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A police car moving at \(22 \mathrm{~ms}^{-1}\) chases a motor cyclist. The police man sounds horn at \(176 \mathrm{~Hz}\), while both of them move towards a stationary siren of frequency \(165 \mathrm{~Hz}\). If the number of beats heard by the motor cyclist per second is zero, then the speed of motorcycle is ( Speed of sound in air \(=330 \mathrm{~ms}^{-1}\))
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Verified Answer
The correct answer is:
\(22 \mathrm{~ms}^{-1}\)
According to the question, we draw the following situation,
Given, velocity of car, \(v_{s_1}=22 \mathrm{~ms}^{-1}\)
speed of the sound in air, \(v=330 \mathrm{~ms}^{-1}\)
apparent frequency by cyclist from the police car,
\(\begin{aligned}
f_1^{\prime} & =\frac{v-v_0}{v-v_s} f_1 \Rightarrow f_1^{\prime}=\frac{330-v_0}{330-22} \times 176 \\
& =\frac{330-v_0}{308} \times 176
\end{aligned}\)
apparent frequency by cyclist from the stationary siren,
\(f_2^{\prime}=\frac{v+v_0}{v} f_2=\frac{330+v_0}{330} \times 165\)
Number of beats heard by the cyclist,
\(\begin{aligned}
& n=f_1^{\prime}-f_2^{\prime}=0 \text { (Given, } n=0 \text { ) } \\
& f_1^{\prime}=f_2^{\prime} \\
& \Rightarrow \quad\left(330-v_0\right) \frac{176}{308}=\left(330+v_0\right) \frac{165}{330} \\
& \Rightarrow \quad v_0=22 \mathrm{~ms}^{-1}
\end{aligned}\)
\(\therefore\) Hence, the speed of motorcycle is \(22 \mathrm{~ms}^{-1}\). So, the correct option is (b).
Given, velocity of car, \(v_{s_1}=22 \mathrm{~ms}^{-1}\)
speed of the sound in air, \(v=330 \mathrm{~ms}^{-1}\)
apparent frequency by cyclist from the police car,
\(\begin{aligned}
f_1^{\prime} & =\frac{v-v_0}{v-v_s} f_1 \Rightarrow f_1^{\prime}=\frac{330-v_0}{330-22} \times 176 \\
& =\frac{330-v_0}{308} \times 176
\end{aligned}\)
apparent frequency by cyclist from the stationary siren,
\(f_2^{\prime}=\frac{v+v_0}{v} f_2=\frac{330+v_0}{330} \times 165\)
Number of beats heard by the cyclist,
\(\begin{aligned}
& n=f_1^{\prime}-f_2^{\prime}=0 \text { (Given, } n=0 \text { ) } \\
& f_1^{\prime}=f_2^{\prime} \\
& \Rightarrow \quad\left(330-v_0\right) \frac{176}{308}=\left(330+v_0\right) \frac{165}{330} \\
& \Rightarrow \quad v_0=22 \mathrm{~ms}^{-1}
\end{aligned}\)
\(\therefore\) Hence, the speed of motorcycle is \(22 \mathrm{~ms}^{-1}\). So, the correct option is (b).
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