Answer to Question #132133 in Mechanics | Relativity for Max

Question

a source is moving toward observer with velocity c/2 and observer is also moving in same direction with velocity c/3 if the actual frequency of light emmited by source is v0 what is the apperant frequency of light measured by observer

Answer 1

$\nu = \nu_0 \frac{\sqrt{1-\frac{v^2}{c^2}}}{1-\frac{v}{c}cos\theta}$

where $v$ is a velocity of source relative to the observer, $\theta$ is the angle between directions of source and observer. When source is behind the observer, $\theta=0$; when the source will be ahead, $\theta=\pi$.

To find relative velocity, we use relativistic velocity-addition formula:

$v = \frac{u_2-u_1}{1-\frac{u_1u_2}{c^2}}= \frac{c/2-c/3}{1-1/6}= \frac{c}{5}$

Before the moment of meeting (source is coming closer):

$\nu = \nu_0 \frac{\sqrt{1-\frac{1}{5^2}}}{1-\frac{1}{5}}= \nu_0 \frac{\sqrt{24}}{5}\frac{5}{4} = 1.225 \nu_0$

After the moment of meeting (source becomes more and more distant from the observer):

$\nu = \nu_0 \frac{\sqrt{1-\frac{1}{5^2}}}{1+\frac{1}{5}}= \nu_0 \frac{\sqrt{24}}{5}\frac{5}{6} = 0.816\nu_0$