Question

A police car with a siren of frequency $9\text{kHz}$ is moving with uniform velocity $75\text{km/hr}$ towards a tall building which reflect the sound wave. The speed of sound in air is $340\text{m/s}$. The frequency of siren heard by the car driver is

• A. $8\text{kHz}$
• B. $10\text{kHz}$
• C. $2\text{kHz}$
• D. $14\text{kHz}$

Answer 1

The correct option is B $10\text{kHz}$

Velocity of sound $\left(v\right)=340\text{m/s}$

Velocity of car $\left(v_{\text{s}}\right)=75\times \frac{1000}{3600}=20.83\approx 21\text{m/s}$

Frequency generated by given car siren $\left(f_{\text{o}}\right)=9\text{kHz}$

Let $f_{\text{b}}$ be the frequency observed by building,
Velocity of observer $\left(v_{\text{o}}\right)=0\text{m/s}$

Applying Doppler effect,

$f_{\text{b}}=f_{\text{o}}\left(\frac{v\pm v_{\text{o}}}{v\pm v_{\text{s}}}\right)$

$\Rightarrow f_{\text{b}}=f_{\text{o}}\left(\frac{v}{v-v_{\text{s}}}\right)$$=9\times \frac{340}{340-21}=\frac{9\times 340}{319}\text{kHz}$

Frequency of reflected wave by building,

$f_{\text{b}}=9\times \frac{340}{319}\text{kHz}$

Now let $f_{\text{app}}$ be the frequency heard by police or also known as an apparent frequency.

Doppler's effect for given value,

Here velocity of source $\left(v_{\text{s}}\right)=0$

Velocity of observer $\left(v_{\text{o}}\right)=21\text{m/s}$

$f_{\text{app}}=f_{\text{b}}\left(\frac{v\pm v_{\text{o}}}{v\pm v_{\text{s}}}\right)$$f_{\text{app}}=9\times \frac{340}{319}\times \frac{340+21}{340}$

$f_{\text{app}}=10.184\text{kHz}\approx 10\text{kHz}$

Caution: Be careful, when taking the sign convention in Doppler effect.