Question

A train moves toward a stationary observer at a speed $34\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$. The train sounds a whistle and its frequency registered by the observer is $f_1$. If the train's speed is reduced to $17\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$, the frequency registered is $f_2$. If the speed of sound of $340\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$, then the ratio $\raisebox{1ex}{$f_1$}\!\left/ \!\raisebox{-1ex}{$f_2$}\right.$ is

• A. $\raisebox{1ex}{$18$}\!\left/ \!\raisebox{-1ex}{$7$}\right.$
• B. $\raisebox{1ex}{$21$}\!\left/ \!\raisebox{-1ex}{$20$}\right.$
• C. $\raisebox{1ex}{$20$}\!\left/ \!\raisebox{-1ex}{$19$}\right.$
• D. $\raisebox{1ex}{$19$}\!\left/ \!\raisebox{-1ex}{$18$}\right.$

Answer 1

The correct option is D

$\raisebox{1ex}{$19$}\!\left/ \!\raisebox{-1ex}{$18$}\right.$

Step1: Given data and assumptions.

The velocity of train's sound, $v_s=34m{s}^{-1}$

The velocity of train's sound, $v{\text{'}}_s=17m{s}^{-1}$

The velocity of the sound wave, $v=340m{s}^{-1}$

The actual frequency$=f$

The frequency registered by the observer$=f_1$

The frequency registered after the train speed reduced$=f_2$

Step2: Find the frequency registered by the observer.

Formula used:

$f\text{'}=\left(\frac{v+v_0}{v-v_s}\right)\times f$

Where,

$f\text{'}$is the apparent frequency,

$f$ is the actual frequency,

$v$ is the sound wave velocity,

$v_0$ is observer velocity,

$v_s$ is the sound velocity.

Now,

$f_1=\left(\frac{v+v_0}{v-v_s}\right)\times f$

$\Rightarrow$$f_1=\left(\frac{340+0}{340-34}\right)f$ $\left[\because v_0=0\right]$

$\Rightarrow$$f_1=\frac{340}{306}f$ ..$\left(i\right)$

Step3: Find the frequency registered after the train speed is reduced.

$f_2=\left(\frac{v+v_0}{v-v_s}\right)\times f$

$\Rightarrow$$f_2=\left(\frac{340+0}{340-17}\right)f$ $\left[\because v_0=0\right]$

$\Rightarrow$$f_2=\frac{340}{323}f$ ..$\left(ii\right)$

Step4: Find the ratio, $\raisebox{1ex}{$f_1$}\!\left/ \!\raisebox{-1ex}{$f_2$}\right.$.

We have,

Dividing the equation $\left(i\right)$ by $\left(ii\right)$ we get.

$\Rightarrow$$\frac{f_1}{f_2}=\frac{{\displaystyle \raisebox{1ex}{$340$}\!\left/ \!\raisebox{-1ex}{$306$}\right.}}{{\displaystyle \raisebox{1ex}{$340$}\!\left/ \!\raisebox{-1ex}{$323$}\right.}}$

$\Rightarrow$$\frac{f_1}{f_2}=\frac{323}{306}$

$\Rightarrow$$\frac{f_1}{f_2}=\frac{19}{18}$

Hence, option D is correct.