# Rearranging redshift formula for v

• Pretty Pony
In summary, the conversation involves rearranging the formula for z in terms of v. The steps to solve the equation are outlined, including adding 1 to both sides, multiplying by the denominator, and squaring both sides. The resulting quadratic equation can then be solved using the quadratic formula.
Pretty Pony
Hey there
I would just like some help in rearranging the formula:
$$z=\frac{1+\frac{v\cos \theta}{c}}{\sqrt{1-\frac{v^2}{c^2}}}-1 [\tex] for v. Thanks, PP That is [tex]z= \frac{1+ \frac{v cos(\theta)}{c}}{\sqrt{1- \frac{v^2}{c^2}}}- 1$$

You solve an equation by "undoing" what was done to the unknown, step by step. The last thing done there is "-1" so we undo that by adding 1 to both sides:
$$z+ 1= \frac{1+ \frac{v cos(\theta)}{c}}{\sqrt{1- \frac{v^2}{c^2}}}$$

Now, multiply on both sides by the denominator:
$$(z+1)\left(\sqrt{1- \frac{v^2}{c^2}}\right)= 1+ \frac{v cos(\theta)}{c}$$

Get rid of that square root by squaring both sides (which might introduce false solution- you will need to check for that after finding v):
$$(z+1)^2\left(1- \frac{v^2}{c^2}\right)= 1+ 2\frac{v cos(\theta)}{c}+ \frac{v^2 cos^2(\theta)}{c^2}$$

Now that is a quadratic equation in v. Combining like terms,
$$\left(\frac{1}{cos^2(\theta)}+ \frac{(z+1)^2}{c^2}\right)v^2+ \frac{2vcos(\theta)}{c}v+ 1- (z+1)^2= 0$$

which can be solved using the quadratic formula.

:)

## 1. How do I rearrange the redshift formula for v?

The redshift formula for v can be rearranged by dividing both sides of the equation by z and taking the square root of both sides. This will result in the equation v = c * (z + 1) / (z - 1), where c is the speed of light.

## 2. What is the significance of rearranging the redshift formula for v?

Rearranging the redshift formula for v allows us to calculate the velocity of an object relative to the observer based on its redshift value. This can provide valuable information about the object's motion and distance.

## 3. Can the redshift formula for v be rearranged for other variables?

Yes, the redshift formula can be rearranged for other variables such as the redshift value or the speed of light. This can be useful in different scenarios depending on the information that is known or being sought.

## 4. Is there a specific method for rearranging the redshift formula for v?

Yes, the specific method for rearranging the redshift formula for v is to use algebraic principles to isolate the variable v on one side of the equation. This involves performing the same operation on both sides of the equation, such as dividing by z, until the desired variable is alone on one side.

## 5. Can the redshift formula for v be applied to all celestial objects?

Yes, the redshift formula for v can be applied to all celestial objects that exhibit redshift, which is the majority of distant objects in the universe. However, it may not accurately represent the actual velocity of an object if it is affected by other factors such as gravitational lensing.

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