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Rearranging a Relativity Equation

  1. May 7, 2008 #1

    TFM

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    [SOLVED] Rearranging a Relativity Equation

    1. The problem statement, all variables and given/known data

    Rewrite equation:

    [tex] f = \sqrt{\frac{c + u}{c - u}}f_0 [/tex]

    to find the relative velocity u between the electromagnetic source and an observer in terms of the ratio of the observed frequency and the source frequency of light.

    What relative velocity will produce a 3.0 decrease in frequency and

    2. Relevant equations

    Rearangement of:

    [tex] f = \sqrt{\frac{c + u}{c - u}}f_0 [/tex]

    3. The attempt at a solution

    I seem to get stuck when trying to rearrange the equation:

    [tex] f = \sqrt{\frac{c + u}{c - u}}f_0 [/tex]

    [tex] \frac{f}{f_0} = \sqrt{\frac{c + u}{c - u}} [/tex]

    [tex] (\frac{f}{f_0})^2 = \frac{c + u}{c - u} [/tex]

    [tex] (c - u)(\frac{f}{f_0})^2 = c + u [/tex]

    But I am not sure where to go from here.

    Any suggestions?

    TFM
     
  2. jcsd
  3. May 7, 2008 #2

    Doc Al

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    Staff: Mentor

    You're doing fine; don't stop now.

    [tex] c(\frac{f}{f_0})^2 - u(\frac{f}{f_0})^2 = c + u [/tex]

    Next: Move all the terms containing u to one side.
     
  4. May 7, 2008 #3

    TFM

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    Would that give:

    [tex] c(\frac{f}{f_0})^2 - c = u(\frac{f}{f_0})^2 + u[/tex]

    TFM
     
  5. May 7, 2008 #4

    TFM

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    Do uou now factor out the u:

    [tex] c(\frac{f}{f_0})^2 = u ((\frac{f}{f_0})^2 + 1) [/tex]

    Tnhe divide over to get:

    [tex] u = \frac{c(\frac{f}{f_0})^2 - c}{(\frac{f}{f_0})^2 + 1} [/tex]

    Does this look right?

    TFM
     
  6. May 7, 2008 #5

    Doc Al

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    Looks good! (I would factor out the c to make it more readable.)
     
  7. May 7, 2008 #6

    TFM

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    That would give:

    [tex]u = \frac c((\frac{f}{f_0})^2) - 1}{(\frac{f}{f_0})^2 + 1}[/tex]


    What relative velocity u will produce a 3.0 % decrease in frequency

    For the actual question, do I put f_0 as 1 and f as 0.97?

    TFM
     
  8. May 7, 2008 #7

    Doc Al

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    :confused: Redo this. (Probably a formatting error.)
     
  9. May 7, 2008 #8

    TFM

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    That definately isn't right :

    [tex]u = \frac{c((\frac{f}{f_0})^2 - 1)}{(\frac{f}{f_0})^2 + 1}[/tex]

    TFM
     
  10. May 7, 2008 #9

    Doc Al

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    Looks good.
     
  11. May 7, 2008 #10

    TFM

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    The actual question itself is:

    What relative velocity u will produce a 3.0 % decrease in frequency

    For this, should I put f_0 as 1 and f as 0.97?

    TFM
     
  12. May 7, 2008 #11

    Doc Al

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    Makes sense to me.
     
  13. May 7, 2008 #12

    TFM

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    It gives, the right answer, Thanks

    One last thing, what is the question actually asking to find:

    What relative velocity u will produce an increase by a factor of 3 of the observed light?

    Does it want the freuqncy to be incresed by three factors?

    TFM
     
  14. May 7, 2008 #13

    Doc Al

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    I'm no more a mind reader than you are! :smile: But since the problem seems to be talking about frequency and Doppler shifts, I would assume they mean that the observed frequency is three times the original.
     
  15. May 7, 2008 #14

    TFM

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    I calculated it to be 0.8, which is the answer, so I should say it is, as well.

    Thats the second poorly worded question I've had this week:rolleyes:

    Thanks for all the help,

    TFM
     
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