Question

A car moves towards a hill with speed $v_e$. It blows a horn of frequency $f$ which is heard by an observer following the car with speed $v_0$. The speed of sound in air is $v$.

• A. The wavelength of sound reaching the hill is $\frac{v}{f}$
• B. The wavelength of sound reaching the hill is $\frac{v-v_c}{f}$
• C. The wavelength of sound of hom directly reaching the observer is $\frac{v+v_c}{f}$
• D. The beat frequency observed by the observer is $\frac{2v_c(v+v_o)f}{v^2-v_c^2}$

Answer 1

Answer: (B), (C), (D)

The correct options are
B The wavelength of sound reaching the hill is $\frac{v-v_c}{f}$
C The wavelength of sound of hom directly reaching the observer is $\frac{v+v_c}{f}$
D The beat frequency observed by the observer is $\frac{2v_c(v+v_o)f}{v^2-v_c^2}$


Frequency of horn directly heard by observer =$f^{/}$= $f\left(\frac{v+v_0}{v+v_c}\right)$
$⟹{\lambda }^{/}=\frac{v}{f^{/}}=\frac{v+v_c}{f}$
Frequency of echo $f_1$ $=f\left(\frac{v}{v-v_c}\right)$
$⟹{\lambda }_1=\frac{v}{f_1}=\frac{v-v_c}{f}$
Frequency of echo of horn as heard by observer $f_2$ =
$$\frac{v}{v-v_c}f\cdot \left(\frac{v+v_0}{v}\right)$$
Frequency of Beats = $f_2-f^{/}$

$=(v+v_0)f\{\frac{1}{v-v_c}-\frac{1}{v+v_c}\}=\frac{2v_c(v+v_0)}{(v^2-v_0^2)}$