Question

There are two horns $H_1$ and $H_2$ on a car. When sounded together, the driver records $35$ beats in $10$ seconds. With horn $H_2$ blowing and car moving towards a wall at a speed of $5{\text{ms}}^{-1}$, the driver noticed a beat frequency of $5\text{Hz}$ with the echo. When frequency of $H_1$ is decreased, the beat frequency with two horns sounded together increases. Calculate the frequencies of the two horns $H_1$ and $H_2$ respectively.

[Speed of sound = $332{\text{ms}}^{-1}$]

• A. $167\text{Hz},170.5\text{Hz}$
• B. $167\text{Hz},160\text{Hz}$
• C. $160\text{Hz},163.5\text{Hz}$
• D. $170.5\text{Hz},167\text{Hz}$

Answer 1

The correct option is C $160\text{Hz},163.5\text{Hz}$

Given,
$f_1=f_2\pm 3.5.......\left(1\right)$
Velocity of sound, $v=332\text{m/s}$
Velocity of source (car), $v_s=5\text{m/s}$



Consider wall as observer. Frequency received by the wall when only the second horn is sounded, is

$f^{\prime }=f_2\left(\frac{v}{v-v_s}\right)=f_2\left(\frac{332}{332-5}\right)$

Now wall acts as a source of frequency $f^{\prime }$. Frequency of echo received by the observer (in car) is

$f^{\prime \prime }=f^{\prime }\left(\frac{332+5}{332}\right)$

$\Rightarrow f^{\prime \prime }=f_2\left(\frac{332+5}{332-5}\right)=f_2\left(\frac{337}{327}\right)$

Given,

$f^{\prime \prime }-f_2=5$

$\therefore f_2[\frac{337}{327}-1]=5$

$\Rightarrow f_2=\frac{327}{2}=163.5\text{Hz}$

From $\left(1\right)$,

$f_1=163.5\pm 3.5=167\text{Hz}\text{or}160\text{Hz}$

Given, when frequency of $H_1$ decreases, it gives more beats with $H_2$.
$\Rightarrow f_1<f_2$ initially

$\therefore f_1=f_2-3.5=160\text{Hz}$