 So now let's look a little bit closer at the case for perfectly inelastic collisions. In a perfectly inelastic collision, we're still dealing with a collision in an isolated system. And for now, we're going to limit it to just one dimension. Now what makes something perfectly inelastic is that the objects stick together after the collision. And that means they've got the same final velocity. If they're stuck together, they have to move at the same speed. Well, mathematically, I can express this as the final velocity of the first object is equal to the final velocity of the second object. Since both of those velocities are equal, I can use just a single vf to show the final velocity of the system. So now we can take a look at the equation. I'm going to start with the general equation. And again, this is the general equation for a two-body collision in one dimension. Now my v1f and my v2f over here again are the same quantity. So we can replace that with just a single vf in that equation. And because it's the same vf on both of those terms, I can get my special equation where I factor that out. So now over here on my left-hand side, I have the total initial momentum of the system. And on the right-hand side, I've got the total final momentum of the system. But it's like I have just a single object which has mass 1 plus mass 2 moving at that final velocity. Now when I get ready to actually solve this equation, what I realize now is that I only have five variables. My original general equation had six variables. And those five variables include the two masses, the initial velocity of each object, and now the single final velocity. Because I only have five variables, that means I only need four knowns to find the last unknown. A very common setup of the problem, but certainly not the only one that's possible, is to give the masses and initial velocities which specifies the entire initial condition of the system. And then ask for the final velocity. But you could solve it for any one of the unknowns as long as you knew the four other quantities. So here's an example. I've got a 20 kilogram mass moving at 4 meters per second. And it's going to collide with a 60 kilogram mass moving at 2 meters per second, such that they stick together. What is their speed after the collision? Well, if I write down my knowns here, I can put in my two masses. We're going to call this first one to be mass 1. Second one is mass 2. I can then associate that with my initial velocities for each object. And then I'm solving for my final velocity. Because this is a perfectly inelastic collision and I've got those as my knowns, I can take my special case equation here and plug in my knowns into that equation. In this case, I've got 20 kilograms times 4 meters per second plus 60 kilograms times 2 meters per second. And that's going to equal the total of 20 kilograms plus 60 kilograms times my final velocity. So now starting with that equation, I can start to actually do my algebra here. And I'm going to run through this kind of quick, but you can go through and pause the video if you need to. Multiplying through my masses and my velocities and then adding up to my two masses, I get 80 kilogram meters per second plus 120 kilogram meters per second. And that's equal to 80 kilograms times my final velocity. So that means my initial momentum is a total of 200 kilogram meters per second. And on my right-hand side, I still have the same thing. So solving for my final velocity then, I've got 200 divided by 80, which is going to give me 2.5 meters per second. Now if I take a step back for a minute, what I also see here is a pattern that's going to be true for any of our perfectly inelastic collisions. And that's that my final speed is somewhere between my two original velocities. And it's generally going to be closer to the velocity of the higher mass object in the system. So that's the case here. My 2.5 is between 2 and 4 and closer to my 2. So that's your introduction to perfectly inelastic equations. You'll solve those for any of the problems where the two masses stick together after the collision.