 Okay, let's do a momentum example, or I'm sorry, a collision example. So I'm just going to make up a problem and then we'll just do it from there. So suppose that I have two cars, this one's a thousand kilograms, and it's moving this way, ten meters per second, and then I have a truck, I'm just making these numbers up as usual, two thousand kilograms, let's call that south and east at 15 meters per second, and they collide and they stick together, and you know, it's ice, so there's no friction, that's why they crash, right, because they couldn't stop, if they would stop they would stop, but they couldn't. So the roads are completely icy, happen sometimes, no one's hurt, everyone's okay, car is pretty messed up, here's the truck, like that, and it's going to move it some down like that, okay, so there's no friction. So we have this collision like this, I'm trying to stop, my inlet breaks everything, okay, so the question is what's the final velocity vector for this wreckage, combined wreckage. Okay, so when we're dealing with a collision, the important thing is that if the interaction happens over a short time, or there's no external forces, then the force this exerts on that one, it's the same as the force this exerts on that one for the same amount of time, so you remember, this marker, not good, I had another marker there just, so you remember F, oh that one's even worse, the blue one, we're going blue today, F delta t equals change in momentum, so if the force on this one for the same time is actually opposite force, opposite direction of the force on this one, on that one, then they have opposite changes in momentum. We can also write that as delta P total equals zero, or we could write that as P1 total equals P2 total, so the initial momentum equals the final momentum, and that's your concentration momentum, and I derived this in my book, so you can look at it there, I don't want to spend too much time on that, I want to actually use it. Okay, so let's call this the x direction, and the y direction, I can get my total initial momentum, and then I'll set that equal to my total final momentum, and everything should work out okay. I already put in numbers, but let me put some variables along with this, let me call this mass one, and let's call this V11, yeah, because that's the velocity of object one at the beginning, okay, we don't need that in this case, but that's M2 V21, and then after they collide, we'll call this mass, and then I'll have a velocity vector V, I guess 3, 2, see I didn't really need that because it's a new object, but okay, so if I write down the initial total momentum, I get P1 total or initial momentum, it's going to be the momentum of this, which is going to be M1 V1 plus M2 V11, M2 V21, let me go ahead and put in the numbers because otherwise it doesn't get any different than that, so here I have V11 is in the x direction, and V21 is in the negative y direction, so this is going to give me P1 total is going to be M1, a thousand kilograms, I'll leave off the units just for simplicity, it really should be there, V1 is going to be 10 meters per second x hat minus 2,000 times 15 meters per second y hat, so you see I have that one is in the x hat direction, that one's in the negative y hat direction, okay, I can't add that into one number because they're in different directions, okay, so I can get this as a value though, that'll be 10, 1, 2, 3 kilogram meters per second x hat minus, let's see, 5, 2, 30, 3, 30,000, 30,000 kilogram meters per second y hat, that's it, okay, so I'm actually done because that's my initial momentum, that's my final momentum, P2 total, forgive my coffee pot, it's going off, P2 total is going to be the same thing, and but I could also write that as M1 plus M2 V32, okay, so what if I want to, if I just want to find V32, I can divide both sides by M1 plus M2, and remember P2 total is P1 total, okay, so M1 plus M2 is going to be 3,000, so let's divide both sides here by 3,000, and I get V32 equals 10,000 for 3,000, and that's kilogram meters per second divided by kilograms, give me meters per second x hat minus 30,000 over 3,000 meters per second y hat, so it's going to be 10 over 3 meters per second x hat minus 10 meters per second y hat, that's my final velocity, that was it, okay, does that make sense? First, should it be moving faster in the x or y direction? Well, this has a larger mass and a larger velocity, so that's going to make it have a velocity more downward than that way, and that's, agrees with this right here, right, it's three times as much. Is it going, okay, let's see, what if I wanted to find this angle theta that it goes at in the speed? Well, I could find the magnitude of the speed V32, the magnitude would be the square root of 10 squared over 3 squared plus 10 squared, and that would give me the magnitude of that vector. If I wanted to find that angle theta from the net below the x-axis, then I would just say here's my vector, that's V32y, this is V32x, so that angle theta is going to be tangent of theta, it's going to be opposite over adjacent, so it's V32y or 10, just use the 10 because I've already drawn the direction over 321x, which is 10 over 3. I wash, I was weird, okay, but there you see it. There is one other thing that we can do and I'm not going to do it, but the other thing to do is say how much, how much energy was lost? How much kinetic energy was lost, or what was the change in energy of this whole thing, you know, it got, it had to take configuration, change in energy, all the things got smashed together, there was sound, there was increase temperature and all that stuff, so where did that come from? Well, kinetic energy is not conserved, the initial kinetic energy is not the final kinetic energy, but the energy is, because there's no external work, so I can say E1 equals E2 or K kinetic energy of object 1 to begin with plus kinetic energy of object 2 equals kinetic energy of object 3 afterwards plus other energy, actually there should be a delta other. So I can find that, I know the speed, I can find the, so I could find the energy, okay, but kinetic energy is not conserved. Momentum is conserved if the collision is either short or there's no external forces, energy is conserved only if it's an elastic collision.