[MUSIC] Before going on, we'll have a little detour to discuss finite objects, systems of objects, and their center of mass. We formally limited the syllabus of this course to particles, objects small enough that we can represent their position with a point. But for objects of finite size, what point would we use? The end of this stick has a complicated path. But is there a point in the stick that traces out the parabolic path that we started in projectile motion? Yes, there is such a point and is called the center of mass. We shant to the formal development here, but we do give you a link to it. However, here's a neat trick. Provided that the gravitational field is uniform, the center of mass is the same the center of gravity or the balance point. For this stick from symmetry the center of mass lies on its axis. And it's at the point where it balances. So let's label it. Now we see that the center of mass traces out a parabola. The center of mass behaves like a particle. So, for a finite or extended object, Newton's second law is just total external force equals mass times the acceleration of the center of mass. What about the center of mass if two or more objects? Where is the center of mass of these two objects. Obviously the balance point is going to be closer to the larger mass. Treat these two cars as a system and suppose that they have equal mass. So the center of mass is halfway in between them. If their initial speeds are equal, the center of mass is stationary and it remains stationary afterwards. This is an important application of Newton's second law or conservation of momentum. If the total external force has negligible impulse then the velocity of the center of mass is unchanged during a collision. With the air track, we reduce friction almost to zero and I'll use two sliders that have the same mass. So the center of mass is halfway between them. Here I put putty on one so that they stick together in a collision. I want you to imagine what the center of mass will do after the collision. And what about this case? And this one? Okay, make your predictions. So, in each of these cases, kinetic energy was lost. In fact, the only kinetic energy left after the collision, was that due to the motion of the center of mass, I'll call that the center of mass kinetic energy. And the center of mass kinetic energy can't be lost if external forces are negligible. As we saw before, the velocity of the center of mass is unchanged. We call all of these collisions completely inelastic collisions. Because the objects travel together after the collision, they lose as much kinetic energy as is possible. Remember that the centre of mass kinetic energy cannot be lost, not without external forces. So, you know what's coming next. An elastic collision loses no kinetic energy. Spring forces are conservative so let's replace the putty with a spring. Again, I ask you to predict what the masses will do after the collision. And what about this case? And this one? Okay, once again, make your predictions. That velocity equation was interesting, it had two solutions. One solution is v1 = 0 and v2 equals v. [SOUND] The other solution is v1 = v and v2 = 0. Well, that just means it missed. Possible in algebra, but unlikely on the air track. So far we've looked only at collisions in one dimension. If external forces transfer negligible impulse during the collision, momentum is conserved. Further, because momentum is a vector, that give us two scalar equations, we shall do much with those. But we shall give you a link if you'd like to go further. But for now, let's go to a quiz. [MUSIC]