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