Chapter 33, Problem 27P

To determine

The speed of a galaxy.

Answer to Problem 27P

The speed of a galaxy with $z=0.60$ is $0.44c$ .

Explanation of Solution

Given:

The value of $z=0.60$ .

Formula used:

The expression for red shift parameter zis

  $z=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}-1$

Here, v is the speed of the galaxy and c is the speed of light.

Calculation:

Assume the speed of the light is denoted by $c$ .

The speed of the galaxy with $z=0.60$ is,

  $\begin{array}{l}z=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}-1\\ z+1=\sqrt{\frac{\frac{c+v}{c}}{\frac{c-v}{c}}}\\ z+1=\sqrt{\frac{c+v}{c-v}}\\ {\left(z+1\right)}^2=\left(\frac{c+v}{c-v}\right)\end{array}$

Substitute $0.60$ for z .

  $\begin{array}{l}{\left(0.60+1\right)}^2=\left(\frac{c+v}{c-v}\right)\\ 2.56=\left(\frac{c+v}{c-v}\right)\\ 2.56\left(c-v\right)=\left(c+v\right)\\ 2.56c-2.56v=c+v\\ 1.56c=3.56v\\ v=\left(\frac{1.56c}{3.56}\right)\\ v=0.44c\end{array}$

Conclusion:

Thus, the speed of a galaxy with $z=0.60$ is $0.44c$ .