# Rearranging a Relativity Equation

• TFM
In summary, the question is asking for the relative velocity between the electromagnetic source and an observer in terms of the ratio of the observed frequency and the source frequency of light. By rearranging the equation, you can find the relative velocity u between the two.

#### 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

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

Would that give:

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

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

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

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

TFM said:
That would give:

$$u = \frac c((\frac{f}{f_0})^2) - 1}{(\frac{f}{f_0})^2 + 1}$$
Redo this. (Probably a formatting error.)

That definitely isn't right :

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

TFM

Looks good.

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

TFM said:
For this, should I put f_0 as 1 and f as 0.97?
Makes sense to me.

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

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.

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