# Using the approximation formula for the doppler shift, fo=fs(1+Vrel/c), one finds that a source is moving aways at 0.60c. What result would the exact formula yield for the recession speed?

I have not idea how to figure this one out. Help please.

http://en.wikipedia.org/wiki/Relativistic_Doppler_effect

## To find the result using the exact formula for the Doppler shift, we can use the equation:

fo = fs * (sqrt((1 + Vrel/c) / (1 - Vrel/c)))

In this case, we know the source is moving away at a speed of 0.60c. Let's substitute this value into the equation and solve for Vrel:

0.60c = sqrt((1 + Vrel/c) / (1 - Vrel/c))

To simplify the equation, we can square both sides:

(0.60c)^2 = (1 + Vrel/c) / (1 - Vrel/c)

Simplifying further by multiplying both sides by (1 - Vrel/c):

(0.60c)^2 * (1 - Vrel/c) = (1 + Vrel/c)

Expanding the left side and simplifying:

0.36c^2 - 0.60Vrelc + Vrel^2 = 1 + Vrel/c

Rearranging the terms:

0.36c^2 - 1 = Vrel (1 - 0.60c) - Vrel^2

Simplifying:

0.36c^2 - 1 = Vrel - 0.60Vrelc - Vrel^2

Rearranging terms and combining like terms:

Vrel^2 - (0.60c + 1)Vrel + (0.36c^2 - 1) = 0

This is a quadratic equation in Vrel. We can solve it using the quadratic formula:

Vrel = [-(0.60c + 1) ± sqrt((0.60c + 1)^2 - 4(0.36c^2 - 1))] / 2

Simplifying further:

Vrel = [-(0.60c + 1) ± sqrt(0.36c^2 + 1.20c + 1 - 1.44c^2 + 4)] / 2

Vrel = [-(0.60c + 1) ± sqrt(-0.44c^2 + 5.20c + 5)] / 2

From this equation, we can use numerical methods to approximate the value of Vrel. The exact result will depend on the value of c, which is the speed of light. For most practical purposes, we can assume c to be approximately 3 x 10^8 meters per second.

Remember to keep in mind that this calculation assumes you have the tools to solve quadratic equations and access to numerical methods if you need to approximate the result.

## To find the result for the recession speed using the exact formula for the Doppler shift, we can start by rearranging the formula to solve for the relative velocity (Vrel):

fo = fs(1 + Vrel/c)

Rearranging the equation, we get:

fo/fs = 1 + Vrel/c

Next, we can substitute the given values into the equation. In this case, the given value is that the source is moving away at 0.60c:

fo/fs = 1 + 0.60c/c
fo/fs = 1 + 0.60

Simplifying further:

fo/fs = 1.60

Now, let's solve for the recession speed by rearranging the equation again:

Vrel/c = fo/fs - 1

Substituting the given value:

Vrel/c = 1.6 - 1
Vrel/c = 0.6

Finally, to find the recession speed Vrel, we multiply both sides by the speed of light (c):

Vrel = 0.6 * c

Therefore, the exact formula for the recession speed yields a result of 0.6c.