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Rearranging a Relativity Equation

  • Thread starter TFM
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  • #1
TFM
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[SOLVED] Rearranging a Relativity Equation

Homework Statement



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

Homework Equations



Rearangement of:

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

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
 

Answers and Replies

  • #2
Doc Al
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[tex] (c - u)(\frac{f}{f_0})^2 = c + u [/tex]
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.
 
  • #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
 
  • #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
 
  • #5
Doc Al
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Looks good! (I would factor out the c to make it more readable.)
 
  • #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
 
  • #7
Doc Al
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44,882
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That would give:

[tex]u = \frac c((\frac{f}{f_0})^2) - 1}{(\frac{f}{f_0})^2 + 1}[/tex]
:confused: Redo this. (Probably a formatting error.)
 
  • #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
 
  • #9
Doc Al
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44,882
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Looks good.
 
  • #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
 
  • #11
Doc Al
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For this, should I put f_0 as 1 and f as 0.97?
Makes sense to me.
 
  • #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
 
  • #13
Doc Al
Mentor
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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.
 
  • #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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