What Is the Magnitude of the Velocity of the Wind Relative to the Water, in Meters Per Second?
Question
The velocity of the wind relative to the h2o is crucial to sailboats. Suppose a sailboat is in an ocean current that has a velocity of 2.20 grand/s in a management $30.0^\circ$ due east of due north relative to the Earth. It encounters a current of air that has a velocity of iv.l m/s in a direction of $fifty.0^\circ$ south of west relative to the Earth. What is the velocity of the current of air relative to the water?
$v_{wo} = 6.68 \textrm{ yard/due south, } 53.iii^\circ \textrm{ S of Westward}$
Solution Video
OpenStax College Physics Solution, Affiliate three, Trouble 62 (Problems & Exercises) (5:38)
Sign up to view this solution video!
Calculator Screenshots
Video Transcript
This is College Physics Answers with Shaun Dychko. Nosotros're gonna figure out the velocity of the Current of air with respect to the Ocean in this question. So what nosotros're given is the velocity of the Bounding main with respect to the Earth. This is the current and information technology's 2.two meters per second, 30 degrees to the due east compared to due north. And then there's as well the velocity of the Wind with respect to the Earth, which is iv.five meters per 2d 50 degrees to the south of west. So this is a drawing of both those vectors, but we're actually not gonna take the resultant of these two every bit their drawn here. Instead, permit's consider the addition of relative velocities with regards to this pattern of subscripts. What we desire is to know the velocity of the Wind with respect to the Ocean. So to find that nosotros can add the velocity of the Air current with respect to the World to the velocity of the World with respect to the Sea. And when we have it gear up so that the inner subscripts are the same. They're both e for Globe in this instance. That means they will cancel. And we'll be left with the outer subscript velocities hither. And so we take w and o, and then this volition find u.s. the velocity of the Air current with respect to the Sea. And then nosotros're given this term conveniently right here. But this term we're not given explicitly, nosotros're given the velocity of the Ocean with respect to the Earth. What we want is the velocity of the Earth with respect to the Ocean. At present to have these subscripts flipped around yous merely need to put a negative sign in front. And so this velocity that we want is going to exist the negative of the velocity of the Ocean with respect to the Earth that we're given. And to accept the negative of a vector is to accept the same vector but in the contrary direction. So what we really have is this flick here. This is the velocity of the World with respect to the Ocean in the opposite management to the Sea with respect to the Earth. So that's 30 degrees to the west compared to south. And we'll add together to that using the head to tail method, the velocity of the Wind with respect to the earth. And the east subscripts are both inner subscripts, they cancel. And nosotros're left with w and o. And that's what this would be this resultant, the velocity of the Air current with respect to the Ocean. Then we're gonna discover this angle hither to the s of west, and we'll observe this magnitude of this speed here. Okay, and then the 10-component of the velocity of the Wind with respect to the Bounding main; will be the sum of the x-components of these two vectors. And then the Current of air with respect to the Earth, it's x-component is to the left. And and so that's why we take this minus sign here. And along this dotted line is the adjacent leg compared to this angle here. And and so we have cosine of theta w e multiplied by this hypotenuse. And I'm referring to this triangle here when I say that. And so this is the x-component and this is the y-component for the Wind with respect to the Earth velocity. Okay, and so with regards to the World with respect to the Sea velocity. We are finding the opposite leg of this blue triangle. Here'south the blue velocity triangle, and this opposite component hither, we'll find past using the sine of this angle multiplied by the hypotenuse. And that is to the left again, similar this, and so that'south why in that location's a minus sign hither. Then nosotros have negative 4.5 meters per 2nd times cos 50, minus 2.2 meters per second times sine xxx. Giving negative 3.9925 meters per 2nd, is the x-component of the velocity of the Wind with respect to the Ocean. And then for the y-component, it'southward the aforementioned idea, we have to look at these triangles to find out how to find the y-components. And then nosotros're gonna utilise the sine part for the green triangle. And we'll use the cosine function for the blue triangle and both of these y-components are directed downwardly. And then they're both negative. So negative and negative. So we have negative 4.v times sine of fifty minus 2.2 times cos of 30. Giving negative five.3525 meters per second. So nosotros're gonna combine these two components using Pythagoras to go the magnitude of the resultant velocity; And then that's the square root of negative 3.9925 meters per second squared, plus negative 5.3525 meters per seconds squared. Giving six.68 meters per 2nd. And then the management volition be the inverse tangent of the y-component divided by the ten-component. I'm not concerned with negative signs here, because I know that we're finding this angle here. And so the direction is taken intendance of by looking at the picture and knowing that this is to the due south compared to w. And then you meet this central directions hither an angle that'due south here is to the south compared to west. So we taken the changed tangent of the y-component divided past the ten-component. And that gives 53.iii degrees, so our final respond for the velocity of the Wind with respect to the Ocean is 6.68 meters per 2nd, 53.three degrees south of west.
Solutions for issues in chapter 3
Source: https://collegephysicsanswers.com/openstax-solutions/velocity-wind-relative-water-crucial-sailboats-suppose-sailboat-ocean-current
Post a Comment for "What Is the Magnitude of the Velocity of the Wind Relative to the Water, in Meters Per Second?"