Two skaters collide and grab on to each other on frictionless ice. One of them, of mass 70.0 kg, is moving to the right at 4.00 m/s, while the other, of mass 65.0 kg, is moving to the left at 2.50 m/s. What are the magnitude and direction of the velocity of these skaters just after they collide?

Video transcript

Hey everyone welcome back in this video. We have two skaters and they are on a frictionless surface, Okay. And they're going to collide and when they collide they're going to manage the situation by grabbing each other, holding onto each other. Okay? So instead of bouncing off each other and moving separately, they're going to hold on to each other and we're asked to find the magnitude and direction of both skaters velocity after the collision. Okay, So the first thing we want to do with the collision problem like this is go ahead and draw the system both before and after the collision and fill in any of the details we know. So we're going to take to the right to be to the east and we're gonna take that as our positive X direction. Okay, So before we get started, we're just going to do that. All right, before the collision, we know we have two skaters. So we have red skater here, we have blue skater over here. All right. Our first skater is 68 kg and is traveling west. Okay, So because he's going west, they're gonna be going to the left, So we're gonna take the blue skater to be this skater, they're gonna be 68 kg and they're traveling west again, we've taken east to be to the left, so west is going east to be to the right, West is going to be to the left, This will be the velocity of skater one initially, and because they're going to the left and we said to the right, will be positive direction, they are going to have a negative velocity -7.5 m/s. All right now we have our red skater are red skater we're told is kg. They're traveling east, so to the right, they're traveling in the positive direction, so their velocity will be positive And 4.2 m/s. Alright, So that's before the collision. What about after? Well, we're told that they hold on to each other. So instead of having two separate skaters, we have two skaters together. All right now, when they're together they're gonna have a final mass, that's going to be the mass of the first child or this first skater plus the mass of the second skater. Okay, so holding on to each other, they have one total mass And that's going to be 140 kg. Okay, so 72 kg from the red skater, 68 kg from the blue skater gives us 140 kg. We aren't told anything about their final velocity. That's actually what we want to find out. Okay. And it's important to note, So it's asking for the magnitude and direction of both skaters velocity After the collision, that sounds like two quantities, but in this case the skaters are holding on to each other and moving together. So the magnitude and direction of their velocity will be the same. So it's really only one quantity that we need to find. Alright, well, we're told that this is a frictionless surface so we have conservation of momentum. What is conservation of momentum? Tell us? Well, it tells us that the momentum in the system initially is equal to the momentum in the system after the collision before the collision. What is our system contained? Well, it has two skaters. So we need to consider the skater momentum from skater one and the momentum from skater two. So we have the momentum from skater one initially. That's the momentum from skater to initially after the collision the skaters stick together. Okay, so they combined to make one big mass and they're gonna be moving together with one velocity. So their momentum, it's just gonna be momentum, final recall momentum is going to be mass times velocity. So for each of these quantities we get the mass times the corresponding velocity, the mouse in terms of corresponding velocity And on the right hand side we have that final mass, which is the total mass of skater one and skater too, and their final velocity, which is what we are looking for. That speed or the velocity that they're traveling with together. Alright, so let's go ahead and fill in the information that we know We know the mass of skater one is 72 kg. Their initial velocity is 4. m/s. The mass of two Skater to kg and their velocity is -7.5 m/s. Okay, and this is momentum. So we have to be careful, momentum. The direction matters just like with velocity. So when we plug in our velocity here, we need to make sure we're including the negative to indicate the direction. And on the right hand side, that total mass of both skaters 140 kg. And that quantity VF we're looking for. Okay, so the hard part's done, we've drawn our system, we've figured out what's going on, we figured out what equations to use and now we just need to do um a little bit of adding subtracting in order to get these values. So on the left hand side we're going to get 302.4. Okay, The unit is gonna be kilogram meter per second because we've taken a kilogram times by meters per second plus minus 510 same units kilogram meter per second. Okay, And on the right hand side we still have our 140 kg times the quantity V. F. All right, so the left, just simplifying minus 207.6 kg meters per second. On the right hand side. 140 kg, times the f. Alright. And dividing by 140, the unit of kg will cancel and we'll be left with V F is equal to -1. 83 m per second case. So when the kilograms cancel, we're left with the unit of meters per second, which is the unit we want for velocity. So the units check up. Okay. And that is the answer we're looking for now. We need to interpret it. Okay, So we're asked for the magnitude and direction of the velocity. So if we're looking at magnitude, we know that it's going to be 1.483 m per second. Okay. What about direction while the velocity is negative? Okay, We've indicated that to the right is positive and east. So the negative velocity means they're going to be going to the left, which is west. So the answer here is going to be G the magnitude is 1.48 m/s and the direction is West. Alright, thanks everyone for watching. I hope that helps you in the next video.