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

Answer 1

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:
$$\begin{gathered} n=\frac{v+v_{car}}{v}\times n_0 \\ \Rightarrow n=\frac{324+x}{324}\times 400.....\left(1\right) \end{gathered}$$
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:

$$\begin{gathered} 410=\frac{324}{\left(324-x\right)}\times \frac{\left(324+x\right)}{324}\times 400 \\ \Rightarrow 410=\left(\frac{324+x}{324-x}\right)\times 400 \\ \Rightarrow 410\left(324-x\right)=400\left(324+x\right) \\ \Rightarrow 324\left(410-400\right)=810x \\ \Rightarrow x=4\mathrm{m}/\mathrm{s} \end{gathered}$$

The speed of the car is 4 m/s.