Chapter 16, Problem 9STP

To determine

To identify: Whether the object is approaching or receding.

To find: The speed of the object.

Answer to Problem 9STP

The object is approaching with speed $2.97\times {10}^7\text{m/s}$ .

Explanation of Solution

Given:

Wavelength of light, $\lambda =525\text{nm}$

Position of spectral line, ${\lambda }_{obs}=473\text{nm}$

Formula used:

Using Doppler shift formula,

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

Where,

  $c$ is the speed of the light

  $v$ is the speed of the source

  ${\lambda }_{obs}$ is the observed wavelength

  $\lambda$ is the actual wavelength

Calculation:

Difference between the actual and observed wavelength is:

  $\begin{array}{l}\Delta \lambda ={\lambda }_{obs}-\lambda \\ \Delta \lambda =473-525\\ \Delta \lambda =-52\text{nm}\end{array}$

Since the difference is negative. Therefore, the object is approaching the source.

Speed of the object can be calculated as:

  $\begin{array}{l}{\lambda }_{obs}-\lambda =-\frac{v}{c}\lambda \\ -52\times {10}^{-9}=-\frac{v}{3\times {10}^8}\times 525\times {10}^{-9}\\ v=\frac{52}{525}\left(3\times {10}^8\right)\\ v=0.297\times {10}^8\\ v=2.97\times {10}^7\text{m/s}\end{array}$

Conclusion:

The object with velocity $v=2.97\times {10}^7\text{m/s}$ approaches the source.