Question

Wavelengths of light from a distant galaxy are found to be 0.50% longer than the corresponding wavelengths measured in a terrestrial laboratory. Is the galaxy approaching or receding from the earth? At what speed?

Short answer

a) The signal would take 1.8 secs to reach the home planet

b) The wavelength observed by the home planet is 488.3 m

Step-by-step solution

Write the given information

The speed of the spaceship is v = 0.10c

Here, c is the speed of light (3 x10⁸ m/s)

The distance between a spaceship and the planet is d= 54x10⁶ km

The wavelength of the laser signal used by the spaceship λ= 540m

(a) To determine the time taken by the signal to travel to the planet

The time taken by the laser signal to cover the given distance is given by$$\begin{gathered} T=\frac{dis\tan ce}{speed} \\ T=\frac{54\times {10}^6}{0.1(3\times {10}^8)}=1.8sec \end{gathered}$$

Thus, the signal would take 1.8 secs to reach the home planet.

(b) To determine the wavelength of the signal detected by the home planet

The wavelength observed by the home planet is given by

$\lambda \text{'}=\lambda \sqrt{\frac{c-v}{c+v}}$

Substitute the known value

$$\begin{gathered} \lambda \text{'}=(540nm)\sqrt{\frac{c-0.1c}{c+0.1c}} \\ \lambda \text{'}=488.3nm \end{gathered}$$

Thus the wavelength observed by the home planet is 488.3 m