Question

Police in a siren car moving with a velocity 20 ms^– chases a thief who is moving in a car with a velocity v₀ ms^-1. The police car sounds at frequency 300 Hz, and both of them move towards a stationary siren of frequency 400 Hz. Calculate the speed in which thief is moving. (Assume the thief does not observe any beat)

Answer 1

Velocity of sound v = 330 m/s

Velocity of a police siren car v_s = 20 m/s

Frequency of a police siren car f = 300 Hz

Let the velocity of the thief be v m/s

The frequency of police siren heard by a thief is

${\displaystyle {\text{f}}_1=\left(\frac{330-\text{v}}{330-20}\right)\times 300}$

${\displaystyle =\left(\frac{330-\text{v}}{310}\right)\times 300}$ Hz

Frequency of stationary siren = 400 Hz

Frequency of stationaiy siren heard by a thief

${\displaystyle {\text{f}}_2=\left(\frac{330+\text{v}}{330}\right)\times 400}$

If there are no beats then

f₁ = f₂

${\displaystyle \left(\frac{330-\text{v}}{310}\right)\times 300=\left(\frac{330+\text{v}}{330}\right)\times 400}$

(330 - v) × 0.9677 = (330 + v) × 1.2121

330 × 0.2444 = 2.1798 v

${\displaystyle \therefore \text{v}=\frac{0.2444\times 330}{2.1798}}$

= 36.99

= 37 m