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
|
|
Since
the photon energy, E, is equal to hn, where h = Planck's constant =
6.63 X 10-34 j sec, then
Also,
since photon momentum equals h/c,
where c is the speed of light, then
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
This
equation may be simplified to read
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.
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