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.

 

[pmacias]

Hello there,

Proving formulas in physics can be challenging, but there are some general steps you can follow to help guide you through the process. Here are some suggestions for proving the EM Doppler Shift Ratio formula:

1. Start by writing out the given formula for r in terms of f and f0. This will give you an expression for r in terms of v and c.

2. Next, use the given relativistic Doppler frequency formula to substitute for f in your expression for r. This will give you an expression for r in terms of v, c, and f0.

3. Now, you can use algebraic manipulations to simplify the expression for r. This might involve factoring, expanding, or combining like terms.

4. Once you have a simplified expression for r, you can use it to solve for v/c. This will involve rearranging the equation and using algebraic operations to isolate v/c on one side of the equation.

5. Finally, you can compare your expression for v/c to the given formula for v/c = (r^2 - 1) / (r^2 +1). If they are equal, then you have successfully proven the formula.

Remember to carefully follow the rules of algebra and to be thorough in your calculations. Also, don't get discouraged if you don't get the correct answer on your first try. Keep practicing and seeking help if needed. Good luck on your exams!