How Do Police Radars Really Work?

Problem:
A car, traveling at speed, V, approaches a police radar, scanning for speeders with a frequency, . Calculate the approximate frequency shift of the reflected signal.

Solution:
Consider a single photon from the police radar. The photon must interact with the approaching car for a finite time while it is being reflected. Call this time, t. Let an interaction force, ± f, exist between the photon and the car for the time, t. The force exerted by the photon on the car, +f, acts to remove energy from the car. The force exerted by the car on the photon, -f, acts to add energy to the photon. Therefore, we expect the photon frequency to increase. During the time t, the car travels a distance s = V t. We may now write two equations, one for a change in momentum, p, and one for a change in energy, E:

Momentum:

p = f t

1.

Energy:

E = f s = fV t

2.

Since the photon energy, E, is equal to hn, where h = Planck's constant = 6.63 X 10^-34 j sec, then

E = h

3.

Also, since photon momentum equals h/c, where c is the speed of light, then

p = 2h/c + h /c

4.

where the first term on the RHS represents p for an elastic reflection (i.e., one for which E = 0), and the second term takes into account the change in frequency due to the change in energy.

Dividing eq. 1 by eq. 2, and substituting for E and p from eqs. 3 and 4, we find

(2h/c + h /c)/(h ) = 1/V

5.

This equation may be simplified to read

/ = 2V/(c-V) 2V/c

6.

where the final step results from V << c. Now, let V = 60 mph = 27 m/sec, and let = 5 X 10⁸ Hz. Then,

= 90 Hz

7.

The police radar detector easily detects this frequency shift.