Chapter 16, Problem 54A

To determine

The speed with which the galaxy is moving away from Earth.

Answer to Problem 54A

  $1.95\times {10}^7\text{m/s}$

Explanation of Solution

Given:

The actual wavelength of the light from hydrogen, $\lambda =434\text{nm}$

The observed wavelength of the light from hydrogen, ${\lambda }_{\text{obs}}=1.065\lambda$

Formula used:

Doppler effect: The difference between the observed $\left({\lambda }_{\text{obs}}\right)$ and the actual wavelength $\left(\lambda \right)$ of the light,

  ${\lambda }_{\text{obs}}-\lambda =\pm \frac{v}{c}\lambda$

Where $c$ is the speed of light, and $v$ is the relative velocity of the observer with respect to the source.

Now, if the observer and the source are moving away from each other, then the observed wavelength is,

  ${\lambda }_{\text{obs}}-\lambda =\frac{v}{c}\lambda$

And, if the observer and the source are approaching each other, then the observed wavelength is,

  ${\lambda }_{\text{obs}}-\lambda =-\frac{v}{c}\lambda$

Calculation:

The speed of light is known as, $c=3\times {10}^8\text{}\text{m/s}$

Thus for the given case, using the given values in above formula,

  $\begin{array}{c}1.065\lambda -\lambda =\left(\frac{v}{3\times {10}^8}\right)\lambda \\ 0.065\times \lambda =\frac{v}{3\times {10}^8}\lambda \\ v=1.95\times {10}^7\text{m/s}\end{array}$

Conclusion:

Hence, the velocity with which the galaxy moves away from the Earth is $1.95\times {10}^7\text{m/s}$ .