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Question
An operator sitting in his base camp sends a sound signal of frequency 400 Hz. The signal is reflected back from a car moving towards him. The frequency of the reflected sound is found to be 410 Hz. Find the speed of the car. Speed of sound in air = 324 m s−1
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Solution
Given:
Velocity of sound in air v = 324 ms−1
Frequency of sound sent by source \[n_0\]= 400 Hz
Let the speed of the car be x m/s.
The frequency of sound heard at the car n is given by :
\[n = \frac{v + v_{car}}{v} \times n_0 \]
\[ \Rightarrow n = \frac{324 + x}{324} \times 400 . . . . . \left( 1 \right)\]
If \[n_1\] is the frequency of sound heard by the operator, then its value is given by :
\[n_1 = \frac{324}{324 - x} \times n\]
\[410 = \frac{324}{324 - x} \times n\]
On substituting the value of n from equation (1), we have :
\[410 = \frac{324}{\left( 324 - x \right)} \times \frac{\left( 324 + x \right)}{324} \times 400\]
\[ \Rightarrow 410 = \left( \frac{324 + x}{324 - x} \right) \times 400\]
\[ \Rightarrow 410 \left( 324 - x \right) = 400\left( 324 + x \right)\]
\[ \Rightarrow 324 \left( 410 - 400 \right) = 810x\]
\[ \Rightarrow x = 4 \text{ m/s }\]
The speed of the car is 4 m/s.
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