Ships A and B leave port together. For the next two hours, ship A travels at 20 mph in a direction 30° west of north while ship B travels 20° east of north at 25 mph. b. What is the speed of ship A as seen by ship B?

Video transcript

Hey everyone in this problem. During an exhibition game, two jet skis start to move at the same time from the starting line. During the first minute, the two jet skis move at a constant velocity. The blue jet ski moves at a speed of 15 kilometers per hour north of east while the red jet ski moves at a speed of kilometers per hour in a direction of 30 degrees west of north. We're asked to calculate the speed of the red jet ski with respect to the blue. After 30 seconds of motion, we're given four answer choices. Option a 1.9, 4 km/h. Option B5 km/h, option C 15.7 km/h and option D 25 km/h. Let's start by drawing out what we have. So we have this starting point of our jet ski and they move from the starting line and we're gonna draw that as a horizontal line. We have the red jet ski or sorry, the blue jet ski moving at a speed of 15 km/h north of east. And so north is pointing straight up east is pointing to the right and so north of east is going to be directly in between. Ok. That's gonna make an angle of 45° with the horizontal. And this is our blue jet ski and let me draw that in blue, our blue jet ski 15 km/h along that um diagonal. Now, our red jet ski moves at a speed of 10 km/h in a direction, 30° west of north. Ok. So north is straight up west of north and west is to the left. So we're gonna go from the or from the vertical 30° to the left. And this is our red jet ski moving at 10 km/h. Now, let's take up into the right as our positive directions. We're looking for the speed of the red jet ski with respect to the blue. So let's start by finding the velocity. Ok. The speed is gonna be the magnitude of the velocity. We know that the velocity of the red jet ski with respect to the blue, it is going to be equal to the velocity of the red jet ski with respect to the water plus the velocity of the water with respect to the blue jet ski. Now, we know the velocity of the red jet ski with the water or we can calculate it based on our diagram based on the angle we're given and the speed, but we don't know the velocity of the water with the, with respect to the blue jet ski, we do however know the velocity of the blue jet ski with respect to the water. So recall that we can flip this equation, we can write this as the velocity of the red jet ski with respect to the water minus the velocity of the blue jets ski with respect to the water. When we do that, now we can sort out this value. So let's start with the velocity of the red jet ski with respect to the water. So this is our red jet ski in the X direction. First things, first, ST jets is traveling to the west a little bit or to the left. And so this is going to be a negative X component. So we have a negative and the X component of their speed, it is going to be related to our angle 30° through sign because it's the opposite sign, the opposite side, sorry. And so we have negative sign of 30 degrees multiplied by that hypotony, which is 10 km/h and that is in the I hat direction, OK. The X component, then we have the Y component and you know, this guy is moving to the north, that's our positive Y direction. So it's gonna be positive and the Y component is going to be related to our angle through coast. And so we have plus cosine of 30 degrees multiplied by that 10 kilometer per hour. Speed and that is in the J hat direction or the J hat component. If we simplify this, we have that the velocity of the red jet ski with respect to the water is equal to negative five kilometers per hour in the I hat direction plus 8. kilometers per hour in the J A direction. Now I've left quite a few significant digits here. Always check with your professor or the textbook you're using using on how many significant digits they like you to keep. Now we're gonna do the same with the velocity of the blue jet ski with respect to the water, the velocity of the blue jet ski with respect to the water. When we're talking about the X component, let's go to our diagram. The X component is gonna be related through cosign. Yeah, because it's the adjacent side and it's moving in the east direction. So that's gonna be positive. This is equal to cosine for angle 45° multiplied by the hypotenuses or that speed 15 km/h in the eye hat direction. The Y direction the Y component is gonna be related through the sign of the angle. OK? It's moving upwards. So this is gonna be positive. So we have plus sign of 45° multiplied by 15 km/h. OK? We're just doing triangle math here and that is in the Jha direction. Simple of buying this one. We get 10. kilometers per hour in the I hat direction plus 10. kilometers per hour in the J hat direction. OK. Sine of 45° and cosine of 45° are equal. So we get the same magnitude in each of those directions. And that makes sense since we're traveling right in the middle of north and east. Now getting back to the velocity of the red jet ski with respect to the blue jet ski that we're looking for and we have both of these values. Now, we just need to subtract the one. So again, this is the velocity of the red jet ski with respect to the water minus the velocity of the blue jet ski with respect to the water, that's equal to negative five kilometers per hour in the I hat direction Plus 8.6602, 5, 4 km/h in the J hat direction. And then we're gonna subtract and I'm gonna do it underneath just so we don't run out of room. We're gonna subtract the velocity of the blue jet ski with respect to the water, which is 10.6066 kilometers per hour in the I hat direction plus 10.6066 kilometers per hour in the J hat direction. Now, we want to simplify, OK, we wanna combine the I hat and J hat components. So we have one single value for each in the I hat direction. We have negative 5 km/h from the first term and then we have minus 10.6066 kilometers per hour from the second component because we're subtracting that's in the I hat direction in the J hat direction, we have 8. kilometers per hour minus 10.6066 kilometers per hour and this is in the J hat direction. Now be careful that you include the minus here the negative because we're subtracting this entire velocity of the blue jet ski with respect to the water. That negative has to apply to both the I hat and J hat term. So it can be easy to, to miss that second one. When we simplify these values, we get negative .606, 6 km/h in the I hat direction plus negative 1. 346 kilometers per hour in the Jha direction. OK. So according the or the velocity of the red jet ski with respect to the blue jet ski, OK? According to the blue jet ski, the red jet ski is moving in the negative direction. In both the X component and Y component, we go back up to the top. That makes sense. OK. The red jet ski is moving to the left while the blue jet ski is moving to the right. So they're gonna be getting further apart in the X direction. So it's gonna look like he's going in the negative X direction and in the wide direction, the blue jet ski is going faster, they have a larger speed in the wide direction. And so the red jet ski again, it's gonna look like it's going in the negative Y direction because it's not moving as quickly or as much in the wide direction. Now, we found the velocity, OK. We have to remember that the question was asking us to find the speed. So to find the speed, we wanna take the magnitude of the velocity. So the magnitude of the velocity of the red jet ski with respect to the blue jet ski and this is gonna be equal to the square root of the sum of the squares of the components. OK. So we take the square root of the X component or the I AC component 15. kilometers per hour squared plus negative 1. kilometers per hour squared. OK. So we square the I A T component, we square the J hat component, we add them together and then we take the square root. OK. So this is kind of like Pythagorean when we think about if we were to draw these as a triangle or X component, our Y component and our hypotenuse, if we work this out on our calculators, we get that the speed is 15.7275 km/h. And not at the value. The question was asking us to find if we go and compare this to our answer choices. We found that the speed of the red jet ski with respect to the blue jet ski rounding to three significant digits was approximately 15.7 km/h, which corresponds with answer choice. C Thanks everyone for watching. I hope this video helped see you in the next one.