Question

A submarine $\left(A\right)$ travelling at $18{\text{Km hr}}^{-1}$ is being chased along the line of its velocity by another submarine $\left(B\right)$ travelling at $27{\text{Km hr}}^{-1}$. $B$ sends a sonar signal of $500\text{Hz}$ to detect $A$ and receives a reflected sound of frequency $\nu$ . The value of $\nu$ is close to :
$(\text{Speed of sound in water}=1500{\text{ms}}^{-1}$)

• A. $504\text{Hz}$
• B. $507\text{Hz}$
• C. $499\text{Hz}$
• D. $502\text{Hz}$

Answer 1

The correct option is D $502\text{Hz}$

Given that,

$v_A=18{\text{Km hr}}^{-1}=5{\text{ms}}^{-1}$
$v_B=27{\text{Km hr}}^{-1}=7.5{\text{ms}}^{-1}$

As $A$ and $B$ are moving in the same direction,

The frequency received at $A$ is given by,

$f_A=f\left(\frac{v-v_0}{v-v_s}\right)$

$\text{Here,}{v}_0=v_A,v_s=v_B$

$\Rightarrow f_A=500\left(\frac{1500-5}{1500-7.5}\right)$

For the reflected signal from $A$ received by $B$,

The apparent frequency is given by,

$f_B=f_A\left(\frac{v+v_0}{v+v_s}\right)$

$\text{Here,}{v}_0=v_B,v_s=v_A$

$\Rightarrow f_B=f_A\left(\frac{1500+7.5}{1500+5}\right)$

$f_B=500\left(\frac{1500-5}{1500-7.5}\right)\left(\frac{1500+7.5}{1500+5}\right)$

$\Rightarrow f_B=502\text{Hz}$

Hence, option $\left(D\right)$ is correct.