A stationary source is emitting sound at a fixed frequency f0, which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is 1.2% of f0. What is the difference in the speeds of the cars (in km per hour) to the nearest integer? The cars are moving at constant speeds much smaller than the speed of sound which is 330 ms

Engineering

Physics

Dopplers Effect

Question

A stationary source is emitting sound at a fixed frequency f₀, which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is 1.2% of f₀. What is the difference in the speeds of the cars (in km per hour) to the nearest integer? The cars are moving at constant speeds much smaller than the speed of sound which is 330 ms^–1.

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$ρ_{1} = \frac{c + v_{1}}{c - v_{1}} ρ_{0}$

$ρ_{2} = \frac{c + v_{2}}{c - v_{2}} ρ_{0}$

$\frac{Δρ}{ρ_{0}} = (\frac{c + v_{1}}{c - v_{1}}) - (\frac{c + v_{2}}{c - v_{2}}) = \frac{1 .2}{100}$

$= \frac{c^{2} - v_{1} v_{2} + {cv}_{1} - {cv}_{2} - (c^{2} - v_{1} v_{2} - {cv}_{1} + {cv}_{2})}{(c - v_{1}) (c - v_{2})}$

$= \frac{2 c (v_{1} - v_{2})}{c^{2}} = \frac{1 .2}{100}$

$= v_{1} - v_{2} = \frac{1 .2}{100} \times \frac{330}{2} \times \frac{18}{5}$

$= \frac{1 .2 \times 33 \times 18}{100} \approx 7 km / hr .$