Question

Two sources of sound $S_1$​ and $S_2$ produce sound waves of the same frequency $660Hz$. A listener is moving from source $S_1$​ towards $S_2$​ with a constant speed $um/s$ and he hears $10beats/s$. The velocity of sound is $330m/s$. Then, $u$ equals:

• A. $5.5m/s$
• B. $15m/s$
• C. $2.5m/s$
• D. $10m/s$

Answer 1

The correct option is C

$2.5m/s$

Step 1: Given

Frequency of sound: $f=660Hz$

Velocity of sound: $v=330m/s$

Velocity of the listener: $um/s$

Difference in frequencies: $f_2-f_1=10beats/s$

Step 2: Formula Used

Apparent frequency as observer and source go away from each other: $f\text{'}=f\left(\frac{v-v_O}{v}\right)$

Apparent frequency as observer and source go towards each other: $f\text{'}=f\left(\frac{v+v_O}{v}\right)$

Step 3: Find the value of u

Calculate the apparent frequency when observer and source go away from each other by using the formula

$\begin{array}{rcl}{f}_1& =& f\left(\frac{v-v_O}{v}\right)\\ & =& 660\left(\frac{330-u}{330}\right)\\ & =& 2\left(330-u\right)\\ & =& 660-2u\end{array}$

Calculate the apparent frequency when observer and source go towards each other by using the formula

$\begin{array}{rcl}{f}_2& =& f\left(\frac{v+v_O}{v}\right)\\ & =& 660\left(\frac{330+u}{330}\right)\\ & =& 2\left(330+u\right)\\ & =& 660+2u\end{array}$

Calculate the value of u by taking the difference of the frequencies and substituting values

$$\begin{gathered} f_2-f_1=10 \\ \Rightarrow \left(660+2u\right)-\left(660-2u\right)=10 \\ \Rightarrow 4u=10 \\ \Rightarrow u=2.5m/s \end{gathered}$$

Hence, the value of the velocity of the observer is $u=2.5m/s$.