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A plane traveled in the wind. The pilot needed to determine theflying angle to compensate for the effect of the wind. The pilotknew the wind vector (30 m/s, 220 deg...

Question

A plane traveled in the wind. The pilot needed to determine theflying angle to compensate for the effect of the wind. The pilotknew the wind vector (30 m/s, 220 degrees), and was paid to go to adestination at 140 degrees. The pilot wrote the destination vectoras (unknown resultant speed, 140 degree) and the plane vector as(45 m/s, unknown flying angle).Use vector drawing to find the unknown resultant speed andunknown flying angle. Show a vector diagram with scale. Hints Drawthe wind vector first

A plane traveled in the wind. The pilot needed to determine the flying angle to compensate for the effect of the wind. The pilot knew the wind vector (30 m/s, 220 degrees), and was paid to go to a destination at 140 degrees. The pilot wrote the destination vector as (unknown resultant speed, 140 degree) and the plane vector as (45 m/s, unknown flying angle). Use vector drawing to find the unknown resultant speed and unknown flying angle. Show a vector diagram with scale. Hints Draw the wind vector first, then use the end-tip of the wind vector to draw the destination vector as a line at 140 degrees, then use the start-tip of wind vector to draw a circle with radius 45 m/s. Use the intersection of the 45 m/s radius circle and the 140 degrees line to complete the vector triangle.



Answers

A plane flying due east at $200 \mathrm{~km} / \mathrm{h}$ encounters a $40-\mathrm{km} / \mathrm{h}$ wind blowing in the northeast direction. The resultant velocity of the plane is the vector sum $\mathbf{v}=\mathbf{v}_{1}+\mathbf{v}_{2},$ where $\mathbf{v}_{1}$ is the velocity vector of the plane and $\mathbf{v}_{2}$ is the velocity vector of the wind (Figure 30 ). The angle between $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ is $\frac{\pi}{4}$. Determine the resultant speed of the plane (the length of the vector v).

But hello. I hope you're doing well. So for this problem, never graph river plain that's flying in that direction. This is our vector B, and the problem tells us that the angle between north and or the angle um, between this V vector and the Y axis the north direction here is 50 degrees instead of you vector. And then we're told that our W vector is just going straight east like that. Okay, So in order to figure out the vector components of the NW Super expressing thes vectors in terms of their X and Y components, the X component of RV vector is going to be our B times sign 50 degrees. So that's going to be the component that's in this direction the X direction. And then to get the why components of our vector v you're going to multiply the by Kasey 50 degrees. That's going to give the component of the vector B. I'm sorry. That's in this direction. The vertical direction. This is our X component right here. And this is our white component right here. So the final result for the vector of the in terms of its X and Y components would be the sign 50 degrees I plus the coast side, 50 degrees J. So that's and then R W vector would just be whatever the value of W is in the eye direction, because it's just, um, in the X direction. So now once you have that kind of would help us out with part A. So in order to find resulting vector between V and W to add two vectors together, you just add their ex components and why components together. So W is just going to be W I. So you add these two components together to get some number I. Plus, you have the J components together. In this case, there is no J component for your W vectors. You just keep this Vico signed data RV coast on 15 periods, and J. So you just add the components of vectors when you add two vectors and then in order to find the magnitude of two vectors, are the magnitude of a vector. Let's say you have a vector that's V X in the eye direction plus V Y in the J direction. Find the magnitude of the Vector V that's essentially the length of the vector going. Thio essentially do the Pythagorean theory. We're going to take the square root of the sum of the squares of the vector components, so it's going to square root V X Square plus B Y square. So that's how you get the value of the or the magnitude of up your vectors, in this case, to BB plus W. And lastly, if you're wanting to get the direction, let's say you have, um, a V X this direction b y. This direction will make this into a right triangle. And then, if you're wanting to find this angle right here, you know that the tangent of an angle theta is equal. The opposite over adjacent. So opposite in this case is V. Y. Jason's in this case is V X engine data. So that means that to find data, you would take the inverse tangent of the Y over VFW's. Okay, so let's just some review on how to solve, you know, work with factors. So now we're going to go and dive into this problem. So we're told that so again, um, and drop our figure and we have our vector V here with an angle right here. If data is equal to 50 degrees, we're told that V it's flying at a speed of 180 MPH. That's essentially the magnitude of the single, so you want to split it up into its X and Y components. Remembering what we did before. The X component of our vector here is going to be equal to our magnitude 180 times The sign of this angle sometimes sign 50 degrees that will give us the I'm component in the X direction. So now, to get the B y, the component in the Y direction, you're going to do the same thing to take your value 180 this time multiplied by the coastline of the ankle. So coastline of 50 degrees. So that means that our final answer for the vector V is going to be, uh, equal to 180 sign 50 degrees, which, if you plug that into your calculator, you'll end up with 1 37. When 89 that's in the eye direction. Then, plus your Y component will be 180 cosign 50 degrees, which, if you plug that into your calculator you get 1 15.70 that's in the J direction. So it's sort of the vector and now moving on to a W vector. That's equal Thio. So we have r W vector in this direction here. So it's just in the X direction so we don't have to even worry about this Jake opponent. We just take a magnitude of r w vector, and that's in the eye direction. So, um, are w vector. It tells us that the wind is blowing at 40 MPH. So are W vector is just 40 in the attack direction. Remember that I direction is your the same as your extraction and the J is in the Y direction. Okay, so this is our final answer for part A. This is the vector components of alright two vectors BMW that were given. So now we need to find the resulted vector V plus W. This is part a part B is V plus w So essentially add these components together. So you have V is 1. 37.89 I plus 115.7 d j w vector is gonna add that here 40 i plus and there's no J components. Zero j So if you add these together, you end up with V Plus. W is equal to adding these two I terms. Together you get 177.89 I adding these two J terms together you get plus 115.7 d j. And this is our final answer for part B of this problem. So now we need to figure out three ground speed the plane, which is just the magnitude of this V postal. So the magnitude of this view plus w it's going to be cool to the square root of the sum of the squares of these two components. Just like the Pythagorean theorem. You have a component your, um this X component of your vector in that direction. That's why component of your vector in the vertical direction. You're trying to find this high pot news value magnitude value here. So you're going to take 177 89 squared plus 115.70 squared. Take the square pretty. So you end up part C you end up with. When you plug this into your calculator, you get 212 MPH. That is the ground speed of the airplane. All right, so lastly need Thio figure out the bearing of the plains essentially the angle that it makes with north direction. So to do that, we need to take this vector here. Um, kind of draw it out. So our x component is 177.89 So 1 77 like 89 the X direction and the Y direction are white. Component is 115.7, 115.70. Okay, so we know we just got that. The magnitude of that is 212. That's its high pot news. There. We're gonna find out this angle, Fada, right here. So to do that, we know that tangent of the angle Fada is opposite over hypotenuse. Um, I'm sorry. Opposite over adjacent. So it's opposite is 115.70 over adjacent, which is 177.8. So our theta is going to be the art tan of 115 70 over 177.89 You plug that into your calculator, you get data equals 33. Um, there's just about 33 degrees. But the bearing, in order to write out the bearing of the plane, needs to figure out what the angle is with the north direction. So this is our says North. This is east. If this angle here data is equal to 33 degrees, let's say this single here, let's call it Alfa here. Is it going to be called a 90 degrees minus data. So Alfa is equal to 90 degrees minus data, which is 33 degrees, is equal to 57 degrees. That means our final answer for the bearing of this plane is from the angle with the direction of the plane of the plus W um, with the north directions 57 degrees, it's going to be north 57 degrees east. And this is our final answer right here for party. All right, so we have our answers for Parts A, B, C and D for this problem. All right, well, thanks. And I hope that helps

Going to given data. We can see that V's making 50 billion with why exists So we can later be? Yes. Since this is 50 degree angle, that means this Angelus, because this complete handle is 90. So the 1st 40 degree angle, the best one district and night for the a part of the question these equal since the manager is given as 1 80 that it is 1 80 going 40 degree. I bless one baby signed for Hillary De So when we will might apply these 2180. My bad with cost 40. This is a 41 37 point a Did I bless 115 17 D. So this is the director in i n j Careful and w vector. We can like that since this is the group which is so this is making zero degree angle attacks access So we can say that the magnitude of the blues 40 So we can I d says 40 cost of judo. I less for be signed you Jay. So this is equal to 40 cause it was zero is one. So this is 40 I and this part is you know, because I enjoy Rosile. So this is the director, and this is the better. Now, the second part of the person he is, we have to find we that helpless the flu it Then we'll get the resident. Okay, so this is the president. So from this we will get B plus W. This is equal 1 37 point it it I bless 115.7 de plus 40. I sure this is 1/4 1 37 point. It did plus 40. I blessed one under 15.7. Is it? Till then? Will add the stool will get 1 77 point it it I plus 115.7 g. So this is Director V plus vectored. Now the see part of the question is to find magnitude, so build its magnitude of the problem. This rate as manly Jude off three plus w this is equal. Do route 177.8 less 11 played 0.7 squids. So general, calculate this We are going to get this value s here. You get fruit for 50 seven 0.78. It is approximately will do to wondered. Well, mind who will be eternal light. Now the be part of the person is toe find direct, sensible light. Since what is What do will calculate the says cost Tita is a good this part 1 77 point did divided by 200 Well, to that means we'll get 1 77 But hair will get a little salty ties. You called because in verse 1 77.8 it divided by 212. So this is approximately equal Toto Titi daily So this is teeter So we can say that 90 minus trita If we're calculating from north 90 minus Tita, this is you called 90 miners started Lee So this is 57 degrees. So finally we can write down says from north towards East 57 Billy, the distributed dancer

Dough is a bent speed off the blend is the land off the some vector We was evil plus me do Therefore we won equals the one on the geo read weepers we do because by 14 on Vito sign by four This becomes we do into one of our rude too And we doing one upon because we have the diagram. Which song? And this Yeah, it is we do on this is by a wonderful no we compute the sonny the sum So the equals even Plus we do that is equal We won armadillo plus be too upon routes to on the telephone too So this is equal Duke The one plus we do upon rude to and they don't want to No, we have the v waas the record The one plus we do upon to Hollis Fire. Plus we do upon rude to respond to this human perimeter is the one equals 200 on we do It was for d way Bless it in the abdication that is no V becomes on the road off 200 plus Fouty upon route too. What is God class Foodie point Rude too What is bi After solving, we get the as a little golf find 2913.70 to it. This is equal Do dude. 30 point, you know. 97 Get it done. What? This is our regarded intense. It is the answer to the problem.

Base. If we set up the corn and access that the North is in the positive direct, why direction and the East is in the positive X direction we can set that V winning is equal to 50 times. Co sign of 45 I, plus the sign off 45 j. With respect, it's still air. The velocity vector off the plane could be written as V playing is equal to 250 times the co sign of 30 I, plus the sign of 30 j. So therefore, feet total be total, which is viewing. Plus weakling is equal to 50 co sign 45 I. We can actually factor out the ice 0 50,045 plus 250 co side 30 multiplied by I plus Negative 50 sign of 45 Remember its name because it's going in the opposite direction plus 250 sign of 30. This whole thing was multiplied by J, so therefore we can rewrite this as 25 times the square root of two plus 125 times the square root of three I plus 125 minus 25 times the square root of two J, which is approximately equal to 251 with nine i plus 89.6 j. Okay, now the ground speed, which is the magnitude is equal to the square root off 251.9 squared, plus 89.6 squared, which is equal to 267 kilometers per hour. If you wanted to find the angle that it makes the ground, we can say that data is equal to you. The Art kendrick off 89.6, divided by 251.9, which is approximately equal to 20 degrees. Therefore, the true angle the plane is about end off 19 minus 20 or end 70 degrees east.


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