A kayaker needs to paddle north across a 100-m-wide harbor. The tide is going out, creating a tidal current that flows to the east at 2.0 m/s. The kayaker can paddle with a speed of 3.0 m/s. (b) How long will it take him to cross?

Video transcript

Hey everyone in this problem, an athlete swims due east along a straight path of 200 m in the sea. The speed of the swimmer is constant and equal to 1.25 m per second with respect to the water. During the swim, the magnitude of the surface sea currents was 0.5 m per second and their directions were due south. We're asked to calculate the time taken by the swimmer to finish the 200 m. We're given four answer choices, option A 160 seconds. Option B 175 seconds, option C 287 seconds and option D 400 seconds. Now we want to calculate the time taken by the swimmer to finish the 200 m. Now, the swimmer is swimming with a constant speed. So recall that when we have a constant speed, the velocity is equal to the distance over time. So in order to calculate the time you need the speed and the distance of the swimmer, now we're given the distance, but we're given their speed with respect to the water and the distance is with respect to a fixed frame on earth. And so we need to calculate the speed of the swimmer with respect to the earth. Now, recall velocity of the swimmer with respect to the earth will be equal to the velocity of the swimmer with respect to the water plus the velocity of the water with respect to the earth. Now let's throw a quick diagram. We have our sea currents which we know are going do. So I'm gonna draw this as just a little wave. So we know this represents a secret, we're gonna take up into the right as positive. So if they're moving due south and the velocity of the water with respect to the earth is going to be negative 0.5 m per second. And this is going to be in the J hat direction, the Y direction or the J hat direction. Now, we also have our swimmer swimming, they're gonna be swimming straight east with respect to the earth. OK. That's what they wanna do. They wanna swim straight east in this competition. And so their swim with respect to the water is actually gonna be at an angle and it's gonna be an angle theta with the horizontal, we know that they're going 1.25 m per second. And so that's gonna be a long that day, you know, if they swim at this angle theta then the current is going to push down and therefore they're gonna be swimming do east with respect to the frame of reference on earth. All right. So that was a lot to set up. But let's go ahead and write this. So the velocity of the swimmer with respect to the earth is gonna be the velocity of the swimmer with respect to the water. Now we're gonna write this in components. So the X component of the swimmer's velocity is gonna be related by cosine theta because it's the adjacent side. So we have cosine of theta multiplied by 1.25 m per second. And that is in the I hat direction. And then they also have a Y component to their velocity related through sine. So we have sin of theta multiplied by 1.25 m per second in the J hat direction. OK? Or the Y direction. So that's the velocity of the swimmer with respect to the water. And we've already written out that the velocity of the water with respect to the fixed frame on earth is negative 0.5 m per second. Jha. So the velocity of the swimmer with the respect to this fixed frame on earth is going to be coast, the multiplied by 1.25 m per second in the I hat direction plus sine data multiplied by 1.25 m per second in the J hat direction plus negative 0.5 m per second in the J hat direction. Now we can rearrange this and group these terms into their components. So we're gonna put all of the I hat components together and all of the J hat components together. So the velocity of the swimmer with respect to this fixed frame on earth is going to be co theta multiplied by 1.25 m per second in the I hat direction plus science, the multiplied by 1.25 m per second minus 0.5 m per second in the J hat direction. Now we need to use some more information we were given in the problem. Maybe we don't know this value of data which means we don't know the swimmer speed. Now we want the swimmer to be swimming due east. OK? They're gonna travel that 200 m due east in order to travel due east with respect to this fixed frame on earth, the J hat component must be equal to zero or that Y component because the east direction is to the right. So up and down the swim must not be moving. This tells us that sign of data, this why this J hack component is the second part of our equation. So sign of data multiplied by 1.25 m per second minus 0.5 m per second is equal to zero. Now we can solve our sign data. We saw pre signed data, we will know the speed of the swimmer. So sign data is equal to 0.5 m per second, divided by 1.25 m per second. OK. So S sign data is equal to 0.4. If we take the inverse sign, we get that theta is equal to 23.57 82 degrees. OK. So now we know the value of data. So the velocity of the swimmer with respect to this fixed frame on earth is now given by coast of 23.5782 degrees, multiplied by 1.25 m per second in the I hat direction plus zero in the J hat direction. And we've just defined data so that we have zero in the J hat direction so that the swimmer is swimming due east. And if we simplify this, we have that the velocity of the swimmer with respect to the fixed frame on earth is 1.14 56, 4 4 m per second I have. So now recall, we're looking for the time it takes a swimmer to swim 200 m. In order to find the time we needed the velocity and the distance we knew the distance was 200 m. But we needed to find the velocity of the swimmer with respect to the earth and not with respect to the water. We've done that. So now we can find that time that we were looking for. So rearranging that first equation we have that the time T is equal to the distance D divided by the velocity. In this case, the distance is 200 m. The swimmer is swimming straight in the eastward direction. 1.145644 m per second. So we divide by that value and we get that the time it takes the swimmer to complete. This swim is 174.57 seconds. If we compare this to our answer choices, we round to the nearest second we found that the time taken for the swimmer to finish the 200 m is approximately 175 seconds which corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.