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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 - = γ Classical momentum mv is conserved in S
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
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 γ 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.
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 have consistency within the theory) become quite limited. (In fact, we will see that just like time and space are linked in relativity, so is energy and momentum linked.) But, mass and energy will no longer conserve separately, but are “different manifestations of the same thing.”

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H. B. Chan 8 Relativistic momentum, force, 2 nd law and energy ( 29 v m p v γ = momentum force ( 29 dt v m d F v
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