Physics 2D Lecture Slides Lecture 5 April 6, 2009 Lorentz Velocity Transformation Rule S and S’ are measuring ant’s speed u along x, y, z axes x2' ! x1' dx ' In S' frame, u x' = ' = ' ' t2 ! t1 dt v dx = " (dx ! vdt), dt ' = " (dt ! 2 dx) c dx ! vdt u x' = , divide by dt v dt ! 2 dx u c ux ! v u x' = vux 1! 2 c For v << c, u x' = ux ! v ' S S’ v (Galilean Trans. Restored) Velocity Transformation Perpendicular to S-S’ motion v dy ' = dy, dt ' = ! (dt " 2 dx) c dy ' dy ' uy = = v dt ' ! (dt " 2 dx) c divide by dt on RHS u = ' y uy v ! (1 " 2 ux ) c There is a change in velocity in the direction # to S-S' motion ! Similarly Z component of Ant' s velocity transforms as uz ' uz = v ! (1 " 2 ux ) c Inverse Lorentz Velocity Transformation Inverse Velocity Transform: ux ' + v ux = vux ' 1+ 2 c ' As usual, uy replace uy = v ' v⇒-v ! (1 + 2 ux ) c ' uz uz = v ' ! (1 + 2 ux ) c Does Lorentz Transform “work” For Topgun ? y’ Two rockets A &B travel in opposite directions S’ -0.85c y S B 0.75c A x’ O’ O (Earth guy) x An observer on earth (S) measures speeds = 0.75c And 0.85c for A & B respectively What does A measure as B’s speed? Place an imaginary S’ frame on Rocket A ⇒ v = 0.75c relative to Earth Observer S Consistent with Special Theory of Relativity Example of Inverse velocity Transform Biker moves with speed = 0.8c past stationary observer Throws a ball forward with speed = 0.7c What does stationary observer see as velocity of ball ? Speed of ball relative to stationary observer uX ? Place S’ frame on biker Biker sees ball speed uX’ =0.7c Hollywood Yarns Of Time Travel ! Terminator : Can you be seen to be born before your mother? A frame of Ref where sequence of events is REVERSED ?!! u S in fro eo ff ve (,)(,)xtxt 22''22 F * $ v !x ' !t ' = t " t = # , !t " & 2 ) / % c (. + Reversing sequence of events 0 !t ' < 0 ' 1 rri (,)(,)xtxt 11''11 ' 2 Ia m SD S’ It ak S I Can’t be seen to arrive in SF before I take off from SD u S S’ (,)(,)xtxt 11''11 22''22 (,)(,)xtxt * $ v !x ' !t ' = t " t = # , !t " & 2 ) / % c (. + For what value of v can !t ' < 0 v !x v !x v u !t ' < 0 0 !t < 2 01 < 2 = 2 c c !t c v c 0 > 0 v > c : Not allowed c u ' 2 ' 1 Watching an Inelastic Collision between two putty balls Relativistic Momentum and Revised Newton’s Laws Need to generalize the laws of Mechanics & Newton to confirm topmu=!! Lorentz Transform and the Special theory of relativity: Example : S P = mv –mv = 0 P=0 Before v 1 v V=0 2 1 2 After v1 ! v v2 ! v !2v V !v ' v = = 0, v2 = = , V ' = = !v 2 v1v v1v Vv v 1! 2 1! 2 1! 2 1+ 2 c c c c !2mv ' ' ' ' pbefore = mv1 + mv2 = , pafter = 2mV ' = !2mv 2 v S’ 1+ 2 c p'before " p'after ' 1 1 v1’=0 Before v2’ 2 V’ 1 2 After Definition (without proof) of Relativistic Momentum p= mu 1 ! u 2 / c2 With the new definition relativistic momentum is conserved in all frames of references : Do the exercise New Concept Rest Mass is mass of object in frame in which it at rest. ! = 1 / 1 " u 2 / c2 u is speed of object, NOT reference frame. Nature of Relativistic Momentum m u ! p= ! mu 1 ! (u / c)2 ! = " mu With the new definition of Relativistic momentum Momentum is conserved in all frames of references Relativistic Force & Acceleration ! p= ! mu 1 ! (u / c)2 ! = " mu Relativistic Force And Acceleration Reason why you cant quite get up to the speed of light no matter how hard you try! ! % ! dp! d " mu F= = $ ' use dt dt $# 1 ! (u / c)2 '& ) m mu F=+ + + 1 ! (u / c)2 2 1 ! (u / c) +* ) 2 2 2 , mc ! mu + mu . du F=+ 3/ 2 . + 2 2 dt c 1 ! (u / c) .*+ ( ( ) m F=+ + 2 +* 1 ! (u / c) ) 3/ 2 d du d = dt dt du , !1 !2u . du ( ( )( 2 ) 2 c . dt .- ) , . du : Relativistic Force 3/ 2 . dt .! ! du Since Acceleration a = ,[rate of change of velocity] dt ! 3/ 2 F ! )1 ! (u / c)2 , / a= m* ! Note: As u / c 0 1, a 0 0 !!!! Its harder to accelerate when you get closer to speed of light ( ) A Linear Particle Accelerator - F q -E d + Parallel Plates E= V/d F= eE V Charged particle q moves in straight line ! ! in a uniform electric field E with speed u ! ! accelarates under force F=qE ! ! 3/ 2 3/ 2 ! 2 2 du F " u % qE " u % ! a = = 1! 2 ' = 1! 2 ' $ $ dt m# m # c & c & larger the potential difference V across plates, larger the force on particle Under force, work is done on the particle, it gains Kinetic energy New Unit of Energy 1 eV = 1.6x10-19 Joules 1 MeV = 1.6x10-13 Joules 1 GeV = 1.6x10-10 Joules Your Television (the CRT type) is a Small Particle Accelerator ! Linear Particle Accelerator : 50 GigaVolts Accelating Potential ! 3/ 2 ! eE 2 a= "#1 ! (u / c) $% m PEP-II accelerator schematic and tunnel view Relativistic Work Done & Change in Energy x2 X2 , u=u x1 , u=0 W = ! x1 ! ! F .dx = x2 ! x1 ! dp ! .dx dt du ! m mu dp dt p= # = , 3/ 2 dt u2 $ u2 ' 1" 2 1" 2 ) & c c ( % substitute in W, du u m udt dt # W =! (change in var x * u) 3/ 2 0 $ u2 ' &1 " 2 ) c ( % Relativistic Work Done & Change in Energy X2 , u=u u W= x1 , u=0 (" 0 mudu u % $1 ! 2 ' # c & 2 3/ 2 = mc 2 1/ 2 " u % $1 ! 2 ' # c & 2 ! mc 2 = ) mc 2 ! mc 2 Work done is change in energy (KE in this case) K = ) mc 2 ! mc 2 or Total Energy E= ) mc 2 = K + mc 2 But Professor… Why Can’t ANYTHING go faster than light ? K= mc 2 ! mc ( 2 1/ 2 ( K + mc ) " u2 % $1 ! 2 ' # c & " u2 % ( $1 ! 2 ' = m2 c 4 "# K + mc 2 %& # c & ( u = c 1! ( 2 2 ) , + . 2 mc + . =+ 1/ 2 . 2 + "1 ! u % . +* $ c 2 ' .# & !2 K K !2 + 1) (Parabolic in u Vs ) 2 2 mc mc As K / 0 , u / c 1 2 Non-relativistic case: K = mu ( u = 2 2K m 2 Relativistic Kinetic Energy Vs Velocity A Digression: How to Handle Large/Small Numbers • Example: consider very energetic particle with very large Energy E E mc 2 + K K ! = = = 1 + mc 2 mc 2 mc 2 1/ 2 u # 1& = 1 ! Now calculate u from % ( c $ "2' • Lets Say γ = • Try this on your el-cheapo calculator, you will get u/c =1, u=c due to limited precision. • In fact u ≅c but not exactly!, try to get this analytically 3x1011, In Quizzes, you are Expected to perform Such simple approximations When Electron Goes Fast it Gets “Fat” Emc!= Total Energy E = ! mc = K + mc 2 2 2 vAs 1, cApparent Mass approache New Concept Rest Mass = particle mass when its at rest Relativistic Kinetic Energy & Newtonian Physics Relativistic KE K = ! mc 2 " mc 2 # Remember Binomial Theorem & % ( nx n(n-1) % for x << 1; (1+x) n = (1+ + x 2 +smaller terms) ( %$ (' 1! 2! " 1 2 # u2 & 1 u2 ) When u << c, %1- 2 ( * 1+ + ...smaller terms 2 2c $ c ' 1 u2 1 2 2 2 so K * mc [1 + ] " mc = mu (classical form recovered) 2 2c 2 Total Energy of a Particle E = ! mc 2 = KE + mc 2 For a particle at rest, u = 0 " Total Energy E= mc 2 E=mc2 ⇒ Sunshine Won’t Be Forever ! Q: Solar Energy reaches earth at rate of 1.4kW per square meter of surface perpendicular to the direction of the sun. by how much does the mass of sun decrease per second owing to energy loss? The mean radius of the Earth’s orbit is 1.5 x 1011m. r • Surface area of a sphere of radius r is A = 4πr2 • Total Power radiated by Sun = power received by a sphere whose radius is equal to earth’s orbit radius E= mc2 ⇒ Sunshine Won’t Be Forever ! Total Power radiated by Sun = power received by a sphere with radius equal to earth-sun orbit radius( r in figure) r sun Plost = sun Plost P Earth incident Aearth! sun = A = 4.0 # 1026 W P Earth incident A 2 3 2 11 2 4" rearth! = (1.4 # 10 W / m )(4 " )(1.5 # 10 ) sun So Sun loses E = 4.0 # 1026 J of rest energy per second E 4.0 # 1026 J 9 Its mass decreases by m = 2 = = 4.4 # 10 kg per sec!! 8 2 c (3.0 # 10 ) If the Sun's Mass = 2.0 # 1030 kg So how long with the Sun last ? One day the sun will be gone and the solar system will not be a hospitable place for life E = ! mc 2 " E 2 = ! 2 m2 c 4 Relationship between P and E p = ! mu " p 2 c 2 = ! 2 m2u 2 c 2 " E 2 # p 2 c 2 = ! 2 m2 c 4 # ! 2 m2u 2 c 2 = ! 2 m2 c 2 (c 2 # u 2 ) 2 4 m2 c 2 2 m c 2 2 2 2 4 = (c # u ) = (c # u ) = m c 2 2 2 u c #u 1# 2 c E 2 = p 2 c 2 + (mc 2 )2 ........important relation For particles with zero rest mass like photon (EM waves) E E= pc or p = (light has momentum!) c Relativistic Invariance : E 2 # p 2 c 2 = m2 c 4 : In all Ref Frames Rest Mass is a "finger print" of the particle Mass Can “Morph” into Energy & Vice Verca • Unlike in Newtonian mechanics • In relativistic physics : Mass and Energy are the same thing • New word/concept : MassEnergy , just like SpaceTime • It is the mass-energy that is always conserved in every reaction : Before & After a reaction has happened • Like squeezing a balloon : Squeeze here, it grows elsewhere – If you “squeeze” mass, it becomes (kinetic) energy & vice verca ! • CONVERSION FACTOR = C2 • This exchange rate never changes !