Question

The figure shows a transmitter and a receiver of waves contained in a single instrument. It is used to measure the speed $u$ of a target object (idealized as a flat plate) that is moving directly toward the unit, by analysing the waves reflected from the target. What is $u$ if the emitted frequency is $18.0\text{kHz}$ and the detected frequency (of the returning waves) is $22.2\text{kHz}$?
$[$speed of wave in stationary medium $=343\text{m/s}]$

• A. $18\text{m/s}$
• B. $28\text{m/s}$
• C. $36\text{m/s}$
• D. $56\text{m/s}$

Answer 1

The correct option is C $36\text{m/s}$

Before reflection of sound, the transmitter acts as a source and the target act as observer.

From Doppler's effect formula, frequency recieved at the target
$f_1=\left(\frac{v+v_o}{v-v_s}\right)f_o=\left(\frac{v+u}{v}\right)f_o$ .....$\left(1\right)$

After reflection of sound, the target acts as a source emitting frequency $f_1$ and the receiver acts as observer.

From Doppler's effect formula, frequency received at the receiver
$f_2=\left(\frac{v+v_o}{v-v_s}\right)f_1=\left(\frac{v}{v-u}\right)f_1$ ....$\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$,
$f_2=\left(\frac{v+u}{v-u}\right)f_o$
$\Rightarrow 22.2=\left(\frac{343+u}{343-u}\right)18$
$\Rightarrow 7614.6-22.2u=6174+18u$
$\Rightarrow 40.2u=1440.6$
$\Rightarrow u\approx 36\text{m/s}$