 Hi, I'm Zor. Welcome to Unisor Education. So we continue with dynamics. This is a part of the physics for teens course, which is presented on Unisor.com. That's the website. I suggest you to watch this lecture and all others on this website, because it contains very detailed notes for each lecture. It has certain educational functionality for people who want to study under supervision. Of a parent or teacher or whoever. So, and by the way, the site is completely free and no advertising. So I do suggest you to go to Unisor.com. Oh, by the way, it also contains a prerequisite course for this, which is the math for teens. So math for teens. It's prerequisite for physics for teens. Now, we're talking about momentum of motion. This is a completely new concept. Well, you remember we had the concept of space, time, speed, acceleration, velocity, mass. So this is yet another physical concept. Now, in this case, it's not a primary concept. It's derived. So we will define it basically. And it's a very, very convenient characteristic of the motion. And it will be obvious why at the end of this lecture. Okay. So first of all, I would like to consider a relatively simple motion, not the simplest. The simplest motion is when there are no forces and the object just moves along its straight line trajectory with a constant velocity. So this is the most simple. The second simplest way of motion is motion with constant acceleration. Now, since there is a constant acceleration, there is a force, obviously. So we are assuming that we are talking about a case when there is a straight line. This is the trajectory of the movement. The force is exactly along this line. So this is our object. And it has certain mass. And it has certain acceleration, which is directed along the same line of straight line of motion. So this is the very, very simple case. And since it's a one-dimensional case, I don't really have to use the vector. So A is basically a scalar. And let this be the positive direction of the x-axis. So A is positive and f is also positive. And we all know the second law, Newton's second law, which states this. So f is a constant force acting along the straight line trajectory of the object. A is acceleration, m is a mass. Okay, so let's now consider two moments in time. First is when the object was here. This is t0, or just t, that would be even better. And another moment in time, then the object move to this position. This is t plus delta t. So this interval is delta t. Now speed was v. And speed of t plus delta t was obviously v plus delta v. Speed is increasing from the value of v. This object had at this particular moment in time. And by the time t plus delta t, speed has increased, obviously. By how much? Well, we do know the acceleration, right? Now the average increase of the speed per unit of time would be v of t plus delta t minus v of t divided by t plus delta t minus t, right? We have the difference between speeds, difference between times. And this is obviously delta v divided by delta t. And this is the average acceleration. But since acceleration is constant, that's exactly equal to a. The same a. It's a constant acceleration, very simple, right? So what do we have from here if we will use this? Well, this will be m delta v divided by delta t. Or f times delta t is equal to m times delta v. Same thing. So increment of speed times mass is increment of time times force. This is a very interesting equation. It has far-reaching consequences. Now, since this is true, let's just make our delta t smaller and smaller. Obviously, we all know from calculus that whenever it happens, this delta v divided by delta t would be basically a derivative, right? So that goes to m dv by dt or mv prime. So this is a derivative of the speed. Okay, so what we can say is that force is equal to m times dv by dt. Now, what's interesting about this equation, since delta t can be infinitesimally small, we don't really have to depend on constant force and constant velocity or constant acceleration, et cetera. Because all we need is the constant value of these within this delta t. Within the next delta t, it can be different. That's okay. No problems, right? But within the delta t, we assume that if these functions are smooth enough, then basically, whenever we are making this interval smaller and smaller, this is the final value. So f can be actually function of t and m can be function of t. No, not m. m is constant right now. For now, I assume it's constant. But velocity is the function of t. So this is already a little bit more general formula because it does not require the values of force and acceleration to be constants. Okay, that's good. Now, what can we say then next? Well, next is we can obviously extend this formula for f being a vector in three-dimensional space. So not just we have this particular straight line as a trajectory. f can be anything. It's a vector which can be dependent on time, obviously. So it's changing. So a trajectory also can change any way we want. And the formula will still be the same because obviously for each coordinate, the x-coordinate, the y-coordinate, and z-coordinate, the formula is exactly the same. So it can be combined into the vector formula. And so I can actually do this. Vector, vector. So this is the vector of velocity, general vector of velocity of the point, which has a mass m. And the force is f. It's any vector which depends on the time. And obviously that equation would be true. All right, it is important. And again, it can be actually rewritten in differential form equals m dv of t. That's the same thing, but this is the derivative form and this is the differential form. Now, what's also interesting is that since m is a constant, I can multiply m under the derivative, right? Because the factor multiplied by a function can be taken outside of the derivative. So it can be d dt. And here I will have m v dt. That's the same thing. And here also the same thing. Differential of m times v of t. Now, one of the things which can be derived from here, by the way, is that if there is no function at all, so the object is completely by itself. So f of t is equal to zero. Then this is a constant. So if f of t is zero, what follows is that m times v of t is constant. What's also interesting is that in this particular notation, I do not really require m to be a constant. So we obviously derived it in case m is a constant. But again, if my time limit from t to t plus delta t is very small, even if mass is changing, smoothly changing, you will still have exactly the same type of a formula. So m also can be a function of t. There is nothing wrong with that. So in any case, this particular quantity is called momentum of motion. And we see that the momentum of motion is not changing if there are no external forces. That's very, very important. Now, what I will do next is I will exemplify this with a very important example, which will show actually how important this particular quantity is. Here is the problem. Here is this example. I mean, so far, it all seems to be like just manipulation with formulas, which, you know, doesn't make much sense. But this example will show you how deeply this momentum of motion is actually embedded in our world. Let's consider a two-stage rocket. So this is the rocket and it has two stages. Now, it's flying freely. So all the engines have already finished. Everything is okay, no problem. The first stage probably burned. It's filled whatever it is. It ended the job and the rocket is actually flying freely in the space. And let's assume there are no close by planets. So it's just free space, no gravitational fields, etc. So it's basically flying along a straight trajectory according to whatever system of references it is in. And then we have to really get rid of this first stage, which has already, you know, burned. It did its function. Now we have to get rid of it. Let's consider a very ideal kind of a scenario when what happens is there are some mechanisms inside this rocket which basically force this first stage to go back along the same line for simplicity. So let's just assume there is certain internal force F. Just for simplicity, you might consider, okay, this is a spring which is actually tied together and then at a certain moment we release the spring. So it springs and pushes this one down and obviously this one up with the same speed because there is a third law of Newton, right? So there is a certain force which acts during the time t and this force actually allows our first stage to go back and the first stage and second stage continues to go along the same trajectory. Now during this particular time period this force obviously accelerates and then it finishes. That's it. Like spring has already done its job, it's finished and then the first stage goes by its own inertia and the second stage continues going by its own inertia. All right, so let's just consider the following thing. Let's say the first and the second has certain masses and the whole rocket has a speed b in the very beginning. Now what happens when I act with this force during this time with the first stage? Well, it has initial speed v as the whole thing and then I am acting on this particular first stage with this force with this time which means the acceleration is this, right? Acceleration is equal to, in this case, mass is m1. This is acceleration. This is force. So acceleration is this. Now during the time t, if I have acceleration, my speed would increase or decrease depending on the sign of the acceleration by a times t, right? Now in this case since the rocket moves this direction and we are pushing the first stage backwards, so it's minus. So it's minus f divided by m1t and this is my speed at the end of this process. This is the speed of this first stage after we have finished pushing it out. Now if we have pushed really hard, it may go completely in the opposite direction along the same trajectory. If we pushed it slightly, it will just separate from the second stage and it will just move slower. That's it and it will be farther and farther but it will still continue moving this direction but slower than the second stage. Now as far as the second stage is concerned, initially it has also speed v. Now the acceleration it will have is the same force because the same force which goes towards the first stage, it will push the second stage forward along its trajectory but the acceleration would be the same force divided by different mass and the same time t. That's what we know, right? Now what happens next is very simple. I would like to actually convert it into momentum of both stages, both parts of the rocket. So the final momentum is, I will multiply this by m1, so the final momentum is this, final momentum of the second stages. Now this is the first, this is the second, second is equal to m2v plus ft. Right? mv minus mv plus. Okay now let's add it together. What do I have? I will have m1v1 plus m2v2 equals 2. Well this will cancel out obviously and this has the same v so it would be m1 plus m2v. Now m1 plus m2 is the mass of the entire rocket. Now this is a very important equation. What is mv? mv is momentum of motion of an entire rocket before we separated the stages. What is this? This is a sum of momentum of two parts of the rockets into which we are, which we have separated this rocket. So the sum of momentum after the action and this is the total momentum before action. So if the whole rocket with two stages is a system, let's call it a system, right? Then this is the total momentum of the system after this separation happened and this is the total momentum before this separation. So our total momentum is constant. You know? Same thing. We go back to whatever I was talking before about constant momentum if there are no forces. And now there is a force here but it's internal force. It's internal for the system. So if we have a system of objects and there are no external forces upon them like gravitational field or something like that, which definitely distorts the whole picture. So if there are no external forces, if all forces are internal, then obviously different parts of our system will start moving differently but the total momentum will be preserved. So it's conservation of the momentum and we will talk about this later on. I just wanted you to understand how important this momentum actually is. Momentum of motion in case the system is enclosed without any kind of external influence is preserved. That's why it's very, very important. All right. Now, I do suggest you to read the notes for this lecture again. It's on Unisor.com. You go to the website and there is a Physics 14 course on the left in the menu. And then you choose Mechanics. You choose Dynamics. And in the Dynamics you have momentum as a particular topic. And that's where it is. This is the first lecture of this. The whole topic is actually called I think momentum and impulse. All right. Okay. So that's it for today. Thank you very much and good luck.