Hey, guys. Let's take a look at this problem here. So we got these 2 cities. Without any wind, this airliner can make the trip between them in some amount of time, then a wind starts blowing. So this 225 kilometer wind that blows from west to east. We're trying to figure out how long the trip from a to b is going to take now that the wind is blowing. So we're going to solve this because we have some relative motions, some relative to the air, stuff like that, using our relative velocity steps. But there is a little bit of a before and after case that's going on here. So we're going to draw these diagrams, but I'm actually going to draw 2, one for the before situation and one for the after. So let's get to it. The before situation that happens is in this first part of the problem where we're told that without any wind, the trip is, you know, the airliner can make the trip in some amount of time. So the before situation is with no wind. And then the after is going to be when the wind starts blowing. So this is going to be some from the wind. So let's check this out here. So we've got these two cities. Right? So we've got a and b. We know b lies directly west or east of a. And so the delta x, we're told, is equal to 2775 kilometers. And we know the time that it takes to travel between these two cities is 3.3 hours. So what happens is if the plane is just traveling through the air like this, this is the plane like this, then we can actually just straight up figure out what the velocity is just by doing delta x over delta t. So it's just 2775 over 3.3 hours. And so the velocity of the plane through the air is 841 kilometers per hour. So that's the speed of the plane, right, without any wind. But now what happens is the wind is going to start blowing. And so now there's going to be a couple of velocities to consider. There's going to be some relative velocities. Let's draw the same exact diagram. So we've got, again, city a and city b and we know the delta x. We know that this displacement is 2775. What's different now is because there's some wind that's potentially helping or changing the velocity of the plane, this delta t is going to be different. So that's what we're actually asked for in the problem. How long does this trip now take with some wind blowing? So I'm going to call this delta t initial. That's the 3.3 hours. But what I'm really looking for is delta t final. What is the change? What is the amount of time that it takes now that the wind is blowing? So this is really what I'm interested in. Okay. So what's going on here? So now we're just going to go ahead and stick to the steps for solving relative motion problems. So we've got this diagram here. Now we just need to identify the objects and the references. So we've got the plane that is traveling. So we've got this plane over here. That's one of our objects. We're told that the plane in the second part of the problem has the same speed relative to the air, which means that the air is actually itself a reference or an object. So we're going to call the air the object here. And we're told that the air also has a speed. The velocity of that wind that's blowing from west to east is 225. And so it picks you know, it's positive because it points in the same direction. Right? So the velocity of the air. So the next question is what is the velocity of that air measured relative to? Well, so our third reference is going to be the ground. So that's the third reference here, the ground. So we got the plane traveling through the air, but the air is traveling relative to the ground. So those are our 3 objects and references. So now we're going to write each of their velocities with the given subscript notations. So we're told that the velocity of the plane relative to the air is going to be the same as before. So that means that this variable over here was V_PA, and so that's 841. And the air is measured relative to the ground. That's the, you know, the wind speed or whatever. And so that's V_AG. So we've got our two variables here. So we've got V_PA, which we know is 841. We've got V_AG, which we know is 225. So remember, we're going to need a third one to start writing a relative velocity equation. So what's that third variable? Well, the only other thing that we can use is the fact that it covers some distance which is on the ground. Basically the ground distance between a and b in some amount of time. So what this means is that the velocity of the plane through the air and the velocity of the air relative to the ground can combine, and they'll produce a velocity of the plane relative to the ground. So that's V_PG. So both of these things here combine to form V_PG. That's the third variable. So this is actually what we're looking for here, our V_PG. Well, at least we're not given it. So think about this, right? If we're trying to figure out the total amount of time that it takes for the airplane to travel this distance here, then we're going to need to know its plane to ground speed. So if I can figure out what this V_PG is by using my relative velocity equations, then I can actually just go ahead and solve for delta t. So I know this is going to be delta x over t final. So therefore, my delta t final is just going to be delta x over the velocity of the plane relative to the ground. So again, if I can figure out this V_PG, then I can actually go ahead and and figure out what this time is. So now that brings us to the third step, which is we're going to set up the relative velocity equation. So V_PG, if this is what we're looking for, and now we're just going to stick this to our rules. We want the outer subscripts to be p and g. That means I want the first subscript of the first term on the right to be p, And then I want the last subscript to be g. And then the letter that goes inside of the inside subscripts is just going to be the only letter that we haven't used yet, which is a. So this is going to be a and a. So now the inner subscripts are the same, the outer ones are the same as the first term. So now we're just going to go through and figure out if we have all of our terms, if we need to flip anything, stuff like that. So V_PG, that's what I'm looking for. This is going to be V_PA. This is the 841 plus V_AG. So I actually have all of those numbers here. So I can just go ahead and straight up plug them in. So V_PG is just equal to 841 plus the 225. And if you work this out, you get 1066. So if you think about this, the velocity of the plane relative to the ground is just the velocity of the plane relative to the air, plus the velocity that the wind is basically helping it move along the ground. So these things kind of add together to form a faster velocity. Right? So it's kind of like if you were running and, you know, you have wind at your back, it kind of helps you and pushes you along and, you know, go a little faster. So now that we have this velocity here, now I can just plug it back into our equation and solve for delta t. So our delta t final is going to be the same distance, 2775 between the two cities divided now by 1066. And if you work this out, you're going to get 2.6 hours. So that's how long it takes. That makes some sense. If the wind is helping you along, then whereas before it took you 3.3 hours to travel the distance between them, now it only takes 2.6 because the wind is kind of helping you push you a little bit faster. So our answer choice is b, and that's it for this one, guys. Let me know if you have any questions.