⋯∙Predict/Calculate A highway patrolman sends a 28.250- GHz radar beam toward a speeding car. Th

⋯∙Predict/Calculate A highway patrolman sends a 28.250- GHz...

Key Concepts

Approximation Techniques in Physics

Approximation techniques, such as using the binomial approximation (for example, (1+x)^2 ? 1+2x for small x), are often employed in physics to simplify complex equations when certain parameters are small compared to others. These techniques allow scientists and engineers to make reasonably accurate calculations without the need for complex computation, particularly in situations where the speed of an object is much less than the speed of light.

Radar Technology

Radar technology involves transmitting electromagnetic waves toward a target and then receiving the waves that reflect back. By analyzing the frequency or phase shifts in the returned signal, radar systems can determine various characteristics of the target, including its speed and direction. This makes radar a powerful tool in applications ranging from traffic speed enforcement to weather monitoring.

Doppler Effect

The Doppler Effect is the change in the observed frequency of a wave when there is relative motion between the source and the observer. In the context of electromagnetic waves, such as radar beams, this phenomenon is used to determine the speed of an object by measuring the frequency shift of the reflected waves. When an object moves toward the source, the frequency increases (blue shift), and when it moves away, the frequency decreases (red shift).

Reflection in Doppler Systems

In radar applications, the emitted electromagnetic wave reflects off a moving object and returns to the source. This reflection process causes a frequency shift that is typically twice that of the shift due to a single motion, since the wave undergoes a Doppler shift both when approaching the moving object and when being reflected back. This principle is fundamental in determining the precise speed of the object.

Transcript

00:01 Okay, there have been a couple problems in our book like this, so we're not going to go into too many details.

00:06 Look towards other doppler shift problems if you're a little bit confused.

00:10 But we have this patrol car and it's sending out a signal and it's hitting some car that's either moving towards him or away from him at some particular speed.

00:22 And then that car is going to reflect the signal back to the highway patrol car.

00:29 Okay, so the trick here is that we're sending out some original frequency because of the doppler shift we're seeing some different frequency get reflect hit the other car that frequency is getting reflected that shifted frequency is getting reflected and then the patrol car is seen back some frequency that's been double shifted okay, so while the normal doppler shift equation looks like this f times one plus u over c plus or minus depending the double doppler shift equation is going to be the original frequency times one plus or minus u over c squared okay and this problem has helpfully told us that one plus or minus u over c squared when x is pretty small when u over c is small, which is the case that we're interested in, this is approximately equal to f times 1 plus or minus 2 times u over c.

01:44 So this is an approximation, does this actually do this all of the time to make numbers easier to work with? okay.

01:52 And so we need to figure out what the speed of the car.

01:55 So we know the change in frequency and we need to solve.

02:00 For you.

02:02 All right, so the change in frequency, which is delta f prime prime is equal to f prime prime minus f.

02:16 That's going to be equal to this whole quantity, f prime prime, minus f.

02:22 So now we have plus or minus two times u over c times f, is just equal to this double to this double to this delta f prime.

02:38 Okay, and we're trying to solve for you...