Chapter 39, Problem 39.23P

Police radar detects the speed of a car (Fig. P38.15) as follows. Microwaves of a precisely known frequency are broadcast toward the car. The moving car reflects the microwaves with a Doppler shift. The reflected waves are received and combined with an attenuated version of the transmitted wave. Beats occur between the two microwave signals. The beat frequency is measured. (a) For an electromagnetic wave reflected back to its source from a mirror approaching at speed v, show that the reflected wave has frequency

$f^{\prime }=\frac{c+v}{c-v}f$

where f is the source frequency. (b) Noting that v is much less than c, show that the beat frequency can be written as f_beat = 2v/λ. (c) What beat frequency is measured for a car speed of 30.0 m/s if the microwaves have frequency 10.0 GHz? (d) If the beat frequency measurement in part (c) is accurate to ±5.0 Hz, how accurate is the speed measurement?