A police car sounding a siren with a frequency of 1580 Hz is traveling at 120.0 km/h.

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. Here's the formula we're going to be using to answer all these questions. The perceived frequency is going to be the frequency emitted by the object according to somebody who's moving with the object. So, this driver of the police car in other words. Multiplied by the speed of sound plus or minus speed of the observer divided by the speed of sound minus or plus the speed of the source. Now, whether we use a plus or minus depends on whether the observer's motion increases or decreases the perceived frequency. And then also down here we separately think about whether the sources motion increases or decreases frequency. So, when the source is moving towards an object, that would tend to increase the perceived frequency, and we would choose a minus there because when you decrease the denominator, you increase the fraction. So, just like, you know, 1/2 is bigger than 1/3, you know, when you have a smaller denominator, you have a bigger fraction. So, and then when this, when the police car is moving away from something, that would tend to decrease the perceived frequency and we would choose a plus sign in the bottom. And same type of reasoning for the top. And we'll go through each of these scenarios here. So, in part a is the simplest scenario where you have the observer standing still. So, the frequency perceived when the police car's approaching will be the police car's emitted frequency of 1580 hertz times 343 meters per second plus 0 velocity of observer divided by 343 meters per second, and then we'll go minus the speed of the source. And we're choosing a minus because we want to choose whatever sign results in the largest result for this fraction. And you get a bigger fraction when you put a minus in the denominator. So, I put in 33.333 meters per second because that's the converted velocity of the police car after you turn 120 kilometers per hour into meters per second by multiplying by 1 meter for 1000 meters for every kilometer and then multiplied by 1 hour for every 3,600 seconds. OK. So, 1,750 hertz there. And then when the police car is passed the stationary person on the roadside, the police car’s moving away, and so that tends to decrease the perceived frequency in which case we have to choose a plus on the denominators so that we maximize the denominator, and as a result minimize the quotient. And then you get 1,440 hertz. In part b, it's a bit more complicated because now you have the both the observer and the police car moving. So, when the police car is approaching the observer who's also in a car moving towards the police car. We're gonna convert the observer's velocity into meters per second, of course, by dividing by 3.6, which is effectively what's happening here. Multiplied by 1,000, divided by 3,600 is the same as dividing by 3.6. And we're choosing a plus in the numerator because the observer is moving towards a police car. So, the observer is moving towards the sound source which would tend to increase the perceived frequency. And so we'll choose a plus in the numerator. And likewise the police car is moving towards the observer and which we tend to increase the perceived frequency, in which case we want a minus sign at the bottom so that we reduce the denominator. And all this works out to 1,880 hertz. And then once the police car has passed the oncoming car then that means the car is moving away from the police car, and so we have a minus here. And the police car is moving away from the observer, and so we want to a plus there in order to reduce the perceived frequency. We get 1340 hertz. And then for part c this is perhaps the most confusing scenario for choosing signs here. And now we have the observer moving, I mean, the police car is catching up to the observer, right? But the observer has their velocity directed away from the police car. So, that would tend to reduce the Doppler effect frequency and reduce the perceived frequency. So, we're going to choose a minus sign on the top, because the observer's motion is such that it would tend to reduce frequency it's moving away from the police car. I mean, understand that the net result is that the police car's catching up, but we're just thinking about the observer and they're moving away from the police car. So, we have a minus there. And then on the bottom, the police car is moving towards the observer which would tend to increase the Doppler effect perceived frequency. And so we're going to reduce the denominator in order to have an increase in the result. And so we have a minus on both, top and bottom. So, we have 1,640 hertz here. And then once the police car passes, the other car overtakes it. Then, now, the observer is trying to move towards the police car, the siren, the sound source. And that means we have a plus sign there because that'll tend to increase the perceived frequency. However the police car is moving away from the observer and so that would require a plus sign on the denominator so that we have an increased denominator so that there's a reduced quotient. And in the end this works out to 1530 hertz, both of them are plus signs there.