Rearranging a Relativity Equation

[TFM]

[SOLVED] Rearranging a Relativity Equation

Homework Statement

Rewrite equation:

$$f = \sqrt{\frac{c + u}{c - u}}f_0$$

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:

$$f = \sqrt{\frac{c + u}{c - u}}f_0$$

The Attempt at a Solution

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

$$f = \sqrt{\frac{c + u}{c - u}}f_0$$

$$\frac{f}{f_0} = \sqrt{\frac{c + u}{c - u}}$$

$$(\frac{f}{f_0})^2 = \frac{c + u}{c - u}$$

$$(c - u)(\frac{f}{f_0})^2 = c + u$$

But I am not sure where to go from here.

Any suggestions?

TFM

 

[Doc Al]

$$(c - u)(\frac{f}{f_0})^2 = c + u$$

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

$$c(\frac{f}{f_0})^2 - u(\frac{f}{f_0})^2 = c + u$$

Next: Move all the terms containing u to one side.

 

[TFM]

Would that give:

$$c(\frac{f}{f_0})^2 - c = u(\frac{f}{f_0})^2 + u$$

TFM

 

[TFM]

Do uou now factor out the u:

$$c(\frac{f}{f_0})^2 = u ((\frac{f}{f_0})^2 + 1)$$

Tnhe divide over to get:

$$u = \frac{c(\frac{f}{f_0})^2 - c}{(\frac{f}{f_0})^2 + 1}$$

Does this look right?

TFM

 

[Doc Al]

Looks good! (I would factor out the c to make it more readable.)

 

[TFM]

That would give:

$$u = \frac c((\frac{f}{f_0})^2) - 1}{(\frac{f}{f_0})^2 + 1}$$


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

 

[Doc Al]

That would give:

$$u = \frac c((\frac{f}{f_0})^2) - 1}{(\frac{f}{f_0})^2 + 1}$$

Redo this. (Probably a formatting error.)

 

[TFM]

That definitely isn't right :

$$u = \frac{c((\frac{f}{f_0})^2 - 1)}{(\frac{f}{f_0})^2 + 1}$$

TFM

 

[Doc Al]

Looks good.

 

[TFM]

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

 

[Doc Al]

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

Makes sense to me.

 

[TFM]

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

 

[Doc Al]

I'm no more a mind reader than you are! 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.

 

[TFM]

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

Thanks for all the help,

TFM