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Thc cvc of hunricanc passcs ovcr Grana Bahama Island in dircction 60_ 09 north of west with spced of 42.0 km/h: Thrce hours atcr, the coursc of [ne hurrican suddeni...

Question

Thc cvc of hunricanc passcs ovcr Grana Bahama Island in dircction 60_ 09 north of west with spced of 42.0 km/h: Thrce hours atcr, the coursc of [ne hurrican suddeniy shifts duc north, and its speed slows to 27.0 kmh. How far from Grand Bahama the hurricane after passes over the island?

Thc cvc of hunricanc passcs ovcr Grana Bahama Island in dircction 60_ 09 north of west with spced of 42.0 km/h: Thrce hours atcr, the coursc of [ne hurrican suddeniy shifts duc north, and its speed slows to 27.0 kmh. How far from Grand Bahama the hurricane after passes over the island?



Answers

The eye of a hurricane passes over Grand Bahama Island in a direction $60.0^{\circ}$ north of west with a speed of 41.0 $\mathrm{km} / \mathrm{h}$ . Three hours later the course of the hurricane suddenly shifts due north, and its speed slows to 25.0 $\mathrm{km} / \mathrm{h}$ . How far from Grand Bahama is the hurricane 4.50 $\mathrm{h}$ after it passes over the island?

So in this problem we're going to combine the relationship between distance traveled and the velocity and the time which you've been traveling along with vector edition and the components of vectors to figure out how far something has traveled, if it has a certain velocity for a given amount of time and then shifts velocities and travels through another instance of time. So if we start at the origin and we're told that this object moves 60 degrees north of west and if we set up our axes such that north is vertical east and to the right and west is to the left, 60 degrees north of west is the angle between the displacement vector or the velocity vector, and west Will be 60° north of West. And were given that it travels with the given velocity in this direction for three hours. So if we travel along this line will end up say at this point P After three hours, we're then told that the velocity shifts to be directly north And it travels for what will be 1.5 hours. And we'll end up at this point D. And we want to know the final displacement between D and the origin. We want to know really want to figure out what this factor is. And so we need to know the displacement between the origin and P. And we need to know the displacement between P. And D. So I'll call this red vector A. And this blue vector B. So in order to figure um to figure out the magnitude of each of these vectors, we have to know the total displacement. And we're told that along a the The object travels in that direction for three hours With a speed of 41 km, 41 km/h for three hours. We know that the distance traveled is just the velocity, assuming is constant times the time it took time you've been traveling. So this first displacement, so the magnitude of A will be the velocity times the distance at which we've been traveling at that velocity, which we're told is three hours, which if you do that math, that will be 123 km. Just the magnitude of a is 123 km B. We're told that it moves directly north with the velocity of 25 km/h for 1.5 hours. We know it's 1.5 hours because we want to know the position 4.5 hours after it leaves the origin and it takes three hours to get from the origin to point P. So therefore 1.5 hours, So 4.5 minus three, which is 1.5 hours to go from P to the point D. And therefore the magnitude of B is 37.5 kilometers. And we were given also that uh a travels. So if we're gonna figure out the displacement vector D. We need to some A. And B. And in order to do this, we need to know the components of A. And B, not just their magnitudes. And so B be as simple to figure out. He just points in the positive Y direction with a magnitude of 37.5 in the white hat direction A is a little bit different only because we have two components. This time we have an X. Component and we have a wide component. We know that the A. R. The X and Y components are related through trigonometry. So if you do, if you look at this as a right angle the sign. Uh This is the magnitude of a, the sign of this triangle will be the opposite side, which is the magnitude of the Y. Component over the total magnitude. Such that ah for a Uh breaking up into its components, which will be the magnitude of a. In the X. component, will be the times that co sign 60°. Again. Just using the definition of the co sign for this right triangle in X. That direction, plus the magnitude of a climbs the sign of 60 degrees in the Y direction. And these are the two components of a magnitude of a is 123 And then times the coastline of 60. Well give us 61.5 ex happy Plus 123 times the sine of 60. Give us 106 .52 in the white hat direction. And these are both in km. So if you want to add these vectors, we just have to add their components. So a the X component in the way vectors 61.5 X hat Plus 106.52 Why hat? And I should make a, make a note here that the X component while the magnitude might be positive, we know that a points to the left and up. So therefore it has to have a negative X component. And then we add the we add B which only has one component, which is 37.5 in the white hat direction. And so if we add up these these components, We'll see that well, we still have negative 61.5 in the X direction And we have to add these two components. So 106.52 Plus 37.5 Is 144.02 km in the Y Direction. So these are the steps that we have to take to get to the final point. And we want to know the net displacement. We want to know the magnitude of this vector, the magnitude of the vector is just a square to some of the square of the components under a square root, So 61.5 squared plus 144.02 squared. And this is in both. These values are in kilometers, 61.5 Squared plus 1 44.2 squared, All under a square root, will give us about 156.6 km, and this is the net displacement of the object after it Made two different displacements.

Here we will consider the petition off island at the origin of this cottage in corner system. Firstly, the hood again isn't wing at a velocity of 41 kilometer per hour for three hours at 60 degree, not off rest. Now we can calculate distance covered by who again doing this time by using the relation distance Equal toe DeSipio multiplied by time. Here, see peaches. 41 kilometer per hour on time is dollar. So this gives us descends. Gored doing this time equal dough 100 23 kilometer. Now we can write the displacement covered by hood again during this time. And it's accompaniments, as do you on equal do. Do you want X Well s do you, wan? Why do you want extra physical toe? 123 kilometer sign off. 63 d on the unit Director is I had was 123 limiter. Go sign off 60 degree. And here Why? Component is and negative direction. So we have your you know, directories minus Jack. So this gives us a factor. D one equal toe. 106. Why it? Five kilometer? I had minus 61.5 limiter Jay had after three hours. But again genius years apart and moves do not address it. Paid off 25 kilometer for next 1.5 hours Now began Calculate dissents covered by who again doing this direction They do equals toe. Here s a PT in this bodies 25 No matter But all are in tow, sir. Previous one point flight bowler excuse us descends equal toe that is seven fine five kilometer We can write the director as here in a sense guard, by hurt again is along. He's new north. So we have That is 7.5 kilometer on the unit Director New North ease I had Now we can calculate total displacement governed by her again Total displacement equal toe displacement covered in the 1st 3 hours but less Miss Willis Mandic Our in the last 1.5 Ours as displacement. New one is one under six applied five kilometer I had minus 61 wide. Five limiter Jay had plus displacement did to Is that the seven point five kilometer I hired Excuse us. No displacement off her again from island equal toe. 144 kilometer. I had minus 61 point five. Limiter, Jed. Now we can tell cleared magnitude off net displacement. By using that I go to student, that is magnet. Utica will go. So get a load off. So get up X component. Well, us. So get off. Why component? This gives us magnitude up. Net displacement equal dough 100 56 A point fi bad limiter. So this is the dissents off for weekends I from island after 4.5 hours.

In this problem, we're going to find how far hurricane is from where it started after a certain amount of time. During this time, the hurricane will have two distinct motions. Let's look at those two motions here. First part of the motion, it'll be traveling 41 clamors per hour For three hours and at an angle of 60° north of West. Then It travels 25 commerce Prowar 1.5 hours due north. So we want to find out how far at the end of this is the hurricane relative to where it began. Just a note. This is not the distance it travels, as I mentioned that more about that later. So let's draw out the actual displacements. So we started the origin, going to be moving Along the line 60° to a point here. Now, displacement factors by definition are drawn from the starting point To the ending point of that particular motion. So this is T. one dissuasion factor. Now We were always v. one vector always, we're always moving 6° North or West. So that means this displacement factor also Is 60° northwest. Now we travel to north For 1.5 hours we get to this point. So my dissuasion vector here is that vector completely vertical. And again back to the concept are the sweating factor. It's drawn from where you began to where you ended up. So this is the total. Does not care how you got to. End point displacement is drawn directly from starting the final point. Now, if we look at this diagram looks to me, doesn't matter if you use triangle rule this add vectors, parallelogram rule. It looks to me the D. Toto factor is the one factor Plus T. two factors. So what this problem is gonna come down to is being able to elaborate on the components of D one, D. Two and get the components of the total. And then we can find the length of the total. Now, before I go back and find the components, let's just understand we no, from a standpoint of distance, How long the one vector is just from a physics standpoint. T one is going to be for t v one time delta T one. That's going to be links Of the one doctor. And because we always are moving during that interval At an angle 60° that The one and V one are in the same direction and the way we write that maybe right, that is the one factor the one vector delta T. What in this one simple statement, it has The relationship between the magnitudes just wrote T one, B 1, Delta T one. And it also says they are in the same direction. Likewise, D two Would be V2.52 in terms of it For the length of D two vector. And that means D. To vector Would be the two factor Got to T. two. So before we can actually now work on di tutto we need to go back and get our components. So let me go back to the triangle for view one. It's going to be the one why? It's gonna be v. one x. The second quadrant vector, meaning that you must move in negative X. First and then positive. Wide. To construct the vector. So V one X. is going to be negative. The one act is the adjacent to the 60° angle. So it's going to be minus the one co sign 60 And be one. Why? Why is the opposite side to the angle? It's going to be V one sign 60 degrees. The two is easier. I don't want to do anything in extra constructed So V two X 0. The two. Why? To construct the vector? I gotta moving positive. Why? The full length of the vector? So that means V. Two, Y. Is V. Two. So we have our components now. So now we can look at the total the total X. Is going to be D one, X plus T. To uh to Do you want X was 22 x. Which is equal to the one, X. To T. One Plus v. two. X doubt the T. Too. But I told you V two X zero. So this term does not contribute. So this would become -31 Co sign 60° delta T one. And putting in our numbers 41 -41 km Krauer Coast side of 60° three hours. This works out to be -61.5 km Now. Likewise for D. two. He toto. Why? This is gonna be V. One, Y delta T. One Plus v. two. Why Got to T. 2? So this is the one sign 60 degrees, There are 31 plus the two Times Delta T. two. And then we give myself some room here And putting in our numbers 41 kilometers per hour, sign 60 degrees three hours plus 25 km/h, hours. This works out to be 140 for kilometers, that's the Y. Component. So we now have the components, so we really have all the sides of the right triangle to use segments here. Um You know, here is here is D. Toto, you told her why the total X. So just forget about the arrows and anything like that. All we care about our decides now so we can use the Pythagorean theorem to get the total, so t total the man to the vector, the length of the hypothesis is going to be -61.5 km squared plus 1 44 kilometer square, And this works out to be 157 kilometers. That is how far you are relative to where you began again. As I mentioned, it is not the distance you travel, it's not odometer reading, and that's the whole problem.

The first vector can be growing up into its components by doing 41 times the co sign of 60 degrees to find my direction and 41 times the sign of 60 degrees in the J direction. This gives us a vector that is 20.5 in the eye, plus 35.51 in the J in kilometers per hour. However, since it's going west, it means that it is negative 20.5 in the eye. For the second vector, it is only going north. Therefore it has zero in the eye and it has 25 in the J. Therefore, the two is just 25 j in a kilometers report per hour. Next to find the displacement of the 1st 3 hours, we know that the displacement one vector is three times the V one vector because velocity is equal to distance over time. So therefore, distance is equal to velocity times time, so D one would be equal to three times negative. 20.5 i direction plus three times 35.5 in the G direction. This gives us a displacement vector that is equal to negative 61.5 in the eye direction plus one of 6.5 in the J direction and that's in kilometers. Next. For the second displacement, we know that it equals 1.5 times V two. Therefore, D two equals 1.5 times 25 in the J direction. Therefore, D to is equal to 37.5 in the J in kilometers were asked how far the hurricane traveled from its origin in the 4.5 hours. Therefore, we need to find a total displacement vector, which would be which can be found by adding up vector from the display. The displacement were asked to find how far the hurricane traveled from its origin in the 4.5 hours. So we need to find a total displacement vector and that total displayed in vector is equal to the first displacement vector plus the second displacement vector. Therefore, the total displacement is equal to negative 61.5 in the eye +106 point five plus 37.5 in J in Columbus. This gives us a total displacement vector that is equal to negative 61.5 in the eye plus 144 in the J in kilometers. Now we need to find the magnitude of this factor which is equal to the square root of negative 61.5 squared plus 144 squared. This gives us a magnitude of 157 squanders.


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