 Let's consider the special case of a two-body collision in one dimension. Now collisions are generally considered isolated systems. And by this, we mean that the external forces either aren't there or so small they can be ignored. Now this only has to be true during the short time of immediately before to immediately after the collision. So we're not talking about seconds before and seconds but just during the time of the collision. Now there are plenty of internal forces. And of course, these are very important for determining the effects on the individual objects. But at the same time, they're unknown forces. We don't know exactly how long the collision takes or exactly what the force profile is during that collision. But we don't have to know. Because the system is isolated, that means momentum is conserved. And that means that momentum before the collision, the initial momentum, and the momentum after the collision, which we call our final momentum, have to be equal to each other. And each of these are total momentums. So since we have a two-body system, that initial momentum is the initial momentum of object one plus the initial momentum of object two. And our final momentum is, again, object one and object two added together. Now if I take this equation and I expand it out, I realize that each one of these momentums includes a mass and a velocity. Since I've got two bodies, I have two different masses, m1 and m2. And each object has an initial velocity and a final velocity. But this whole equation, everything on the left-hand side, has to balance with everything on the right-hand side. So that means in general, we've got six variables. For mass one, we've got its mass and its initial and final velocities. For mass two, we've got the same thing. It's mass and its initial and final velocities. If you have any five of these six, then we can use our equation to find the one unknown. You might have to do a little algebra in there, but we can find it. So let's look at an example real quick. Let's say I've got mass one hits mass two. And I'm gonna set mass one to be two kilograms and mass two to be three kilograms. And they're moving such that mass one has a velocity of four meters per second and mass two has a velocity of one meter per second. After the collision, we know that mass one has a final velocity of two meters per second. And we wanna know what the final velocity of mass two is. So we have five of our unknowns and we're solving for our last one to give us our unknown sample. Well, if I take my general equation for a two-body collision in one dimension, then I can plug in all the variables I know, leaving v2f as a variable. So let's start with that equation and start simplifying it. First thing I can do is each time I've got a mass and a velocity, I can do that multiplication. So my two times four becomes eight. My three times one becomes a three. My two times two becomes a four. Keeping track of my units in here to realize that these are momentum units now, kilogram meters per second. On our last term, we still just have three kilograms and the unknown velocity two final. Well now because I've got two different momentum's over here, I can add those up to give me 11. And over here on this side, whatever this is, it's gotta balance out so this side is equal to 11 as well. So I can start trying to find out what that is first by moving my four and subtracting it over to the other side. Giving me seven is my three kilograms times my unknown v2. If this whole quantity here is equal to seven, then my two sides of my equations will balance out. So then the last step to solve for v2 final is I'm gonna have my seven divided by three or that gives me 2.33 meters per second. Now let's go back and look at this in total now. I've got all six of my values. So before my collision, I've got a mass one that's a little bit smaller than my mass two, but it's moving much faster. And so that means eventually it's gonna catch up with mass two and collide. Now exactly what happens during the collision, again, we don't have to know the details of that. But after the collision, we still have our two masses, but now this mass one has slowed down and this mass two has been sped up. So when they collided, it slowed down the first mass as it ran into it and gave a little extra kick to my second mass such that this one's now moving away from that one. So that's our basic collision in two dimension. We still have to look at a few other special cases.