A canoe has a velocity of 0.40 m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50 m/s east relative to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

Video transcript

Welcome back everybody. We have a river that is traveling east that flows east and we are told that the velocity of the river relative to the earth equal to 0.95 m per second. We're also told that a boatman taking out a boat for the first time and that he travels exactly northwest and we're told that the velocity of the boat relative to the earth is 0.30 m per second. And we are tasked with finding the velocity of the boat relative to the river, both the direction and magnitude of this factor. Well, in general, it follows that when you have the velocity of an object relative to the earth is going to be the velocity of that object relative to another moving object plus the velocity of that moving object. In this case our river relative to the earth. Now we want this middle term right here. So I'm actually gonna rearrange this equation just a little bit. We're gonna have that the velocity of the boat relative to the river is equal to the velocity of the boat relative to the earth minus the velocity of the river relative to the earth. Now, just conceptually here, we know that the velocity of the boat relative to the river is going to be somewhere in this general direction, but we need to find the magnitude and direction of this philosophy. Now we're going to need to use two important formulas for that. The magnitude of any given vector, let's say for this example a is going to be equal to the square root of its X component squared plus its y component squared. We also know that the tangent of the direction that we are looking for represented by this angle. Theta is going to be just its Y component divided by its X. Component. But what are the Y and X components of the velocity of the boat relative to the river? We are actually going to use this equation right here and we are going to do component addition and subtraction. So A X or the velocity of the boat relative to the river. Its X component is just going to be equal to the velocity of the boat relative to the earth. Its X component minus the X component of the velocity of the river relative to the earth. Well, the ex opponent of the boat relative to the earth Is this factor right here. And since it travels perfectly northwest, we know that this angle is 45°. So let's go ahead and use that. We're gonna have that this factor right here, it's going to the left, so it's in the negative X direction. Going through the magnitude for the velocity times the co sign of this is going to be minus the X. Component of this vector up top. Well, it's only traveling in the X negative extraction. So it's just gonna be negative 0.95 words magnitude. When you plug this into your calculator, you get 0.74 m per second is the X component. And we're gonna do the same thing with the Y component. It follows the same format. So I'm just gonna save us a little time here and just say that we're trying to find the velocity of the boat relative to the river and its y component. Well, the X component of this factor right here, it's traveling in the positive Y direction. Apologies to the Y component is going to be 0.30 times the sine of 45 the minus the Y component of this factor up here, but it's not traveling in the Y direction whatsoever. So it's just going to be zero this into your calculator. You get 0.21 m per second. Now that we have both A X and a Y. We are ready to find the magnitude and direction of our desired velocity vector. The magnitude of the velocity of the boat relative to the river is equal to the square root of its X component or 0. squared plus 0.21. Its y component squared when you plug this into your calculator, you get 0.77 m/s. Now let's go ahead and use this formula right here. Now we actually want to. So I'm gonna isolate theta by doing this. I'm gonna take the arc tangent of both sides and we get that the data or the direction that we are desiring is the arc tangent of the Y. Component over the X. Component. So let's go ahead and plug those in. We have the arc tangent of its Y component, 0.21 divided by 0.74, which when you plug into your calculator, you get that. The desired direction is 15.84 degrees north of east. So now we have found both the magnitude and the direction of the velocity of the boat relative to the river corresponding to answer choice. D Thank you guys so much for watching. Hope this video helped and we will see you all in the next one.