An avant-garde composer wants to use the Doppler effect in his ne...

Question

An avant-garde composer wants to use the Doppler effect in his new opera. As the soprano sings, he wants a large bat to fly toward her from the back of the stage. The bat will be outfitted with a microphone to pick up the singer's voice and a loudspeaker to rebroadcast the sound toward the audience. The composer wants the sound the audience hears from the bat to be, in musical terms, one half-step higher in frequency than the note they are hearing from the singer. Two notes a half-step apart have a frequency ratio of 2¹/² = 1.059. With what speed must the bat fly toward the singer?

Accepted Answer

Everyone in this problem, we have a teacher that wants to investigate the Doppler effect. So they're using a tuning fork to produce the sound of frequency F S. They're doing an experiment and a student riding a bike equipped with a microphone and a sound system is gonna approach the tuning fork at a constant speed. The sound system is gonna emit a sound signal toward the clock which is sitting behind the tuning for. We're told to assume that students hear a sound with a frequency of 1. F S from the sound system. And we're asked to calculate the bikes speed. We're given four answer choices all in meters per second. Option, a 3.5, option B 5.1, option C 7.6 and option D 8.3. All right. So we have a Doppler effect problem. OK. We know that that's what we're gonna be using. And we have to think about this in kind of two stages. The first stage OK is one that bike is moving towards the tuning for. OK. And listening to the tuning. So the tuning fork is going to be the source emitting the sound and the student on the bike with the microphone is gonna be the listener. OK? So we have student approaching tuning tuning fork and this is going to be a Doppler problem where the listener is moving towards the. So let's write out all of the information. We know, we know we're the source, which is the tuning fork, the velocity or the speed of this source, we're gonna call it V S one for the speed of the source in stage one is zero. And the teacher with the tuning fork is not moving. We know that the frequency of the source, the frequency that tuning fork is emitting is just called F S. Now for the listener, which is the student on the bike. OK? We don't know their velocity or their speed. That's what we wanna try to find and we don't know what frequency they hit. Now, this speed of the listener V L one we expect to be positive. OK? Because they're moving from the listener to the source. OK. They're moving towards the source we take from the listener to the source as positive. And so their speed is going to be or their velocity is gonna be in the positive direction. So we do expect that one to be positive. No, we have two unknowns here. OK. And we can write out our equation and we're gonna do that in just a second, just gonna set kind of the stage for what we're gonna be doing, we wanna find this value of Ella. OK. That speed of the bike, we don't know the frequency that they hear, but we do know that this, the frequency that the students hear after the sound system emits it. OK. So the microphone picks up a particular frequency and emits that and then the students hear that frequency. OK. So we're gonna write out the equation here in stage one, Then we're gonna need to move to stage two to find a little bit more information. Put the two equations together. OK. All right. So let's recall the equation for the Doppler effect. We have that the frequency of the listener L one. And so the frequency that the listener hears is gonna be equal to V, the speed of sound plus V L one, the speed of the listener divided by V, the speed of sound multiplied by F S one, the frequency of the source. And this is specifically for the Doppler effect where we have a moving listener and a stationary source. OK? All right. Now, let's substitute in the information we know, hey, this is gonna be equal to 343 meters per second. That speed of sound. What the speed of the listener, which is the bike which we're looking for divided by 343 m per second multiplied by F S, right? And again, we're gonna call this equation one and we're gonna come back to this after we go through stage two. So let's do it. Let's go through stage two and then stage two, we're kind of switching rules. OK. So the student on the bike is gonna be the source, the class is gonna be the listener. So we have that the source is moving toward the listener. So now instead of having a stationary source and a moving listener, do you have a moving listener or sorry, a moving source and a stationary listener and let's write out our info. So we have the source which we said is the student on the bike. OK. So that student on the bike has their microphone, it's listened to the frequency from the tuning fork and now the sound system is emitting that frequency as the source. So V S two, this is gonna be the speed of the source in two. Well, this is gonna be the same speed as before the student's riding consistent speed on his bike. It's gonna be the same speed, but the direction is gonna be opposite. Hey, remember we always take the positive direction from the listener to the. So we have the bike moving towards the listeners that's gonna be in the negative direction in this case. And so that's going to be negative, the speed that the bike was traveling in stage one and they were the listener there. So this negative V L one and I can see how we're gonna relate these two equations. We have this V L one value that's gonna be in both of them. And the frequency F S two that, that source is emitting. OK. That, that sound system is emitting is just gonna be equal to the frequency F L one that they listened to in the first stage. OK. That frequency that the microphone heard is now what they're emitting. Now for the listener in this second stage, the listener is going to be the class. This time we know that they're stationary, they're not moving. And so their speed V L two is going to be zero and the frequency they hear F L two we're told in the problem is 1. F S. All right. So we have all our information. We're gonna write out our equation again. We're taking the Doppler equation in general and recall that that's gonna be F L the frequency of the listener. And in this case, F L two gonna be equal to V, the speed of sound. What's V L two? The speed of the listener divided by V, the speed of sound plus V S two, the speed of the source all multiplied by F S to the speed of the or sorry, the frequency of the source. All right. So substituting in our values for this equation, we have 1.3 F S is equal to 343 m per second. OK. The frequency of the listener is zero. So we don't have to worry about that term. In the numerator, we divide by 343 m per second minus V L one multiplied by F L one. And we're gonna call this equation two. OK? And now let me scroll up a bit. So you can see equation one as well. Now, we have two equations with two unknowns. OK? We don't know F L one and we don't know um V L one. All right. So we are going to substitute equation one into equation two and try to solve or via one that speed of the bike we are looking for. OK. So substituting in equation what? Right on the right uh the left hand side here just in case we run out of room, we'll have some more room on the right. So we get that 1.3 at best is equal to 343 m per second, divided by 343 m per second minus V L one all multiplied by 343 m per second plus V L one divided by 343 meters per second, multiplied by F S. All right. So a couple couple things that we can simplify right off the bat is 343 m per second. Will divide out numerator and denominator. We can divide both sides by F S and that frequency is going to divide out as well. We're gonna be left with 1.3 is equal to 343 m per second plus V L one divided by 343 m per second. Minus V 01. We can multiply both sides by this 343 minus V 01. You get 1.3 multiplied by 343 m per second minus V L one is equal to 343 m per second plus V +01. Expanding on the left hand side 0.29 m per second minus 1.3. V 01 is equal to 343 m per second plus V L one. OK? We're trying to solve for V +01. So we're gonna move this negative 1.3 V +01 to the right by adding it. We're gonna move the 343 m per second to the left by subtracting it. And we're trying to get all the V +01 terms on one side, all of the constant terms on the other. I'm just gonna do this up here. We get 10.29 m per second is equal to 2.3 video one and finally dividing by 2.3, we get V L one is equal to 5.69 m per second. OK. And remember that V 01 that was the velocity or the speed of the listener in stage one, which is the speed of the bike that we were looking for. One more thing I want to make a comment on before we wrap up is in our equation, we had this F S this frequency of that original source that divided out. So this result actually didn't depend on what that original frequency was just on what the ratio or the relationship between the original frequency and the frequency that the class heard was going up to our answer choices and comparing what we found. We want a round to one decimal place and we can see that the correct answer is option B. Thanks everyone for watching. I hope this video helped see you in the next one.

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