Hey, guys. So in the last couple of videos, we were taking a look at how we calculate the total momentum when we have a system of objects. And the reason we thought it was important was because when objects are interacting, it's not the individual momentums of each object that we're looking at, but rather the total momentum of the system that we're actually concerned with. And we said that the total momentum of the system, p system, is going to be conserved. So in this video, I'm going to show you what the conservation of momentum equation looks like and how we use it to solve problems. So let's check this out. What does conserved actually mean? We've seen this before when we talked about energy. Conserved just means that whatever you start off with is whatever you have to end up with. In the case of energy, it was e_{initial} and e_{final} most of the time. So this is the same idea with momentum. What happens is the momentum here, this p system has to be the same value from initial to final. So what we say here is that p_{system}initial is equal to p_{system}final. So whenever you have multiple objects that are moving around like this and you calculate the total momentum for that system, that number has to remain the same from initial to final. So we can say here is that for 2 objects, remember that p_{system} is just p_{1} + p_{2}. So we can rewrite this and say that p_{1}initial + p_{2}initial = p_{1}final + p_{2}final. What gets conserved is not the individual momentums of each object, but rather the combination or the sum of the momentums that has to remain the same from the left and right side of the equation. So now what we can do is we can write this using p = mv, and say that this is m_{one}v_{one}initial, plus m_{two}v_{two}initial = m_{one}v_{one}final + m_{two}v_{two}final. So this right here is known as the conservation of momentum equation. We're gonna use it every time we have multiple objects that are interacting with each other. This is one of the sacred laws of the universe that cannot be violated. Whatever momentum you start off with has to be the momentum that you end up with. The left and right side have to be equal to each other. That's really all there is to it. Let's go ahead and take a look at an example here. So we have 2 balls that are rolling towards each other. And basically, what happens is you have ball a that's moving to the right, b is moving to the left, and they're gonna collide, and then something's gonna happen afterward. So in these problems, the first thing you're gonna want to do is draw a diagram for the before and after. So here's what's happening before the collision. We have these 2 balls that are rolling towards each other. So this is a 3 kilogram ball that's moving to the right with 7, and then we have a 4 kilogram ball that's moving to the left with 5. So immediately, what I can do here is because I have 2 different directions, I'm gonna choose a direction of positive. So my right direction is gonna be to the is gonna be positive, which means that this is +7, and this is -5. Anytime you're writing velocities in your problems, always make sure to keep track of the signs. So now what happens is these things are gonna collide. Right? They're gonna hit each other, and what we're told here is that after the collision, ball b, right, this 4 kilogram ball, is actually now moving to the right. It's moving at 2 meters per second to the right like this. So now what happens is this is 2, and because it points to the right, it's positive. And I want to figure out what happens to this 3 kilogram ball. I want to figure out the magnitude and the direction of ball a's velocity. So this is the magnitude and direction right here. So we're gonna figure out basically what is v_{a} final here. So how do we do that? Well, once we figured out our sort of diagrams for before and afters, these are really helpful at figuring out like what exactly is going on in the problem. Now we're gonna write our conservation of momentum equation. So this is gonna be m_{one}v_{one}initial, plus m_{two}v_{two}initial equals m_{one}v_{one}final, plus m_{two}v_{two}final. So we're gonna write this a lot in our problems. This is gonna become kind of tedious at some point. So one thing I like to do is actually just already replace all the values for the mass that we know. We know we're dealing with a 3 4 kilogram ball. So this is gonna be 3 times some velocity + 4 times some velocity equals 3 + 4, and then we're just real we really just have to figure out what is going on and what values we plug inside each of the of the parenthesis. And to do that, we just look at the at the diagram. Right? So we have this 3 kilogram ball that's initially moving at 7. So this is what goes into the initial for ball a. And then the 4 is going at, at negative 5 because it's going to the left. So we're gonna plug it in with the correct sign. Now what happens is this 3 kilogram ball, we're trying to figure out the final velocity. So that's actually what goes inside here and this is really our target variable. This is v_{a}final. And then this 4 here is going to the right with 2, so this is gonna be +2. So, really, we just have one value, that's unknown, and we're just gonna go ahead and solve for this. Right? So this is gonna be 21 + -20 equals 3v_{a}final + 8. Now when you simplify this, what happens is this is gonna be 1 kilogram meter per second. So this is gonna be you're just gonna get 1 on the left side. And on the right side, you're gonna get 3 v_{a}final + 8. Now remember, what we said is that momentum conservation means whatever you start off with is whatever you end up with. On the left, the initial momentum of the system here, once you've added this all up together, is actually just equal to 1 kilogram meters per second. So that's actually what we have to get on the right side. So all we have to do is just solve for this missing variable right here. So what we're gonna do is we're gonna move this over to the other side. This is gonna be 1 - 8 equals 3 v_{a}final, and this is gonna be -7/3 equals v_{a}final, and what you get is you're gonna get -2.33 meters per second. So the magnitude here is 2.33 meters per second. But what is this negative sign that we got here? Well, hopefully, you guys realized that because we've chosen the direction of positive to be to the right, this negative sign here really just means that this velocity actually points to the left. So what happens is this velocity here is gonna be 2.33 meters per second, except this ball is actually gonna be going to the left like this. Oh, sorry. I meant to say дto the leftđ. So what happens here is that you have these 2 balls that are colliding with each other, and then they're actually gonna rebound and go backward. The initial momentum here is equal to the final momentum here. So that's how you solve these kinds of problems using conservation of momentum. Let's go ahead and take a look at some other problems.