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Lect_32_note

# Lect_32_note - Chapter 39 Relativity Lorentz...

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Chapter 39 Relativity

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Lorentz Transformations 2 2 2 2 1 1 where 1 p p t t γ t v c γ v c == = () 2 ' ' x γ xvt v t γ tx c ∆= ∆−∆ ⎛⎞ ∆= ∆− ∆ ⎜⎟ ⎝⎠
2 22 1 and 11 ' '' ' ' x x x y z yz xz dx u v u uv dt c u u u γγ cc == ⎛⎞ −− ⎜⎟ ⎝⎠ Lorentz Velocity Transformation u x u’ x

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Momentum u x u’ x m z Classic definition P = m P = m u x P’ = m u’ x u r
Relativistic Linear Momentum is the velocity of the particle, m is its mass u pu 2 2 1 m γ m u c ≡= r r r u r u x u’ x m

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Relativistic Linear Momentum u pu 2 2 1 m γ m u c ≡= r r r P = m u u
Relativistic Form of Newton’s Laws z The relativistic force acting on a particle whose linear momentum is is defined as p r p F d dt = r r

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Speed Limit z F goes infinite large when u reaches c . z The speed of light is the speed limit of the universe z It is the maximum speed possible for energy and information transfer z Any object with mass must move at a lower speed
Relativistic Kinetic Energy dp dW F dx dx dt == Let’s begin with classical concepts. The differential work done is: v dW dp dx dp dt dt dt dt Dividing by dt at both sides: vv v dW d dp d dd t dd t = In terms of velocity derivatives: v dW dp = Canceling the d v/ dt ’s: v dp dW d d = or

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Relativistic Kinetic Energy vv v dp dW d d = The kinetic energy will be equal to the work done starting with zero energy and ending with W 0 , or from zero velocity to u : 0 0 W K dW = 0 v u dp d d =
0 00 vv v v v uu u dp Kd p p d d == 22 2 0 0 v v1 v / 1v/ u u p um d p c c c =− = + 2 2 2 1 2 2v/ v dc cc c d c ⎡⎤ −− = ⎣⎦ ( ) 2 1/1 1/ 2 2 u mu m c u c uc ⎛⎞ = +− ⎜⎟ ⎝⎠ 1 c γ=

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Lect_32_note - Chapter 39 Relativity Lorentz...

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