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| | [[File:TF_UCM_GalileanTans_RefFrame.png | 200 px]] | | [[File:TF_UCM_GalileanTans_RefFrame.png | 200 px]] |
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| − | Consider the motion of an object using two different coordinate systems <math>S</math> and <math>S^{\prime}</math> | + | Consider the description of an object in motion using two different coordinate systems <math>S</math> and <math>S^{\prime}</math>. |
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| − | The C.M. frame is often chosen to theoretically calculate cross-sections even though experiments are conducted in the Lab frame. In such cases you will need to transform cross-sections between two frames.
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| − | The total cross-section should be frame independent
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| − | :<math>\sigma_{C.M.} = \sigma_{Lab}</math>
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| − | or
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| − | : <math>\sigma(\theta) d \Omega = \sigma(\psi) d \Omega^{\prime}</math>
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| − | where
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| − | <math>\theta</math> is in the CM frame and <math>\psi</math> is in the Lab frame.
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| − | ;A non-relativistic transformation:
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| − | : <math>\sigma(\theta) d \Omega = \sigma(\psi) d \Omega^{\prime}</math>
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| − | : <math>\sigma(\theta) 2 \pi \sin(\theta) d \theta = \sigma(\psi) 2 \pi \sin (\psi) d \psi</math>
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| − | : <math>\Rightarrow \sigma(\psi) = \frac{\sin(\theta)}{\sin(\psi)} \frac{d \theta}{d \psi} \sigma(\theta)</math>
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| − | The transformation is governed by the dependence of <math>\theta</math> on <math> \psi</math> <math> \left( \frac{d \theta}{d \psi} \right )</math>
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| − | Lets return back to our picture of the scattering Process
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| − | [[Image:SPIM ElasCollis Lab CM Frame.jpg | 500 px]]
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| − | if we superimpose the vectors <math>\vec{v}_1</math> and <math>\vec{v}_1^{\prime}</math> we have
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| − | [[Image:SPIM ElasCollis Lab CM Frame_Velocities.jpg]]
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| − | Trig identities (non-relativistic Gallilean transformation) tell us
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| − | <math>v_1 \sin(\psi) = v_1^{\prime} \sin(\theta)</math>
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| − | <math>v_1 cos(\psi) = v_{cm} + v_1^{\prime} \cos(\theta)</math>
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| − | solving for <math>\psi</math>
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| − | <math>\tan(\psi) = \frac{\sin(\psi)}{\cos(\psi)} = \frac{v_1^{\prime} \sin(\theta)/v_1}{\frac{v_{CM}}{v_1} + \frac{v_1^{\prime} \cos(\theta)}{v_1} }
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| − | = \frac{\sin(\theta)}{\cos(\theta) + \frac{v_{CM}}{v_1^{\prime}}}</math>
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| − | For an elastic collision only the directions change in the CM Frame: <math>u_1^{\prime}= v_1^{\prime}</math> & <math>u_1^{\prime}= v_2^{\prime}</math>
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| − | ;From the definition of the C.M.
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| − | ;<math>\vec{v}_{CM} = \frac{m_1 \vec{u}_1 + m_2 \vec{u}_2}{m_1+m_2} = \frac{m_1}{m_1+m_2} \vec{u}_1</math>
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| − | ;conservation of momentum in CM Frame <math>\Rightarrow</math> :
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| − | :<math>m_1 u_1^{\prime} = - m_2 u_2{\prime}</math>
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| − | :<math> \Rightarrow v_1^{\prime} = u_1^{\prime} = \frac{-m_2}{m_1} u_2^{\prime}</math>
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| − | ; Gallilean Coordinate transformation:
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| − | ;<math>\vec{u}_1 = \vec{u}_1^{\prime} + \vec{v}_{CM} = \vec{u}_1^{\prime} + \frac{m_1}{m_1+m_2} \vec{u}_1</math>
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| − | :<math>\Rightarrow u_1{\prime} = \left [ 1 - \frac{m_1}{m_1 + m_2} \right ] u_1 = \frac{m_2}{m_1+m_2}u_1</math>
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| − | :<math>\Rightarrow v_1^{\prime} = u_1^{\prime} =\frac{m_2}{m_1+m_2} u_1</math>
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| − | ; another expression for <math>\psi</math>
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| − | using the above gallilean transformation we can do the following
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| − | :<math>\frac{v_{CM}}{v_1^{\prime}}= \frac{\frac{m_1}{m_1+m_2} u_1}{\frac{m_2}{m_1+m_2} u_1} = \frac{m_1}{m_2}</math>
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| − | or
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| − | : <math>\tan(\psi) = \frac{\sin(\theta)}{\cos(\theta) + \frac{m_1}{m_2}}</math>
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| − | after a little trig substitution
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| − | <math>\Rightarrow \frac{m_1}{m_2} = \frac{sin(\theta - \psi)}{\sin(\psi)} =</math> constant
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| − | now use the chain rule to find <math>\frac{d \theta}{d \psi}</math>
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| − | : <math>f \equiv \frac{sin(\theta - \psi)}{\sin(\psi)} =</math> constant
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| − | :<math>df = 0 = \frac{ \partial f}{\partial \psi} d \psi + \frac{ \partial f}{\partial \theta} d \theta </math>
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| − | : <math>\Rightarrow \frac{d \theta}{d \psi} = \frac{-\frac{ \partial f}{\partial \psi} }{\frac{ \partial f}{\partial \theta} }</math>
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| − | :<math>-\frac{ \partial f}{\partial \psi} = \frac{\cos(\theta - \psi)}{\sin(\psi)} + \frac{\sin(\theta - \psi)}{\sin(\psi)}</math>
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| − | :<math>\frac{ \partial f}{\partial \theta }= 1 + \frac{\sin(\theta - \psi) \cos(\psi)}{\cos(\theta - \psi) \sin(\psi)}</math>
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| − | after substitution:
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| − | : <math>\sigma(\psi) = \frac{\sin(\theta)}{\sin(\psi)} \frac{d \theta}{d \psi} \sigma(\theta)</math>
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| − | : <math>=\frac{\sin(\theta)}{\sin(\psi)} \left [ 1 + \frac{\sin(\theta - \psi) \cos(\psi)}{\cos(\theta - \psi) \sin(\psi)} \right ] \sigma(\theta)</math>
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| − | For the above equation to be more useful one would prefer to recast it in terms of only <math>\psi</math> and masses.
| + | Assume that <math>S^{\prime}</math> is a coordinate system moving at a CONSTANT speed <math>v</math>. |
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| − | :<math>\sigma(\psi) = \frac{\left [ \frac{m_1}{m_2}\cos(\psi) + \sqrt{1-\left ( \frac{m_1 \sin(\psi) }{m_2} \right )^2 }\right ]}{\sqrt{1 - \left ( \frac{m_1 \sin(\psi)}{m_2}\right )^2 }}\sigma(\theta)</math>
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| | [[Forest_UCM_NLM#Galilean_Transformations]] | | [[Forest_UCM_NLM#Galilean_Transformations]] |
