The siren of a fire engine that is driving northward at 30.0 m>s emits a sound of frequency 2000 Hz. A truck in front of this fire engine is moving northward at 20.0 m>s. (b) What wavelength would this driver measure for these reflected sound waves?

Video transcript

Hey everyone in this problem. We have an ambulance behind a wide load truck driving in the same direction that emits an alert tone at 1500 hertz. The ambulance moves at 28 m per second and the truck moves at 14 m per second and were asked to determine the wavelength of the waves reflected by the truck's load measured relative to the ambulance. Alright, so let's try to unpack this. So we're looking for the wavelength of waves reflected by the truck's load measured relative to the ambulance. The ambulance is emitting a sound. It's going to reflect off of the truck back towards the ambulance and we want to know what the wavelength is of that sound that the ambulance hears. Let's start with the first part of the problem. So we have an ambulance Ambulance and they're going to be the source of the noise initially. And they're emitting sound at a frequency of 1500 Hz. We're also gonna have the truck and they're initially going to be the listener. Now, we have this problem. We have both the source and the listener moving. This is a Doppler effect problem. We want to take the positive direction to be from the listener to the source. So from the listener to the sources to the left. So that's going to be a positive direction. And then when we look at the speed of each of these, we have the speed of the ambulance, which is coming up behind the truck. So it's moving to the right To the right is in the negative direction. And so the speed of the ambulance is going to be negative 28 m/s. Now for the truck we're told that these are moving in the same direction. So the truck is also moving to the right which is going to be the negative direction. And so the speed of the listener, the speed of the truck is going to be negative meters per second. Alright, so given the information we have here, what we can find is we can find the frequency of the sound that the listener hears and that's going to be the frequency that is reflected. Okay, so that frequency that is reflected is going to be the frequency that the listener hears and recall that for a moving source and moving listener, the frequency of the listener is given by V. The speed of sound plus V. L. The speed of the listener over V. The speed of sound plus V. S. The speed of the source, all times the frequency of the source. And if we put in the values for our problem, We have 343 m/s -14 m/s, Divided by 343 m/s -28 m/s, All Times 1500 Hz. Now there's 343 m/s. The speed of sound. That is one of those standard values. And you can look that up in your textbook if you need a reminder on what that value is. Now, when we work this out, We are going to get meters per second divided by 315 m per second, all times 1500 hertz, The unit ofm per second will divide out. And we're going to be left with 1,566. Hz. So that is going to be the frequency that is reflected by the truck. Now we have the frequency reflected by the truck. We want to know the wavelength of the waves relative to the ambulance. Now recall that the wavelength and the frequency are related. Okay, so let's start by finding the frequency because we know how to do that through this Doppler effect type problem. Okay, so let's find the frequency that the ambulance hears of that reflected way. So what we're doing now is we're kind of reversing the problem. So now we have the ambulance as the listener, they're still moving to the right and we have the truck as the source. And the source of that noise is the reflected Sound. And so the frequency of the source now is one of the the frequency of the reflected sound that we just found. 1,566.6667 Hz. And again the truck is moving to the right as well. And we take from the listener to the source as our positive direction. So now our positive direction is to the right. And so both of the speeds of the listener and the speed of the source are going to be positive now. So we get positive 28m/s And positive 14 m/s. Now when we find the frequency of the listener, same as in the previous part of this problem, it's going to be equal to V. The speed of sound plus V. L. The speed of the listener over V. The speed of sound. Pleasant V. S. The speed of the source all times F. S. The frequency of the source plugging in the values for this problem. We have 343 m/s plus 28 meters per second. Over 343 m per second plus 14 m per second. All times 1566. hertz. And when we work this out we get that the frequency that the ambulance hears from that reflected sound is going to be 1628 0. hertz. Alright, so we've got our frequency. But remember we were asked to find the wavelength, How are the two related? Well recall that the wavelength lambda is equal to the speed over the frequency. You know that the speed of sound is 343 m/s. Okay, that's that standard value over the frequency. We just found the frequency 1628. hertz. And this is going to give us a value of 0. m. And so that is the wavelength of the waves reflected by the truck's load measured relative to the ambulance. And if we go back up to our answer choices, we see that that matches with answer choice D. And so D. Is the correct answer. We have a wavelength of 0.211 m. Thanks everyone for watching. I hope this video helped see you in the next one.