At the center of a 50-m-diameter circular ice rink, a 75 kg skater traveling north at 2.5 m/s collides with and holds on to a 60 kg skater who had been heading west at 3.5 m/s. (b) Where will they reach it? Give your answer as an angle north of west.

Video transcript

Hey, everyone in this problem. Two swimmers are in a circular pool with a diameter of 60 m. One swimmer weighing 50 kg is swimming in the north direction at a speed of two m per second. While the other swimmer weighing 70 kg is swimming in the west direction at four m per second. If they collide and hold on to each other, at what point on the edge of the pool will they reach? In other words, in what direction will they end up after they collide and stick together? Hey, we're told to give our answer as an angle measured clockwise from due east. We're given four answer traces. Option a 19.7° north of West. Option B 15.4° north of west, option C 27.3 degrees west of north And option D 22.4° west of north. Now we're gonna start by drawing a little diagram. So we have an initial situation. OK? And we have our two swimmers, we have swimmer A know what we're gonna draw in red and we have swimmer B so we're gonna draw in bully. Now, swimmer A, we're told is traveling in the north direction. So we're gonna draw that straight up, OK? Swimmer A is 50 kg and they're traveling At two m/s, ok? So we're gonna take up and to the left as our positive directions, ok? So this is a positive velocity for swimmer A And then we have swimmer B who is 70 kg they're traveling west. So they're going straight to the left with the speed of four m per second. And since we've chosen the left as our positive direction, that's a velocity of positive four m per second. So we have this initial situation and then we have a final situation after they collide. Now the swimmers are gonna collide and stick together and they're gonna hold on to each other. So we're gonna have the two swimmers stuck together. Swimmer B and swimmer A and their final mass is gonna be the total mass of the two of them, ok? Because they're stuck together. They're kind of like one object. They're gonna be moving together. That means they're gonna have one speed. It's gonna be the same speed for both of them, same direction for both of them because they're stuck together. And so this total mass or this final mass is gonna be the mass of swimmer A plus the mass of swimmer B And so this is gonna be 120 kg, 50 kg plus 70 kg, 120 kg. And we wanna know in which direction they're moving. Now, we have no net external forces acting on this system. So we know that we have a conservation of momentum. What does that mean? Well, if we have a conservation of momentum, then the initial momentum P not is going to be equal to the final momentum. Let me read it further to the left is going to be equal to the final momentum. OK. Initial momentum is equal to the final momentum. Now, in this case, we have motion in two directions. OK? We have motion in the north, south direction up and down and we have motion side to side in the east west direction. So we need to consider the momentum first in the X direction. So we have the initial momentum in the X direction is equal to the final momentum in the X direction. And then we're gonna do the same thing in the wider direction. We're gonna give ourselves some more room to work here. Now what is momentum equal to momentum is equal to the mass multiplied by the velocity. OK. Well, initially, what objects do we have in our system that we have to consider the momentum of, we have the momentum of swimmer a initially in the X direction plus the momentum of swimmer B oops initially in the X direction as well. OK. In the final case, we just have that one momentum. OK? Because the swimmers are holding on to each other. They're stuck together. They're treating, we can treat that as one object, they're moving like one object. All right. So we've mentioned this, the momentum is mass multiplied by the velocity. So for each of these terms, we have the corresponding mass. So the mass of swimmer a multiplied by the velocity initially of swimmer A in the X direction plus the mass of swimmer B multiplied by the velocity of swimmer B initially in the direction. And this is equal to that final mass and F multiplied by the final velocity in the extraction. Now, swimmer A is moving straight north. OK. So they only have Y component of their velocity, OK? They're not moving in the horizontal component at all. And so their initial X speed is just going to be zero. So that entire momentum term for them goes to zero. We're left with the max of Swimmer B which is 70 kg multiplied by the initial speed of swimmer B in the X direction. OK? They're swimming straight west. So their entire velocity, initial velocity is going to be in the X direction in an X component. OK? We've chosen left to be positive. So we have a positive velocity of four m per second. And this is gonna equal our final mass, which is that total mass of our two swimmers that are holding onto each other and stuck together multiplied by their final velocity in the X direction. And again, they're stuck together, they're holding onto each other, they're moving with one speed. Now, if we solve, so we want to solve for V FX. So we're gonna rearrange, we're gonna isolate V FX and we get the V FX. The velocity, final velocity of our swimmers in the X direction Is 2.3, 3 m/s. All right. So we're gonna put a box around that. We're gonna come back to that. Remember, we're looking for the direction that they're traveling. So as we need the X component, but we also need the Y component of their speed. So we're gonna do the same thing conservation of momentum in the Y direction. So the initial momentum in the Y direction is equal to the final momentum in the Y direction. The initial momentum is made up of the momentum of swimmer A, what's the momentum of swimmer B? And this is equal to that final momentum of the two of them together. We have mass multiplied by velocity for each momentum. So the mass of swimmer a multiplied by the velocity of swimmer A initially in the Y direction plus the mass of swimmer B multiplied by the velocity A swimmer B initially in the Y direction. And this is equal to that final mass that mass of the two swimmers together multiplied by their final velocity in the y direction. Now, swimmer B, we said when we were doing the X component that their entire initial velocity is in the X direction because they're moving west, that means that the Y component of their velocity initially is zero. And so the momentum initially of swimmer beat in the y direction, that entire term goes to zero. Subsequuting in the rest of the information, we know we have the massive swimmer a 50 kg multiplied by their initial velocity 2m/s. And this is equal to that final mass 120 kg multiplied by the final velocity in the Y Direction. We can divide by 120 kg to get their final velocity Of 0.8, 3, 3 m/s in the Y Direction. All right. So we've found the X component of their velocity, we found the Y component of their velocity. Now we need to find the direction. So we've said up into the left is our positive directions, we have a positive X velocity and a positive Y velocity. So that means our swimmer is moving to the left and up. OK. So if we consider the X component and they're traveling to the left At 2.3, 3 m/s, and if we can consider the Y component, they're traveling up At 0.8, 3, 3 m/s. So we can make a triangle with these and the hypo is going to be their speed that final speed. And we have an angle theta between that hypotenuse and the horizontal direction. All right, how can we find data. Well, we know that tan theta is gonna relate the opposite and adjacent sides. So we have tan theta is equal to 0. m per second divided by 2. m per second. And this gives us a data value A equal to the inverse Tangent 0.0ops 833 m/s divided by 2.33 m/s, Which gives us an angle of 19.65°. OK? So the direction of their motion is gonna be 19.65°. And now that angle is going from this horizontal direction which is west and it's going up towards the north. OK? So this is gonna be 19.65° north of west, OK? If we round that to three significant digits, we have 19.7° north of West which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.