Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass 7.50 kg, is sliding to the left at 5.00 m/s, while the other, of mass 5.75 kg, is slipping to the right at 6.00 m/s. They hold fast to each other after they collide. (b) How much mechanical energy dissipates during this play?

Video transcript

Hey everyone welcome back in this problem. We have kids playing on a frozen lake, we're told that it's frictionless. The kids are gonna be sliding towards each other, they're going to collide and after the collision they're gonna hold on to each other. All right now, we're asked to find the loss and mechanical energy during the interaction. We have a collision problem like this. The first thing we want to do is go ahead and draw a diagram of the system before and after the collision and fill in the details we know. Now for our diagrams, let's take right to be the positive extraction. All right. Before the collision we know we have two kids, so we have a over here in red and we have Kid B over here in blue. All right now, today we're told the kids are moving towards each other, so kid a is going to be going to the right, Okay, so we have kid a is going to be 14.5 kg 14.5 kg. They're gonna be moving right at 3.2 m per second. Okay, You know, we've chosen right to be the positive extraction so if they're going right, the velocity of Kid A initially is going to be 3.2 m per second. Alright? Now, for Kid B while we're told that there are 16.8 kg And they're sliding left, 3.6 m/s. Okay, so that works out, they are going towards each other like we wanted Now because child B is going to the left and our positive extraction is to the right there velocity is going to be negative. So we're going to have negative 3. m/s. All right now, what about after the collision? Well, again, we're told they're going to hold on to each other. So instead of having the two kids separate, we're going to have the two kids holding onto each other like this. Okay, if they're holding onto each other, they're moving together with the same speed. So let's go ahead and find the total mass. So the final mass I will say is going to be the massive child A plus the mass of child be. Okay, so we just want to know the total mass of those two kids together. And that's gonna be 31.3 kg. Okay, so 14.5 plus 16.8 is gonna give us 31. kg. Now, we're not told anything about the speed of those Children when they're holding onto each other. Alright, so we have our diagram, what are we looking for? We're looking for the loss of mechanical energy. Now, let's recall mechanical energy as kinetic energy and potential energy. In this case we don't have potential energy. We don't have any springs to have potential energy. The kids are on the ground. So we don't have any gravitational potential energy. So this question is going to essentially ask us just to find the change in kinetic energy delta K. Which is gonna be the final kinetic energy minus the initial kinetic energy. Alright, so let's start with the initial kinetic energy. Well, initially we have two kids, they're both moving. So we're gonna have kinetic energy coming from each of them. So that's gonna be the kinetic energy kid, eh, initially plus the kinetic energy, it could be initially. Now recall kinetic energy is given by 1/2 MV two. So writing out the corresponding terms one half M A V A not squared plus one half M B V B not squared. Okay. And filling in what we know. So we know Massive a is 0.5 kg. The velocity of a initially is 3.2 m per second. That's all going to be squared. And similarly with b 16.8 kg. And when we're talking velocity here, or sorry, when we're talking kinetic energy here, we just need the speed, not the velocity. Okay, so we're just gonna have 3.6 m per second. Okay, and this term is squared. So even if you had the negative, you would get the same result. Alright, so working this out, what we're going to get is really at 183. jules. Okay, how do we get jules? Well, our unit is kilogram meters squared per second squared. So, up here, we're going to do a little aside, we have kilogram meters squared per second squared. Well, what we can do is pull out one of the meters so we have kilogram meter per second squared times meter. Okay, well kilogram meter per second squared. That's a newton. Okay, so we end up with Newton times meter and a newton meter. Well, that's a jewel. Okay, so that's how we went from our killer meters squared per second squared to jules. That's the unit of energy. So that's good. Our units work out here, which is what we always want. Now, what about the final kinetic energy? K. F. Okay, well, again, after the collision, instead of having to objects moving, we have one object moving. Okay, it's going to be the two kids stuck together. It's gonna be like one thing moving with one big mass and with the same speed. So we're gonna have 1/2 the final mass, which is again the mass of the two kids together, The sum of their masses and their final speed. Well, we don't know the final speed. Okay, that was the thing in our diagram, we didn't know we don't know via. So in order to find delta K. The change in kinetic energy we're looking for, we need to find this quantity of ef, let's go ahead and use our conservation of momentum. We know we have a frictionless lake, we have no net external forces. So we have conservation of momentum. Okay, conservation of momentum tells us that the momentum of the system initially is going to equal the momentum of the system at the end of the collision. Well, the momentum in the system initially we have the momentum of person A and the momentum of person B. So that's gonna be the momentum of A. Initially plus the momentum of be initially. And on the right hand side while we just have one we have those kids together moving together. So we're gonna have that final momentum. P. F. All right, let's recall momentum is mass times velocity. So for each of these terms mass and corresponding velocity. Okay, so massive be velocity of B initially and on the right hand side the mass final which is from our diagram we can see the total mass of the two kids together and V. F. Okay, and this V F. That relates to our speed. That is what we are looking for. Alright, plugging in the values we know we have the mass of a is 14.5 kg. The velocity of a initially is 3.2 m/s, Massive B is 16.8 kg And its velocity is -3. m/s. All right now, we want to be careful here, this is momentum when we're talking about momentum direction matters. We're using a velocity and so we need to have the negative to indicate the direction On the right hand side. The total mass of the two Children. 31.3 kg. And that quantity VF Alright working this out on the left hand side, we're gonna get -14.08. We're going to have a kilogram meters per second. Okay, so we have kilograms times meters per second, kilogram meter per second. On the right hand side, we still have 31.3 kg times v. Okay, dividing by 31.3. The unit of kg will cancel. We're gonna be left with V. F. Is equal to -0. 984 meters per second. Okay? And again, that unit works out. That's the unit of velocity. That's exactly what we were looking for. Alright, so let's scroll down just a little bit now. We found the velocity. But what we wanted to do is use the speed. Okay, so for the speed we just have to recall that speed is going to be the absolute value of that velocity. And that's going to give us the positive 0. meters per second. Alright, so now we've found the speed that we need we can get back to our energy. So again, we're looking for delta K. We found K. Naught but we need to find K. F. So we can use the value we found the final mass. The mass of the two kids together is 31.3 kg. The speed. We were looking for 0. m/s. That's all going to be squared. That's going to give a value of 3. jewels. Okay. We're going to get to the jewels the same way we did with um Kay not Okay. We have kilogram meter square per second squared. That's going to be a jewel. Alright, so last but not least our delta K. We now have K final and K initial substituting those values, we have 3.1663 minus 183.104 jewels. Okay. And that's gonna give us minus 179. jewels. Okay, that's the delta K we were looking for and if we round we see that we can match to answer A. Alright. Thanks everyone for watching. See you in the next video.