Problem — Doppler radar and the car's speed

Problem

A traffic police radar emits waves at frequency $f_0$ and measures the double Doppler shift reflected from an approaching car. The total measured shift is $\Delta f / f_0 = 1{,}47\cdot 10^{-7}$. Calculate the car’s speed (speed of light $c = 3\cdot 10^8\;\text{m/s}$). Does it exceed $50\;\text{km/h}$?

Solution

Setup. As in the “acoustic mirror” exercise, the radar undergoes a double Doppler effect (the car receives, then re-emits). For $v \ll c$ the total shift is:

$\frac{\Delta f}{f_0} = \frac{2v}{c}$

Speed.

$v = \frac{c}{2}\cdot\frac{\Delta f}{f_0} = \frac{3\cdot 10^{8}}{2}\cdot 1{,}47\cdot 10^{-7} = 1{,}5\cdot 10^{8}\cdot 1{,}47\cdot 10^{-7}$

$\ev{v \approx 22\;\mathrm{m/s}}$

Comparison. $22\;\text{m/s}\times 3{,}6 \approx 79\;\text{km/h} > 50\;\text{km/h}$: yes, the car exceeds the limit.

Collegamenti

Argomenti: Waves Concetti: Doppler Effect