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Flying in the Wind

Patrick Ford
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Hey, guys, let's take a look at this problem here. So we got these two cities without any wind. This airliner could make the trip between them and some amount of time, but then the wind starts blowing, so there's 225 kilometer wind that blows from west to east. We're trying to figure out how long the trip is from A to B is gonna take now that the wind is blowing. So we're gonna solve this because we have some relative motions and 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 gonna draw these diagrams. But I'm actually gonna draw to one for the before situation and one for me after, So let's get to it so that 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 could make the trip in some amount of time, so the before situation is with no wind, and then the after is gonna be when the wind starts blowing. So this is gonna be some from the wind. So let's check this out here. So we got these two cities, right? So we've got a and B, we know be lies directly west of our east of a on DSO. The Delta X, we're told, is equal to 27 75 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 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 27 75/3 270.3 hours. And so the velocity of the plane through the air is 8.89 point 41 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 gonna start blowing. And so now there's gonna be a couple of velocities to consider. There's gonna be some relative velocities, so let's draw the same exact diagram. So we've got again City A and city be and we know the Delta X over Delta T or Sorry, we know the Delta X. We know that this displacement is 27 75. 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 gonna 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 gonna 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 gonna go ahead and stick to the steps for solving relative motion problems. So you've got this diagram here. Now we just identify the objects in 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 were 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 gonna 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, you know, it's positive because it points in the same direction, right? So velocity of the air. So the next question is, what is the velocity of that air measured relative to? Well, so are third reference is gonna be the ground. So that's the third reference here. The grounds we got the plane is traveling through the air, but the air is traveling relative to the ground. So those there are three objects and references. So now we're gonna 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 P. A. And so that's 841. And the air is real, is measured relative to the ground. That's the wind speed or whatever. And so that's V A G. So we've got our two variables here. So you got V EPA, which we know is 841. We've got V A G, which we know is 225. So remember, we're gonna need a third one in order to start writing a relative velocity equations. 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 to the air and the velocity of the air relative to the ground can come and combine, and they'll produce a velocity of the plane relative to the ground. That's V P. G. So both of these things here combined form V P. G. That's the third variable. So this is actually what we're looking for here, R v p g. Or at least we're not giving 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 gonna need to know it's plain to ground speed. So if I can figure out what this V p. G is by using my relative velocity equations, then I can actually just go ahead and solve for Delta T. So I know this is gonna be Delta X over T final. So therefore, my Delta T final, it's going to be Delta X over the velocity of the plane relative to the ground. So again, if I can figure out this V p. G, then I can actually go ahead and figure out what this this time is. So now that brings us to the third step. We're just we're going to set up the relative velocity equation, so v p g, if this is what we're looking for and now we're just gonna stick this to our rules, we want the outer subscript to B, P and G. That means I want the first script subscript of the first term on the on the right to BP, and then I want the last subscript two BG. And then the letter that goes inside of the inside subscript is just gonna be the only letter that we haven't used yet, which is a so this is gonna be a and A. So now the intercept scripts of the same the outer ones of the same as the first term. So now we're just gonna go through and figure out if we have all of our terms if we need to flip anything, stuff like that. So, um bbd that's what I'm looking for. This is gonna be V P. A. This is the 841 plus V a G. So I actually have all of those numbers here, so I could just go ahead and straight up, plug them in, so V p g is just equal to 841 plus the 225. And if you work this how you get 10 66. 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, that kind of helps you and pushes you along and, you know, going a little faster. So now that we have this velocity here now, I could just plug it back into our equation and solve for Delta T. So a Delta T final is gonna be the same distance 27 75 between the two cities, divided now by 10. 66. And if you work this out, you're gonna 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 3.30 hours travel the distance between them. Now it only takes 26 because the wind is kind of helping you push you a little faster. So we're answer. Choice is B and that's it for this one. Guys, let me know if you have any questions.