Police Radar Equipment

Consider a car moving towards a radar emitter at a speed v. Assume further that the radar beam, which consists of microwave radiation moving at the speed of light, is directly in line with the approaching car. The car can be considered as an observer moving towards a stationary source, so the frequency detected by the car, f_o, will be:

f_o = f_s(c + v)/c

where f_s is the frequency of the radiation emitted by the radar. This radiation is reflected back towards the radar equipment (which is fitted with a receiver). The car can now be considered as a moving source, approaching a stationary observer. The frequency, f_r picked up by the receiver will then be:

f_r = f_oc/(c - v)

Combining these two equations gives f_r = f_s(c + v)/(c - v).

The fractional change in frequencies, Δf/f will then be given by Δf/f = (f_r - f_s)/f = f_r/f - 1

from which we get Δf/f_s= (c + v)/(c - v) - 1 ≈ 2v/c when v << c.

Thus, v = cΔf/2f_s. Δf can be measured accurately by the equipment, so, since the emitted frequency f_s is known, the speed of the car, v can be calculated.