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\bullet On a training flight, a student pilot flies from Lincoln, Nebraska to Clarinda, Iowa, then to St. Joseph, Missouri, and then to Manhattan, Kansas (Fig. 1.30). The directions are shown relative tonorth: $0^{\circ}$ is north, $90^{\circ}$ is east, $180^{\circ}$ is south, and $270^{\circ}$ is west. Use the method of components to find (a) the distance she has to fly from Manhattan to get back to Lincoln, and (b) the direction (relative to north) she must fly to get there. Illustrate your solutions with a vector diagram.

$10.4^{\circ}$

Physics 101 Mechanics

Chapter 1

Models, Measurements, and Vectors

Physics Basics

Cornell University

University of Michigan - Ann Arbor

University of Sheffield

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this's Chapter one problem. 64 in this problem were given three different vectors. The represent three legs of a journey that the student pilot is taking. And then we're looking for the magnitude and direction. Those are asked in part A and Party B separately of the fourth vector needed to complete the journey should end up at her starting point. So, in other words, the four vectors we have are these four factors that air drawn in the middle of the screen. Right now we're given the magnitude and direction Director's A and B and C, and we're looking for Dr De. We also know that if we add these up together, add them as vectors. We have a plus plus C plus de equal zero. And so we can also write that as negative Abel's people see equals after D, because whatever the sum of A B and C is that's going to be. That sum is going to be the vector pointing here in opposite direction rate from tale all the way to tip, and we're looking for the rector. That's the opposite of that. So we basically have two since you can adventurous with components we've got to find the accent. Why Components The vectors A, B and C add them up. Like in this equation Appear. Take the negative and that's going to get you the x and Y components of after D. And then you can put it back together to find Matt itude Indirection. One tricky thing here is that the angles given are measured in a different way than usual. Were used to angles being given as counterclockwise from the X axis. But here the angles are given as measured clockwise from north. So what I suggest doing you, Khun, define your axes. Um, in a different way, if you choose. But I was just calling North X in this case and calling east. Why? Because that way our angles air still defined as zero degrees on the X axis, 90 degrees at the Y axis and continuing on from there. So that means we can still use our same old equations for X and Y components of a vector. So if you've got a vector, are the ex component is going to be directors? Magnitude are Times co signed data worth data is defined as above. In this case, it's defined clockwise from north and are why is he going to our assigned data? All right, so now that I've got all this set up, we're going to use the equation at the top to find the components and Dr D So the ex component of after d call us D. X actually here D X is equal to minus the sum of the ex components of the other rector's. So, Mr A, we've got 147 kilometers and I want to just write the unit's kilometers at the end times the co sign of 85 degrees plus 106 kilometers times that co sign 1 67 degrees plus 1 66 kilometers times the co sign of 235 degrees. And remember, these all have units to this entire quantity. Everything we added up each had units of kilometers. So the entire quantity is goingto have units also of kilometers as we expect. And so what you got if you put that in a calculator and do the arithmetic is a positive 185.7 kilometers. And to some of the some of those terms inside, differences are night are negative. You have a moment to take the negative of the whole thing. You get a positive number, all right, so that is the ex component of the blue after D For the white components, we do basically the same thing, except it'll be same magnitude and same angles as in the expression above. Just everywhere, there's a co sign now. Instead, you're going to have a sign, and when you add the's all up and take the negative, you end up with negative 34.3 kilometers and let's check in. The diagram of that makes sense, given the magnitude of after A B and C having 185.7 kilometers as the northward component of after D. Seems reasonable and you can see from the diagram. Thie East West component should be a lot smaller, and it looks like it's probably pointed a little till the West meeting. Since positive wise the East, we expect the wife component to be negative, and it is so that's good, that little tricks out now. Our last step here is we found the extra key components of after D, but we were asked for the man with you in direction, so I've got to convert that into magnitude and direction. So for that use that the magnitude of a vector is equal to the square root of the sum of the squares of its components. Uh, whips columnist Cindy. So magnitude D X squared plus d y squared, and you put in the DX and Iwai from below, and you get 189 kilometers so you answer to per day is going to be 189 kilometers for part B. Want the direction relative to north so you can use that the angle measured in the same way as all the other angles. It's going to be inverse tangent of the Y, over DX and again, you plug in your same values for why and the X here, and you end up finding negative 10.5 degrees, or depending on your calculator settings. You make it minus 10.5 degrees. You might get positive 349.5 degrees. Either way, this means thie direction is pointing slightly counterclockwise of north. There, suddenly to the left of north. Or, in other words, this is you. Could you could write. This is minus 10.5 degrees. You could also just write it as 10.5 degrees west of north, and that's that would indicate the direction of the spectre.

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