A river flows due south with a speed of 2.0 m/s. You steer a motorboat across the river; your velocity relative to the water is 4.2 m/s due east. The river is 500 m wide. (a) What is your velocity (magnitude and direction) relative to the earth?

Video transcript

Welcome back everybody. We have a river that is flowing north with a velocity relative to the earth of 1.8 m/s and we are told that a boatman john traveling across the river and he has a velocity relative to the river of seven m per second. And we are tasked with finding the direction and magnitude of john's velocity relative to the earth. Well, john's velocity relative to the earth is just going to be the summation of the other two velocity vectors of john's velocity relative to the river plus the velocity of the river relative to the earth. So just to be clear, we are looking exactly at this vector right here. This is john's velocity relative to the Earth. Now, in order to find both the direction and magnitude, we are going to have to find this angle data right here. Now, this is coming into contact with this imaginary vector right here And then we're also going to have to find the length, the length of this vector as well. Well, let's write down some important formulas that will be able to use the magnitude of any given vector will say vector. A in this case is just 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 angle that we are looking for is equal to the Y component divided by the X component, but what are the y and X components? Well, as a matter of fact, they are just going to be the vectors that we used to get, john's velocity relative to the earth. So the X component is going to be john's velocity relative to the river of seven m per second. And the Y component is just going to be the velocity of the river relative to the earth of 1.8 m per second. Now that we have those values, let's go ahead and plug into our formulas. I'm actually going to start with the magnitude of our velocity vector magnitude is going to be equal to the square root of the X component squared seven squared plus Y component squared. And when you plug this into your calculator, you get 7.23 m/s. This is going to be our magnitude. Now let's go ahead and use this formula over here. Now we actually want to find data. So I'm gonna do something real quick to isolate data. I am going to take the inverse tangent of both sides. This yields the formula that theta is equal to the arc tangent of our Y component over our X component. So let's plug in those values, we have that. This is equal to the arc tangent of the Y component of 1. divided by the X 0.7. Which when you plug into your calculator, you get 14.42 degrees. Now we're almost done here but we still have to deal with this last term. Right, is it? North of east is north of west While we're looking at this angle right here in this angle makes the velocity vector go north of this west direction, so this is going to be north of west. Now we have found our magnitude and the direction of the velocity vector that we are wanting to look at or responding to answer choice. C. Thank you guys so much for watching. Hope this video helped. We will see you all in the next one.