Question

A police car moving at $22$ m/s chases a motorcyclist. The policeman sounds his horn at $176$ Hz, while both of them move towards a stationary siren of frequency $165Hz$. Calculate the speed of the motorcycle, if it is given that he does not observe any beats :

• A. $33m/s$
• B. $22m/s$
• C. $zero$
• D. $11m/s$

Answer 1

The correct option is B $22m/s$

Let $f_h^{\prime }$ and $f_s^{\prime }$ be the frequencies of the horn and siren respectively heard by the motorcycle man.

Given: $f_h=176Hz{f}_s=165Hz{v}_{policecar}=22m/s$

Doppler effect for police car and motorcycle:

$f_h^{\prime }=f_h[\frac{v_{sound}-v_{man}}{v_{sound}-v_{policecar}}]$ $=176[\frac{340-v}{340-22}]$ ............(1)

Doppler effect for siren and motorcycle:

$f_s^{\prime }=f_s[\frac{v_{sound}+v_{man}}{v_{sound}}]$ $=165[\frac{340+v}{340}]$ ............(2)

As the motorcycle man doesn't hear any beat, i.e $f_h^{\prime }=f_s^{\prime }$

$176[\frac{340-v}{318}]=165[\frac{340+v}{340}]$

$⟹v=22.3m/s\approx 22m/s$