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Need help with derivation

  1. Apr 22, 2004 #1
    [SOLVED] Need help with derivation

    I have four formulas from SR, and need to relate them. Two are the Lorentz Transform, one is a simple formula for velocity, and the fourth is the formula for the addition of velocity in SR.

    [tex]1... x^\prime = { x-vt \over \sqrt { 1-{v^2 \over c^2} } } \quad 2... t^\prime = { { t-{v \over c^2 } x } \over \sqrt { 1-{v^2 \over c^2 } } } \quad 3... x^\prime = wt^\prime \quad 4... W={ v+w \over 1+{vw \over c^2 } }[/tex]

    Dr. Einstein says:

    (Hope that made sense.)

    I have tried this, and got here:

    [tex]w = { x^\prime \over t^\prime }[/tex]

    [tex]\Rightarrow w = { { x-vt \over \sqrt { 1- { v^2 \over c^2 } } } \over { { t - { v \over c^2 } x } \over \sqrt { 1- { v^2 \over c^2 } } }[/tex]

    [tex]\Rightarrow w = { x-vt \over t - { v \over c^2 } x }[/tex]

    ...so then...

    [tex]W=v+{ x-vt \over t - { v \over c^2 } x }[/tex]

    ...and that's as far as I got. I am quite a ways away from equation (4), above.

    Can anyone help me here?
     
    Last edited by a moderator: Apr 22, 2004
  2. jcsd
  3. Apr 22, 2004 #2

    DW

    User Avatar

    Yeah, as he said W is NOT v+w. Instead W is x/t. So continuing from
    [tex]w = { x-vt \over t - { v \over c^2 } x }[/tex]
    divide the top and bottom by t
    [tex]w = { \frac{x}{t}-v \over 1 - { v \over c^2 } \frac{x}{t} }[/tex]
    [tex]w = { W-v \over 1 - { Wv \over c^2 } }[/tex]
    Solve for W and you will get
    [tex]W = { w+v \over 1 + { wv \over c^2 } }[/tex]
     
  4. Apr 22, 2004 #3
    DW,

    Thanks for a quick and thorough response. I was kind of afraid to go there, because it seems to me that there is a pun here between v=x/t and W=x/t.

    Clearly both are true in SOME sense, and it is also clear that v<>W (though in the abstract, W is a kind of v).

    Do you think that you can spare me another moment and clear that up for me?

    Thanks!
     
  5. Apr 22, 2004 #4

    DW

    User Avatar

    In this context, v is not x/t. W is. There are three velocities being related. There is w which is the velocity of some "thing" according to measurements made from the primed coodinate system. There is W which is the velocity of that same "thing" according to measurements made from the unprimed coordinate system. And then there is v which is the speed of one of the coordinate systems according to measurements made from the other.
     
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