lecture6

lecture6 - Physics 126 Lecture 6 Suggested Reading Chapter...

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M.Loy, H. B. Chan 1 Physics 126 Lecture 6 Feb 25, 2011 E= γ mc 2 : what does it mean? Space-time and energy-momentum Useful formulae and example Suggested Reading: Chapter 1.8 – 1.10
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H. B. Chan 2 Frame S’ (move at V x relative to Frame S) Collision of two particles with identical mass m (0,-V y ) (0, V y ) B (mV x ,0) (mV x ,0) P total (V x ,V y ) (V x ,-V y ) A Final Initial Frame S Frame S - γ -1 V y γ -1 V y u y - γ -1 )mV y ( - γ + γ -1 )mV y P y, total γ V y - γ V y u y Final Initial Frame S’ Lorentz transformation: After Lorentz transformation, momentum mv is not conserved in Frames S’ - = 2 1 c u v u u x x y y γ 2 2 1 1 c v x - = Classical momentum mv is conserved in S
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H. B. Chan 3 Lorentz transform, - = 2 1 c u v u u x x y y γ u y γ u y ’= u y / γ Transformation of v y y y u u = v x u y ’= u y Galilean transform, -v x u y Independent of V x Depends on v x , u x v x v x -2v x u y ’’= u y u y γ u y ’’= γ u y In these 2 cases, u y is the same in the original frame. But in the new frame, they become different.
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M. Loy, H. B. Chan 4 Definition of momentum: Newton and Einstein As defined by Newton, p = m v, with v defined as (distance)/( time duration ). That is OK for Newton, with time being universal. But in Einstein’s world, time duration depends on velocity (through γ29 in that reference frame . time duration t depends on the reference frame. Event A : (x A , t A ) Event B : (x B , t B ) t = t B – t A = γ τ AB AB
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M.Loy, H. B. Chan 5 Definition of velocity v between A and B v = (distance between A and B)/ {(time at location B) – (time at location A)} Time(B) – Time(A) = γ ∆τ Because of γ , mv y will depend on v x . In the reference frame where v x is 0, the momentum mv y will be maximum. In reference system when v x is close to c, mv y will be much, much reduced. Therefore, momentum, defined as mv y = m y/ γ τ , have different values in different frames.
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M.Loy, H. B. Chan 6 Relativistic momentum Re-defined momentum so that it will conserve in all frames by using proper time ∆τ AB , instead of t =(t B – t A ): ( 29 ( 29 v m t r m t r m r m p v v v v γ τ = = = = The definition of velocity remains unchanged: v = (distance traveled)/(difference of clock readings from two clocks separated by that distance) While the value of t depends on motion , ∆τ is INVARIANT In this way, momentum in the y direction no longer depend on velocity in the x direction when being transformed into another frame.
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M.Loy, H. B. Chan 7 ‘Force’ in Relativity • Force is defined as: F = dp/dt = d( γ mv)/dt What about conservation of mass and conservation of energy? Once momentum has been redefined, the ‘choices’ available (to
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This note was uploaded on 03/31/2011 for the course PHY 126 taught by Professor Hobunchan during the Spring '11 term at HKUST.

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lecture6 - Physics 126 Lecture 6 Suggested Reading Chapter...

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