Question

An observer is moving with half the speed of light towards a stationary microwave source emitting waves at frequency $10\text{GHz}$. What is the frequency in GHz of the microwave measured by the observer?
(speed of light $=3\times {10}^8{\text{ms}}^{-1}$). Answer the nearest integer.

Answer 1

Speed of observer $V=\frac{C}{2}$
where speed of Microwave is $C$
Original frequency of source is given as $f_0=10\text{GHz}$
Using relativistic Doppler effect in case observer moves towards the stationary source.
Apparent frequency measured by observer $f^{\prime }=f_0\sqrt{\frac{C+v}{C-v}}$
$f^{\prime }=10\sqrt{\frac{C+\left(C/2\right)}{C-\left(C/2\right)}}$
$f^{\prime }=10\sqrt{\frac{\left(3/2\right)C}{\left(1/2\right)C}}$
$F^{\prime }=10\times 1.73\text{GHz}$
$F^{\prime }=17.3\text{GHz}$