(II) What is the ratio of ...

Question

(II) What is the ratio of

(b) the amplitudes, of an earthquake P wave passing through the Earth and detected at two points 15 km and 55 km from the source?

Accepted Answer

Hey, everyone in this problem, a group of students is trying to measure the ratio of the amplitude of a spherical sound wave at 20 kilometers and 30 kilometers away from the source. And we are asked to calculate that ratio. So we have more answer choices here. Option A two thirds, option B the square root of two thirds, option C the square root of three halves and option D three halves. So we're looking at some information about amplitude, OK? About distance from the source. We're looking for this ratio. And so there's two things that are gonna help us here and they both relate to the intensity. OK? So we know that we can relate the intensity to the amplitude and we can also relate that intensity to the distance from the source. OK. So recall that the intensity I is proportional to the amplitude squared of a wave, OK? A sound wave. So what we can say is that the intensity of the wave 20 kilometers an hour. So I 20 will be equal to some constant. We can call C multiplied by the amplitude of that 20 kilometer an hour, sorry 20 kilometer away wave. Squared. OK. And similarly, I 30 the intensity of the wave that is 30 kilometers from the source will be equal to that same constant C multiplied by a 30 squared. OK. So we've managed to introduce the amplitude into our equation here, we're looking for the ratio of those amplitudes. So that's gonna come in handy. But the other information we have is this distance. So how can we get this distance into this equation? And we can similarly recall that the intensity is proportional to the inverse square of the distance. So the intensity is proportional to one divided by R square where R is the distance from the source and the radius of that sound wave. OK. So just like we did for the amplitude, we can write a similar equation in this case. So we can say that I 20 will be equal to some constant, let's call it K divided by the distance away from the source for this 20 kilometer and a 20 kilometer away source. So 20 kilometers square and I 30 will be equal to that same constant K divided by its distance from the source which is 30 kilometers. And then we square that. Yeah. So we have these two sets of equations. We have I 20 we have I three. And what we can do is we can equate the two. OK? And we can equate the two. If we look at the ratio, we want the ratio of the amplitudes. So let's look at the ratio of the intensities that's gonna give us that ratio that we want. OK. So we are gonna start on the left hand side with our intensities related to our amplitudes. And we're gonna look at that ratio I 20 divided by I 30. That's just gonna be C A 20 squared divided by C A 30 squared. Those CS will divide out and we can just write this as the ratio of the amplitudes squared. So that's what we have using the first equation. Then we can use the second equations. We found dealing with the distance from the source. And we're gonna look at that same ratio I 20 divided by I 30. OK? And in this case, those cases are gonna divide out and we're gonna end up with 30 kilometers squared divided by 20 kilometers squared, OK? And notice that those distances get kind of flipped, OK? Because they're in the denominator. So we end up with 30 on top and 20 on the bottom. OK. So we have these two equations now for this ratio and we can equate them because we know that the ratio of the intensity I 20 divided by I 30 must be equal to itself. We know that I 20 divided by I 30 must be equal to I 20 divided by I 30. OK? Writing that we can say that the ratio of the amplitude squared. So a 20 squared divided by a 30 square must be equal to 30 kilometers squared divided by 20 kilometers squared. So on the left hand side, we want the ratio of just the amplitudes, not the ratio of the amplitude squared. So we can take the square root on both sides. And what we get is that the ratio a 20 divided by a 30 it's going to be equal to 30 kilometers divided by 20 kilometers, which is just equal to three halves. OK? So even though we didn't have a whole bunch of other information about this problem, what we had was enough to sort this out. OK. So using what we know about the relationship between the intensity and the amplitude and then when we know about the intensity and the distance from the source, we were able to figure out that the ratio of the amplitudes is going to be three halves. Looking at our answer choices, we can see that this corresponds with answer choice. D thanks everyone for watching. I hope this video see you in the next one.

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(II) What is the ratio of
(b) the amplitudes, of an earthquake P wave passing through the Earth and detected at two points 15 km and 55 km from the source?