Let's look at the relationship between momentum and Newton's laws. So open your Principias to page XVII and let's first look at Newton's first law. "An object in motion remains in motion." Here's an object, it has velocity v, it's in motion or, "An object at rest remains at rest." So if you think, well this has velocity, also has momentum, so it's retaining its momentum. Here this has no momentum, it's retaining a state of no momentum. So what this one is really saying in terms of momentum, is that momentum is constant. He didn't say the part about the isolated system, but it's basically implied. So momentum is constant or conserved. So the first one is really just saying conservation of momentum. Let's look at Newton's second law. So Newton's second law is basically if you have a mass here and you apply a force to it, it accelerates. We know it as F equals ma. Well, if it's accelerating, it's increasing its velocity, and if it's increasing its velocity, its increasing its momentum. So let's write this that way. Let's see, we could write it as m dv/dt. If we're always working where the mass is constant, which we usually are, we're usually applying Newton's laws to just point particles or finite particles that don't change mass, then we can put the mass inside the derivative. It's just a constant, it can go in there. So that's equal to d(mv)/dt. What is mv? It's p. Dp/dt. So Newton's second law really isn't F equals ma, it's that the force equals the time rate of change of the momentum. So you can write that in terms of momentum as well. Interesting. Let's look at Newton's third law. Let's see. So Newton's third law, "For every action, there's an equal and opposite reaction." So for this one, let's draw a couple of masses interacting with each other. So here we have m_1 and here we have m_2. I don't know, let's say they're going to feel a force on each other. It can be a gravitational force that's attractive, it can be an electrostatic force that's repulsive, don't really matter. Let's make it repulsive. So let's say this would be the force F_1-2, the force that 1 applies to 2. Well, according to Newton's third law, this one would have to feel an equal and opposite force that we would probably label F_2-1, the force that 2 applies to 1. It can be electrostatic, it could be gravitational, it could be that they touch, it could be a contact force. Any interaction between these two, the two forces have to balance. So what does that mean? Well, we could also say this one has momentum p_1, and this one has momentum p_2. If they're going to feel these forces, then this one is going to have a change of momentum dp2/dt, and this one is going to have a change in momentum, dp1/dt. These are all vector quantities. The forces are vectors, the momentum is a vector, the rate of change of momentum is a vector. So we know that the change of momentum here is equal to this force. The change of momentum here is equal to that force. Well, then what this means, if we add these two vectors and they have to be zero, we could add these two vectors and they have to be zero. Dp1/dt plus dp2/dt equals zero. Because the sum of the two forces is equal to zero, the sum of the two momentum is equal to zero. If that's equal to zero, what is the momentum of this one plus momentum of that one? It's the momentum of the whole system. This must also be equal to dp_total/dt. The system only has those two particles and that is its total momentum, p_1 plus p_2. So what it's saying is for this isolated system, the momentum is constant. Or you can say that momentum is conserved. So this one is just a statement of conservation of momentum just like that one is. Number three and number one are both conservation of momentum. So there's still Newton's laws and it's like what we said earlier, that you can solve problems with forces and kinematics, or you can solve problems with momentum. Some things lend themselves to forces and kinematics, some things lend themselves to momentum, and you can write these laws in terms of kinematics or momentum. It's really the same information. I just wanted you to see that you can think of it that way. Newton wasn't real specific. He said this, "The state of motion." Maybe he meant momentum. Who knows?
