# Methods of Proofs

1. May 4, 2010

### 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!

2. May 4, 2010

### 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.

3. May 4, 2010

### uart

You wont 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.

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

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

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

Last edited: May 4, 2010
4. May 4, 2010

### 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.

Last edited: May 4, 2010