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A highway truck has two horns $A$ and $B$. When sounded together, the driver records 50 beats in 10 seconds. With horn $B$ blowing and the truck moving towards a wall at a speed of $10 \mathrm{~m} / \mathrm{s}$, the driver noticed a beat frequency of $5 \mathrm{~Hz}$ with the echo. When frequency of $A$ is decreased the beat frequency with two horns sounded together increases. Calculate the frequency of horn $A$. (Speed of sound in air $=330 \mathrm{~m} / \mathrm{s}$ )
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Verified Answer
The correct answer is:
$75 \mathrm{~Hz}$
Here, Iet frequencies of hom $A$ and $B$ are $n_A$ and $n_B$, respectively.
As given,
$$
\begin{aligned}
& n_A-n_B=\frac{50}{10} \\
& n_A-n_B=5 \text { beats }
\end{aligned}
$$
When, the hom $B$ is blown while moving towards the wall the apparent frequency,
$$
n_B^{\prime}=n_B \frac{v+v_0}{v-v_s}
$$
Since, both the observer and source are moving with truck.
So,
$$
\begin{aligned}
& \text { So, } \quad v_s=v_0=10 \mathrm{~m} / \mathrm{s} \\
& n_B{ }^{\prime}=n_B\left(\frac{330+10}{330-10}\right) \\
& n_B{ }^{\prime}=n_B 1.0625 \\
&=\quad n_B{ }^{\prime}-n_B=0.0625 n_B
\end{aligned}
$$
As given, $n_B^{\prime}-n_B=5$
$$
\begin{aligned}
0.0625 n_B & =5 \\
n_B & =80 \mathrm{~Hz}
\end{aligned}
$$
$\because$ Given, that if frequency of $A$ is decreased then beat frequency is increases.
So,
$$
\begin{aligned}
& \therefore \quad n_A=n_B-5=80-5 \\
& \Rightarrow \quad n_A=75 \mathrm{~Hz} \\
&
\end{aligned}
$$
As given,
$$
\begin{aligned}
& n_A-n_B=\frac{50}{10} \\
& n_A-n_B=5 \text { beats }
\end{aligned}
$$
When, the hom $B$ is blown while moving towards the wall the apparent frequency,
$$
n_B^{\prime}=n_B \frac{v+v_0}{v-v_s}
$$
Since, both the observer and source are moving with truck.
So,
$$
\begin{aligned}
& \text { So, } \quad v_s=v_0=10 \mathrm{~m} / \mathrm{s} \\
& n_B{ }^{\prime}=n_B\left(\frac{330+10}{330-10}\right) \\
& n_B{ }^{\prime}=n_B 1.0625 \\
&=\quad n_B{ }^{\prime}-n_B=0.0625 n_B
\end{aligned}
$$
As given, $n_B^{\prime}-n_B=5$
$$
\begin{aligned}
0.0625 n_B & =5 \\
n_B & =80 \mathrm{~Hz}
\end{aligned}
$$
$\because$ Given, that if frequency of $A$ is decreased then beat frequency is increases.
So,
$$
\begin{aligned}
& \therefore \quad n_A=n_B-5=80-5 \\
& \Rightarrow \quad n_A=75 \mathrm{~Hz} \\
&
\end{aligned}
$$
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