Police radar detects the speed of a car (Fig. P39.19) as...

Police radar detects the speed of a car (Fig. P39.19) 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 $$ f=f_{\text {source }} \frac{c+v}{c-v} $$ where $f_{\text {source }}$ is the source frequency. (b) When $v$ is much less than $c,$ the beat frequency is much smaller than the transmitted frequency. In this case use the approximation $f+f_{\text {source }} \approx 2 f_{\text {source }}$ and show that the beat frequency can be written as $f_{\text {brat }}=2 v / \lambda .$ (c) What beat frequency is measured for a car speed of $30.0 \mathrm{m} / \mathrm{s}$ if the microwaves have frequency 10.0 GHz? (d) If the beat frequency measurement is accurate to $\pm 5 \mathrm{Hz}$, how accurate is the velocity measurement? (FIGURE CANNOT COPY)

Police radar detects the speed of a car (Fig. P39.19) 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
$$ f=f_{\text {source }} \frac{c+v}{c-v} $$
where $f_{\text {source }}$ is the source frequency. (b) When $v$ is much less than $c,$ the beat frequency is much smaller than the transmitted frequency. In this case use the approximation $f+f_{\text {source }} \approx 2 f_{\text {source }}$ and show that the beat frequency can be written as $f_{\text {brat }}=2 v / \lambda .$ (c) What beat frequency is measured for a car speed of $30.0 \mathrm{m} / \mathrm{s}$ if the microwaves have frequency 10.0 GHz? (d) If the beat frequency measurement is accurate to $\pm 5 \mathrm{Hz}$, how accurate is the velocity measurement?
(FIGURE CANNOT COPY)

Key Concepts

Doppler Effect

The Doppler effect describes the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the source of the wave. In the context of electromagnetic waves, when a wave is reflected off a moving object, the motion of the object alters the frequency of the reflected wave in a way that depends on the relative speed of the object and the speed of light.

Electromagnetic Waves

Electromagnetic waves, including microwaves used in radar systems, travel at a constant speed (the speed of light) in a vacuum. Their properties, such as frequency and wavelength, are fundamentally linked by the relation c = f?. This relationship is essential for understanding how radar systems convert changes in frequency (due to the Doppler effect) into speed measurements.

Beat Frequency

Beat frequency arises when two waves of nearly identical frequencies interfere with each other, resulting in a modulated signal whose amplitude oscillates at a frequency equal to the difference between the two original frequencies. In radar applications, the interference between the transmitted and the reflected waves produces beats that can be measured to determine the velocity of moving objects.

Small Velocity Approximation

When the speed of the moving object is much less than the speed of light (v << c), the Doppler shift is very small compared to the original frequency. This allows for simplifying approximations, such as assuming that the sum of the transmitted and reflected frequencies is approximately twice the transmitted frequency. This simplification leads to a linear relationship between beat frequency and velocity, which makes calculations more tractable.

Measurement Uncertainty

Measurement uncertainty refers to the potential errors in the recorded value of a quantity, which in this case is the beat frequency. In radar systems, knowing the uncertainty in the measured beat frequency is crucial because it directly affects the accuracy of the calculated speed of the object. Error propagation techniques are used to relate the uncertainty in the frequency measurement to the uncertainty in the determined velocity.

Transcript

00:01 This problem, we're going to talk about the relativistic doppler effect.

00:04 So consider that we have a wave source that is emitting light waves, and we have an observer.

00:16 And the light source emits waves with a frequency f0.

00:23 Now consider that the observer is moving towards the light source.

00:28 Or the light source is moving towards the observer.

00:31 It doesn't matter.

00:31 All that matters is that the motion, the relative motion between them is that they are approaching each other.

00:39 The relativistic doppler effect tells us that the frequency is equal to, the frequency observed is equal to the original frequency times the square root of c plus or minus v, divided by c minus plus v, where the upper sign is used when the objects are approaching, while the lower sign should be used when they are receding.

01:11 And here i'm always talking about the relative motion.

01:17 So in our problem, we're going to use doppler effect to construct a police raider for car speeds.

01:28 So consider that we have a police car that's suppose that the car is at rest, and then we have another car that's traveling.

01:39 I'm just going to drive it as a box.

01:42 Another car that's traveling towards the police block with a speed v.

01:49 Thus, the police emits a wave towards the car with a frequency f0.

01:59 The wave reflects back from the car and reaches the police again with a shifted frequency.

02:08 And with this frequency, the police should be able to calculate what is the velocity of the car.

02:19 And our goal is to show in question a that the frequency f obtained by the police is equal to f0, plus the frequency of the source, times c plus v divided by c minus v.

02:36 So notice that the frequency that's going towards the car is observed by the observer inside the car as f prime that is equal to f0 times square root of c plus v divided by c minus v.

03:00 Okay.

03:02 This is the frequency observed by the car and this is the frequency that is emitted by the car when the wave reflects from the car...

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