Question

A stationary source is emitting sound at a fixed frequency $f_0$, which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is $1.2\%$ of $f_0$. What is the difference in the speeds of the cars (in km per hour) to the nearest integer? The cars are moving at constant speeds much smaller than the speed of sound which is $330m{s}^{–1}$.

Answer 1

Frequency of sound reflected by the car is

$f=\left(\frac{v+v_c}{v-v_c}\right)f_0$
As $v_c<<v\Rightarrow f=\left(\frac{1+\frac{v_c}{v}}{1-\frac{v_c}{v}}\right)f_0=(1+\frac{2v_c}{v})f_0$
$\Delta f=\frac{-2f_0}{v}\Delta v_c$
$\frac{1.2}{100}{f}_0=\frac{-2f_0}{v}\Delta v_c$
Difference $|\Delta v_c|=\frac{1.2}{100}\times \frac{300}{2}m/s$
$=\frac{1.2}{100}\times \frac{330}{2}\times \frac{18}{5}\approx 7km/h$