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Chapter 24: Q41PE (page 889)

Police radar determines the speed of motor vehicles using the same Doppler-shift technique employed for ultrasound in medical diagnostics. Beats are produced by mixing the double Doppler-shifted echo with the original frequency. If 1.50×109-Hzmicrowaves are used and a beat frequency of 150Hzis produced, what is the speed of the vehicle? (Assume the same Doppler-shift formulas are valid with the speed of sound replaced by the speed of light.)

Short Answer

Expert verified

The velocity is3m/s .

Step by step solution

01

Define frequency

The number of complete cycles of waves passing a spot in unit time is defined as the frequency.

02

Evaluate velocity

The apparent frequency and the original frequency are known to be connected via the Doppler effect, which is given by

f=f0v+v0v

Where f0is original frequency,fis apparent frequency, vis the velocity of light, and v0is the velocity of the observer.

Also, calculate the apparent frequency by multiplying the original frequency by the beat frequency.

f=f0+150Hz=1.5×109Hz+150Hz≈1.5×109Hz

Substitute the values in the above equation,

1.5×109Hz=1.5×109Hz3×108m/s+v03×108m/s1.5×109Hz1.5×109Hz≈3×108m/s+v03×108m/sv0≈3m/s

1.5×109Hz=1.5×109Hz3×108m/s+v03×108m/s1.5×109Hz1.5×109Hz≈3×108m/s+v03×108m/sv0≈3m/s

Therefore, the velocity is 3m/s.

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