 Hi, I'm Zor. Welcome to Unisor Education. Today I would like to start a new chapter of this course, which is called Physics for Teens. It's presented on Unisor.com. That's where all my lectures are, including the prerequisite for this course, which is Math for Teens. And I do recommend you to use the Unisor.com as an entrance point, because it actually contains the course rather than individual lectures. All individual lectures are included in this course, and they are stored on YouTube, which you might have found this particular lecture. But I do encourage you to go to Unisor.com and take the course actually from the beginning. It's in logical sequence and every lecture is supplemented with very detailed, textual explanation of everything. Plus the course includes different problems to solve and even examinations. The site is completely free. There are no advertisements at all, and you don't even have to sign in. I mean, if you do, it's fine, but if you don't, you have exactly the same material to learn. So, kinetic energy. That's my first lecture in the chapter called energy. Well, obviously, as the person with mathematical background, I would like to start with the definition of what we are talking about. So, what is the kinetic energy? Well, in many cases, you can find an explanation that this is an energy of motion or something like this, which is kind of true. But again, being from the mathematical background, I cannot define a term called kinetic energy using the word energy, which is not defined. So, I'm not going to define energy as a general right now. I'm talking about kinetic energy only. And I would like to define it. I was actually thinking about how to define it more or less, I would say, strongly and strictly and logically. So, I define it as follows. The kinetic energy is a quantitative characteristic of an object in motion. So, we know what the object is. We know what the motion is. And so, we are talking about certain quantitative characteristic associated with this motion. Now, what's important is that this particular quantitative characteristic, it indicates that a certain amount of work can be done if this particular object in motion interacts with other. So, this amount of work actually depends on the object in motion, more than on other objects which it interacts with. And I'm going to explain what it is. So, basically kinetic energy is an ability of the object in motion to do some work, to perform some work when this object interacts with other objects. Now, what I'm going to do is I'm going to start with a very simple example. I have an object in motion which has mass and speed and we consider it's a uniform motion along a straight line. And let's assume that I have one particular constant force acting against this motion along the same trajectory. So, this is our object. It moves this way, mass m, speed v. And now, let's consider it's... There is a force which acts exactly against this motion and it tries to, obviously, to slow it down. Now, force will do certain amount of work when it slows down this object from the initial speed v to ending speed 0. So, I would like to find out what is the amount of work f which is needed to stop this object. Now, what is the source of the f? Well, it can be for instance air resistance or friction or, I don't know, some kind of a spring or whatever when it actually hits the spring when, for instance, in some cars they have an ending point and then they go this way and then go back along the same route. They sometimes have a spring which basically stops them. So, whatever the force is, it actually represents certain interaction of this object with something. But this is the simplest form of interaction. Just the constant force acting against the motion. So, how can they find out the work which this particular force actually is doing? Well, this is a product of this force times the distance it acts. So, I have to find the distance. I know that we have initial speed v. We know the ending speed is 0. So, there is a deceleration. And I know the formula. This is the formula of a distance. If the object is changing its speed with acceleration a during the time t, then that basically represents that particular distance which this object covers. So, it's basically the same formula as if you are going from the speed 0 to certain maximum speed achieved using this acceleration in this time. Or vice versa, if it has certain speed in the beginning can be decelerated down to 0. So, the formula is exactly the same. So, in one case a would be positive, another is negative. So, right now we just disregard this particular sign and we assume that this is just a positive constant and that represents the distance. So, I have to find this constant. Well, I know that a is related to force and mass as this is a consequence from the second Newton's law, right? F is equal to mass times acceleration. That's a Newton's second law. So, that's why we can find acceleration. So, acceleration I know. How about the time? Well, that's also relatively easy. If you have on one end the speed is equal to 0. This is V end and V begin equals to V. So, during the time t I have this acceleration which I know nowadays, right? Since I have determined it and we know that V begin is equal to a times t, right? Oh, sorry. Yes, yes, that's exactly what it is. When we are decelerating from this down to this, this is basically the plane kinematics equation for the speed after acceleration or in this case it's deceleration actually. So, from this I can find t. It's equal to V which is the beginning speed divided by a. Now, a I know what it is. It's f divided by m so m goes to the top. So, this is my time. So, I know acceleration. I know the time. Now I can find the distance. The distance is equal to a which is f divided by m times t square which is V square m square divided by f square and divided by 2 which is equal to m f. So, it's m V square divided by 2f. So, what's my work? Work is equal to f times s. The force times the distance. So, if I were multiplied by f that would be m V square divided by 2. So, that's actually quite a remarkable result. Look at this formula. What it depends on? It depends on mass and the initial speed of my object. It does not depend on force f. This is what's remarkable about it. It means that no matter what kind of resistance this particular moving object faces, the work which that force of resistance actually should spend, should perform to slow down my motion to zero basically is independent on the force itself. Now, if you have a strong force it will have, you see it's in denominator. The strong force will have a shorter distance. The weaker force will have a longer distance but the work will still be exactly the same. So, my point is that this work which basically is something which kind of performed by outside force relative to the motion of the object really is completely irrelevant in this particular case. There is certain quantitative characteristic of motion which is this one and I call it kinetic energy. So, this kinetic energy, this is the definition basically of the kinetic energy of the moving object of mass m with uniform speed V. This is certain amount of future work, if you wish, that anybody who wants to basically stop this motion should perform. No matter how you stop this motion, you will still have to perform this amount of work which depends only on the moving object. So, that's extremely important part of this. Kinetic energy of the moving object depends only on this object and its motion. It does not depend on anything which resists this motion. No matter how you resist, the amount of work will be exactly the same. That's why it's very important to understand that kinetic energy of the object again is characteristic of the object and its motion. It's independent of any outside forces and this is basically the formula which kind of justifies my initial definition that kinetic energy is a quantitative characteristic of the object in motion. It's a very mechanical kind of thing. We are talking about velocity or speed or whatever, so it's mechanics. So, it's a characteristic only of the object in motion. It's not a characteristic of anything outside of this object, though the work which is needed obviously depends on objects which interact with this object. Okay, fine. So, this is something which is basically a very, very obvious thing, what it is. Now, let's consider a slightly more difficult situation. Very, very slightly. For instance, we are not stopping an object. We just slow it down. But again, we are slowing down our object from speed V, which is beginning speed, down to speed V end. Now, if my definition of the kinetic energy is correct, then there is a certain amount of kinetic energy in the moving object when it moves with the speed V and there is a certain amount of kinetic energy which is associated with the ending speed V end. Now, if I would like to slow down the object from this speed to this speed, I have to spend a certain amount of energy. The same as in the previous example, I spend a certain amount of energy to decelerate the object from some speed V down to zero. Now, I'm just decelerating to the value of the speed V end. Now, if again, the kinetic energy is a characteristic of the object in motion, it means there is a certain amount of kinetic energy in the beginning. There is a certain amount of kinetic energy which still remains in the object when its speed has been decelerated to this value, which is not equal to zero in this particular case. So, the difference between these two must be equal to the work which needed to be spent to do this, right? So, I will basically do very, very similar calculations and I will find the amount of work based on whatever the force F is. And eventually I should come up with the formula that this amount of work should be equal to a difference between beginning and ending kinetic energy of the moving object, right? I mean, if my definition is correct, if kinetic energy is really a characteristic of the object in motion, then this is the amount of kinetic energy I have, which means potentially it needs to be performed certain work to slow it down, let's say, to zero. But I still have this, so I don't really need all the amount of work. I only need the amount of work to slow it down to the end. So, the difference between them must be equal to the work. Well, let's check it out. A. Now, F is the same thing, so it's F divided by M, right? Now, the T, again, we know that the end is equal to the beginning, which is our speed V, plus acceleration T. This is the general formula of kinematics. Now, we are, we would like to have an absolute value of deceleration, which means, so I'll just consider it positive, but it doesn't really matter. So, the T is equal to the difference between the speed divided by acceleration, right? So, my speed, whichever is bigger is V. So, it's V minus V end divided by acceleration, which is this. So, that's my formula for T. And formula for S is not exactly this one. Formula for S would be, in this particular case, since we are decelerating, it would be V times T minus A T square over 2. In this case, I assume that A is actually positive. Otherwise, I would have to put plus and consider A as negative. It doesn't really matter. So, that just changes the order of this. So, we are subtracting from bigger to smaller, and that's why A is positive. That's why I put minus here. So, this is the formula. So, let's just basically do some elementary calculations here. You don't need this. You don't need this. You don't need this. And let's see what happens. So, S is equal V times T, which is this one. V minus V end times M divided by F minus A T square over 2. Now, A again, A is my F over M. Now, divided by 2 means here. And now, T square, which is this square, which is V minus V end square times M square divided by M square. Right? Hope I'm right. So, what happens here is this. This goes out and this goes out. Now, we got this. So, what do we have? We have M times V times V minus V end minus again M divided by 2 V minus V end square. And the whole thing should be divided by F. This is S. Well, first of all, you immediately see that F times S already independent of the force. Right? This is my work. So, all we have to do is that we will have something like this if I will open all the parentheses. Right? So, what happens here is the following. So, let's put W, which is work, which is equal to S times F. It's equal to, let me put M over 2 outside of the parentheses. So, I will have 2 V square minus 2 V V end minus V square plus 2 V V end minus V end square. So, what we have is this goes against this, this and this are out. And what do I have? I have M over 2 V square minus V end square, which is exactly what this actually is. So, as we see, to slow down from one speed to another and moving the object of the mass M, again, it's just exactly the same amount of work which is needed by this force F, regardless of the force F. Again, the shorter or longer distance will be with stronger or weaker forces, but their product F times S, the work which is supposed to be done is exactly the same, which means that we have this certain amount of kinetic energy in the beginning. And when we spend certain amount of work to slow down, well, the kinetic energy is diminishing to the ending value. But the difference between kinetic energy in the beginning and kinetic energy at the end is equal exactly to the amount of work which our resisting force is supposed to spend, regardless of what this resisting force is, which proves again that our kinetic energy is characteristic of the object in motion and it's independent of outside forces. Okay. So, these are basically illustrations of the fact that my definition of kinetic energy makes sense. Kinetic energy as a quantitative characteristic of the moving object, which basically characterizes how much work we have to perform when we have to change this motion in some way or another. But in this particular case, we were talking only about very specific kind of motion. It's a uniform motion along the straight line and the force acting exactly opposite to this to slow it down, right? So, there are two different things which I would like to pay attention to right now. First of all, we all understand that from the definition of work, it follows that it's additive characteristic of whatever we are performing because if I would like to do something and then independently, I would like to do something else. I perform a certain amount of work here and a certain amount of work here and if I would like to accomplish this and this, I have to basically perform this and this and this. The work is actually added together. It basically follows from the definition of what is exactly the work. So, I have one force maybe acting on one object, another acting on another object. Both perform a certain amount of work and if you would like to achieve a certain result, in both cases, I have to spend basically both amounts of work and add them together, which means work is additive, which means that energy, kinetic energy, is additive. So, basically what I would like to say is the following. So, if I have n objects, now each of them have a certain amount of certain mass and each of them moves with certain speed. Now, kinetic energy of one particular object is this. Kinetic energy of the entire system of all the object is this. It's sound of this i from one to n. So, I'm adding together kinetic energies of each object and then basically gives me the kinetic energy of the entire system. Imagine, for instance, a room with molecules of air. Each molecule is moving in some chaotic direction and each of them at any given moment has a certain speed and has a certain mass. So, each molecule has certain kinetic energy. So, kinetic energy which is concentrated in the entire room would be a sum of all these kinetic energies, which is a lot. Okay, so kinetic energy is additive and this is a kinetic energy of an entire system because, again, if I would like to do something with this entire system, let's say, slow down all these molecules to a state zero so they don't really move, I have to spend amount of work to slow down each of them to zero, which means I have to spend this amount of work on each molecule and that's why we're talking about the whole kinetic energy being an additive characteristic of the system of objects. That's kind of obvious but I just wanted to tell about this. Another thing is, well, I'm talking about speed here, not about velocity because I really don't really care about the direction of this because at any given moment we can say that the speed is a certain scalar. Now, when, obviously, there is a vector characteristic of this which is called velocity but it's not really part of this thing. This is just a scalar value of the speed which is important in this particular case because even if the trajectory is some kind of a curve each particular infinitesimal piece of that curve can be considered as a straight motion and basically at that particular moment you can calculate what is the kinetic energy of this particular object is. And then as the speed actually is changing the kinetic energy is changing, obviously. Now, my last example is related to a little bit more, again, complicated case when instead of force acting exactly opposite to direction of the movement, let's say I'm trying to act at an angle. Now, what happens in this particular case? Just let me exemplify it. For instance, for instance, I have a movement of this kind, this is my object, but the force acts at a certain angle. Let's say maybe it's some kind of a car which is moving and this is the force of the wind which is trying to kind of attack this car at a certain angle. Now, what happens? Well, obviously, again, the speed will change in some way or another. I mean, obviously, depending on the force of the of the wind it will change differently. But however, I would like to basically, again, I would like to find out what happens if my force is acting a certain amount of time at angle to the speed. What happens? Well, again, my final result should be that, again, it doesn't really depend on anything except the beginning and the ending mass and speed of this particular object. So, how the object would move in this particular case? Well, it will be this probably kind of a trajectory, right? When the force moves, when the force acts this way and the object moves this way, then this would be probably something like a trajectory, right? Now, I'm talking about the movement on the plane. Obviously, whatever I'm doing right now on the plane can be transformed into a three-dimensional space, so it doesn't really matter. But let me do it in a two-dimensional case. So, let's consider I have x and y coordinates here and my initial vector is v along the x and 0 along the y axis, right? So, this is my vector of the velocity. In this case, I'm talking about velocity. So, this is my x component and this is my y component in the very beginning. So, this is some value and this is 0. Now, let's consider that during certain time t, now let's say t is given in this particular case, the force acts at certain angle, which means I have vector which has coordinates like this. So, this particular force can be represented as fx and fy. These are two vectors. These are two scolars which have the value of the magnitude of these two vectors, right? So, that's basically what it is. Now, the fx component acting during the time t would force the object to change its speed along the x coordinate and fy would be acting against the y component, right? So, my acceleration also will be a vector which is equal to fx divided by m, fy divided by m. So, this is my acceleration vector. This is acceleration along the x and this is acceleration along the y. Now, obviously, based on the time and based on the initial speeds along the x and y and based on acceleration along the x and y, I can find the distance, right? So, my fx would actually drive our object along the sx on the x axis and that would be a certain amount of work which I can call wx. You see, again, forces are additive, energy is additive, work is additive. That's why I'm basically dividing it into parts. And the force which is acting along the y axis will act at the distance sy and this would be amount of work which my force will spend along the x direction. This will be amount of work along the y direction and I will see if their sum is equal to change in the kinetic energy of the body of the subject. Now, in the beginning, kinetic energy was mv square divided by 2. At the end, it will be correspondingly piece along the one axis and piece along another axis. So, there is an energy which this particular object has in one direction and energy in another direction. Now, it's just plain arithmetic or algebraic. I know acceleration. I know the time. So, now we can find out the distance. So, sx is equal to, now along the x, my initial speed is v and acceleration is at square over 2 ax, which is fx square, sorry, fx t square divided by 2m, right? fx divided by m is acceleration. So, it's ax times time square divided by 2. That's the formula from kinematics. This is the distance. My force, which is equal to fx, is acting along the horizontal axis, right? Now, my sy is equal to initial speed along the y speed is 0. So, I have only ay t square over 2, which is fy to m t square. Okay, what's next? Well, what's next is that I have to compare some of these, which is two amounts of work, which my force spends in one direction and another direction, multiplying this and this and summing them together, and compare it with the total amount of energy, which was in the beginning and the total amount of change of the kinetic energy between the beginning and the end. Now, in the beginning, I have mv square over 2 and along the y axis is 0, I mean along the y axis is 0, right? At the end, at the moment t, I will have, at the moment t, along the x axis, I will have mv x t square over 2, where the x t is the horizontal speed at moment t and my y direction energy is mv y t square. Let's put parentheses divided by 2. That's my energy. So, if I will subtract from the beginning, which is from the total energy of this, I will subtract this, I have to actually have some of these. We just have to check if it, you know, if it works. Well, let's just try. Let's just try. So, what is my v x t? At moment t, my speed would be the initial moment in the horizontal, which is v, plus a t, right? a is f divided by m t and my v y t is equal to similar, except that my vertical direction is 0, so it will be just f x, sorry, f y divided by m. So, I have to calculate now this plus this and check if I will have the same as total amount of work. All right, let's check it out. Okay, let's calculate this one. I know this, so I have to square it. So, energy along the x axis would be equal to m divided by 2 and the speed at moment t square, which is v plus f x over m t square. Now, energy along the y direction at moment t is equal to m over 2 times f y t divided by m square. This is mass. This is the speed square divided by 2. Okay, so if I will add them together and subtract my initial value of energy, what will be? So, e x at moment t plus e y at moment t minus e minus e, which is m v square divided by 2 equals. Okay, m divided by 2 v square plus 2 f x t divided by m plus f x square t square divided by m square plus y m divided by 2 f y square t square divided by m square minus initial m v square divided by 2 equals. Okay, this goes against this one. m 2 2 m f x. So, it's f x t, right, plus f x square t square divided by 2 m. m and m 1 m cancel 2 is remaining and plus m over, okay, so that's also f y square t square divided by 2 m, right? Okay, now let's say what happens here. Okay, if I will multiply this times f x by the way, I miss d here. I'm sorry. Yes, d times f x, right, yeah, I see something is wrong here. So, here also we'll have, sorry, here. Fine, so let's multiply this by f x and this is by f y and add them together. What happens? f x d t, f x d t plus f x square t square 2 m, f x square t square 2 m plus f y square t square 2 m. Exactly the same thing. So, again, the quality doesn't really depend on anything. Now, this contains components of the force and this contains components of the force, that's true. But since they are the same on both sides, which means that regardless of what exactly this f x and f y are, what's their direction relative to the trajectory, initial trajectory of our object, the resulting difference between kinetic energy in the beginning and at the end is exactly the same as amount of work force which acts at the angle performs. And here what's very important, I was using additive property of the energy and additive property of the work. I basically decided to represent my force f as two forces, f x and f y, each one separately doing something, each one separately performing certain amount of work. And the result of this is the difference between the total amount of kinetic energy. Well, that's it for today. I do recommend you to read the textual part of this lecture, which is basically the same thing. It's like a textbook and it's on unizor.com. Then I will probably do some problems next time about kinetic energy. So thanks very much and good luck.