustoanalysethemostcomplexinteractionsintermsofsimpleprinciples

https://www.flexptnj.com/the-ride-of-your-life/ http://www.alomabowlingcenters.com/boardwalk/bowl-a-strike-at-boardwalk-bowl/

Relativisticmechanics

Page 5: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

ModifyingNewtonianmechanics

• Topreserveconservationlawsathighspeedsrequiresustomodifythedefinitionsofenergyandmomentum

• Toseewhy,considerasimplecollisionintwoframes:

Before:

After:

Frame𝑆′

𝑢 𝑢𝑚𝑚

2𝑚

Frame𝑆 [sittingwith2nd particle](classical)

2𝑢𝑚𝑚

2𝑚

𝑢

Momentumconserved Momentumconserved

Page 6: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

𝑢

ModifyingNewtonianmechanics

Before:

After:

Frame𝑆′

𝑢 𝑢𝑚𝑚

2𝑚

Frame𝑆(relativistic)

2𝑢1 + 𝑢-/𝑐-

𝑚𝑚

2𝑚

Momentumconserved Momentumnotconserved

• Topreserveconservationlawsathighspeedsrequiresustomodifythedefinitionsofenergyandmomentum

• Toseewhy,considerasimplecollisionintwoframes:

Page 7: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

ModifyingNewtonianmechanics

• Thisissueissolvedbymodifyingthedefinitionofthemassofaparticlesuchthatitincreaseswithspeed𝒖

• Particlemass𝑚 𝑢 = 𝛾(𝑢)𝑚4 =56

789:/;:�

• 𝛾(𝑢) = 7789:/;:� isthenormal“Lorentzfactor”intermsof

thespeedoftheparticle,𝑢 [notaframetransformation]

• Themassatzerospeed,𝑚4,iscalledtherestmassoftheparticle andisaninvariantinallreferenceframes

Page 8: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

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-�

Page 9: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

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-

Page 10: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

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-

Page 11: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

Relativisticenergy

• Inrelativity,thekineticenergyofaparticle→ ∞ astheparticleapproachesthespeedoflight,𝑣 → 𝑐

Relativity

Classical

https://courses.lumenlearning.com/physics/chapter/28-6-relativistic-energy/

Anacceleratedparticlerequiresinfinite

energytoreachthespeedoflight!

Page 12: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

• 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

Page 13: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

Rest-massenergy

• Whatistherest-massenergyofahuman?

• Isthisalotofenergy,ornot?

http://explorecuriocity.org/Explore/ArticleId/1606/e-mc-squared-einsteins-relativity-1606.aspx

Page 14: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

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

Page 15: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

• 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

Page 16: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

Photons

• 𝐸 = 56;:

789:/;:� :noparticlecantravelatthespeedoflight

• However,quantummechanicstellsusthatlightitselfcanbehaveasparticlesknownasphotons withenergy𝐸 = ℎ𝜈

• Thiscontradictionisresolvedifphotonshavezerorest-mass,𝑚4 = 0

• Using𝐸- = 𝑚4𝑐- - + 𝑝𝑐 -,wefindforphotons,𝐸 = 𝑝𝑐

• Momentum𝑝 = k;= lm

;

Page 17: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

Relativisticcollisionexample

• Amassof1𝑘𝑔,movingatspeed𝑢 = 0.8𝑐,isabsorbedbyastationarymassof2𝑘𝑔.Atwhatspeed𝑈 doesthecombinedmassrecoil?Whatisthecombinedmass𝑀?

• Writeanequationfortheconservationofmomentum

• Writeanequationfortheconservationofenergy

• Eliminatevariablesbetweenthesetwoequationstosolveforthetwounknowns𝑈 and𝑀

𝑢 = 0.8𝑐𝑢 = 𝑈𝑚4 = 1

𝑚4 = 2𝑚4 = 𝑀

Page 18: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

Relativisticcollisionexample

• Checkingyouranswers:𝑈 = s77𝑐 = 0.36𝑐,𝑀 = 3.42𝑘𝑔

• Asthisexampleshows,massisnotconservedincollisions!!(rest-massbefore= 1 + 2 = 3,rest-massafter= 3.42).Rather,energyandmomentumareconserved

• Question:wheredoestheextramasscomefrom?

Page 19: In this class we will study how the basic concepts of mechanics, …astronomy.swin.edu.au/~cblake/Class5_EnergyAndMomentum.pdf · is the normal “Lorentz factor” in terms of the

Class5:EnergyandMomentum

Attheendofthissessionyoushouldbeableto…

• … recallhowthedefinitionsofmomentum andenergy aremodifiedinrelativity

• … applyconservationlawsofthesequantities,tostudyrelativisticparticlecollisions

• … befamiliarwiththeconceptoftherestmassofaparticle,andhowitrepresentsanequivalentenergy

• ...understandhowtocalculatetheenergyandmomentumofphotons,particlestravellingatthespeedoflight