Proving EM Doppler Shift Ratio: v/c = (r^2 - 1) / (r^2 +1)

[SkovBriscombe]

Hi there,

I have some exams later this month, and some of the previous exam questions are to prove a formula given another formula fx here with EM doppler shift:

define ratio: r= f/ f0
using relativistic doppler frequency for EM: f = square root of: ((c+v) / (c-v)) * f0

Show:

v/c = (r^2 - 1) / (r^2 +1)


Are there any general methods or ways to go about such a question as there are quite a few of them and i find it hard to know where to start, i usually try and rearrange and substitute into each other using the equations given, but never seem to get them right... Please help me!

 

[CompuChip]

So if you plug the equation for f into the equation for r, you will get a direct relation between r and f.

To show the identity, it is probably easiest to substitute for the variable which you have isolated, i.e. calculate (r² - 1) / (r² + 1) and show that you get v/c.
That is usually easier than trying to rework the equation for r to an equation for v.

 

[uart]

Hi there,

I have some exams later this month, and some of the previous exam questions are to prove a formula given another formula fx here with EM doppler shift:

define ratio: r= f/ f0
using relativistic doppler frequency for EM: f = square root of: ((c+v) / (c-v)) * f0

Show:

v/c = (r^2 - 1) / (r^2 +1)

You won't be able to show that equation because it's wrong. In general though the answer to proving something like that is just algebra, algebra and practice.

You've got ((c+v) / (c-v)) = r and you want to find v/c, so start by dividing num and denom on the LHS by c. This gives you,

$$\frac{1+v/c}{1-v/c} = r$$

Straight away it looks much easier to handle, you've now got an equation with just got one variable (v/c) to isolate. From this point onward we will keep all "v/c" terms together as if they were just one variable.

So now just mulitply by (1-v/c) and collect the v/c terms.

$$1+v/c =r - r v/c$$

$$(1+r) v/c =r - 1$$

$$v/c = \frac{r-1}{r+1}$$

 

[uart]

BTW. I should add. This is a maths question pure and simple. The equation chosen was motivated by physics but this is not really a physics question.