 Hi, I'm Zor. Welcome to Unisor Education. Continuing discussion on different aspects of motion, today's lecture is about acceleration. This is part of the course called Physics for Teens, presented on Unisor.com. And that's exactly the website where I suggest you to watch this lecture because it also contains notes for every lecture and exams. It also contains the course of math for teens, which is actually a prerequisite for this course. And also there is even US law for teens. Now the site is completely free and there are no advertisements, so I do suggest you to use it extensively in your studying. So, let's talk about acceleration. Well, the previous lecture about speed and velocity was really fundamental to understand all the aspects of motion in classical mechanics. Now this one, acceleration, is just basically a continuation. If you have understood perfectly well what is a velocity, that this is a derivative from coordinate functions. It's a vector from each coordinate function. What's the difference between velocity and speed? Velocity is a vector, speed is its magnitude of this vector. So if you understand that, then this particular lecture is very, very simple. So, again, velocity is basically an instantaneous speed of change of position, right? So, if you have coordinate functions x, y and z, which describe the vector from the origin of coordinates to a position of our point at time t, then the corresponding velocity vector is an instantaneous change of each parameter. And the instantaneous change, as I have explained, is just a derivative from the corresponding function, coordinate function. So, instant change of the position is velocity. Now, let's just consider a completely generalized function of time, some kind of characteristic of the object which is related to time. Whatever the function is, h of t. T is the time parameter, h is its characteristic, which means anything, basically, which can mean anything. It can mean the position, it can mean speed, it can mean an odd density, whatever. Now, if this is the function of time, then instantaneous change of this particular characteristic is exactly the same as in case of position and velocity, right? So, instantaneous change of time-related characteristic of an object is derivative from this function by the time, by the parameter argument t, right? So, that's a general position. That's exactly what was used here. Now, let's just apply this rule to this vector. Now, this vector is the vector of the position, right? Let's call it p. And instantaneous change of the vector of the position is velocity. Now, what if I want to find out how velocity is changing? Velocity can change. We are not really driving the car with constant speed, right? So, how can we do this? Exactly the same way. Instantaneous change of this particular vector of velocity is derivative of each function which corresponds to each coordinate of this vector, right? Now, so we need to do the derivative from the derivative. And as we know, this is just derivative of the second order. So, we are applying exactly the same logic to find out what is the instantaneous change of velocity. And this is called the vector of acceleration and the story. So, basically we have applied exactly the same logic to find out the instantaneous rate of change of certain parameter. We do the derivative. Now, if you want to know instantaneous rate of change of this parameter, we do the derivative. So, if we have our position described as three coordinate functions, the first derivative describes the vector of velocity, and the second derivative from these original functions describes the vector which we call acceleration. It's definition. So, there is nothing to talk about. It's definition, right? So, alright, now we have to talk about terminology just a little bit. Now, in case of the first derivative, this is the vector, velocity is a vector. But we also have the word speed, right? Which, strictly speaking, we have decided that, okay, velocity is a vector and speed is its magnitude. Now, what's the magnitude of the vector V? Well, the magnitude of the vector is, as we know, square root of squares of coordinates. So, this is basically speed. Now, what's interesting is, if you consider a movement, let's say along the z-axis, which means that x and y are always zero, then this and this is zero. This is the first derivative. Now, this and this will be zero, right? And this square root of this thing would be basically the z of t itself, right? So, in this case, the vector will be z prime of t. Zero, zero, come z prime of t. And the magnitude of this vector would also be, well, if we want it to be precise, it's absolute value because this is the square root and it's a positive. Now, this particular value may be negative, but well, it's almost the same as z prime t, right? So, almost the same as first derivative. But in any case, we have two different words. One is for the vector of velocity, which is a vector actually, and the speed is the magnitude of this vector, which by absolute value is equal to basically the monicomponent if we are moving along a straight line along some kind of either z-axis or x-axis or anything else. Vector itself and its magnitude when we are talking about direction along the straight line are almost the same because we're not really talking about direction very much because direction is obviously along the same straight line, it does not change. We can probably ignore it. But in any case, strictly speaking, in a three-dimensional world, this is a vector called velocity and this is a scalar, the magnitude of this vector and we call it speed. So, we have two different words, velocity for vector, speed for its magnitude. Now, in case of acceleration, there are no two words. There is a vector of acceleration and you can talk about absolute value of acceleration, which is basically, again, the square root of this square plus this square plus this square. So, the word acceleration in some cases assumes that we're talking about the vector and in some cases it assumes that we're talking about the magnitude of this vector. So, it's only the context probably should clarify this issue. But again, if we are talking about the straight line movement which in many beginners' courses of physics is the only kind of a movement which they consider while accepting the rotation. So, in any case, in most cases, addressed in regular course of physics for beginners, we are talking about the straight line, in which case, obviously, all the vectors have exactly the same direction and we can probably not think about the direction anymore and think only about the magnitude of this vector. With a sign, plus or minus, which basically kind of signifies the direction. All right, so that's just a side issue about the terminology. So, we have acceleration and these are second derivative of the coordinate functions. Now, what I wanted to do right now is just to give you a couple of examples of movement, its velocity and speed and its acceleration and its magnitude of the acceleration. All right, so, examples. I have four examples. The first two are absolutely trivial. Now, let's consider you have the movement which is defined by the following coordinate functions. Now, what does it mean? It means that at any moment of time our object, our point actual where the object is located is always in the beginning of the coordinates, in the origin, and it doesn't move. So, basically, it's a body at rest. It's the object which doesn't move. Its initial position is at the origin of coordinates, zero, zero, zero, and it doesn't move since at all. Well, that's the easiest actually kind of movement, no movement at all, right? Now, what's the first derivatives? Well, it's a constant, so the first derivative is zero. Obviously, the magnitude of this vector. So, the velocity is a null vector. I hope you remember what null vector is. Vector is coordinates equal to zero, all components. Now, its magnitude is obviously zero as well. Now, if we want to go to acceleration, and it's obviously zero as well, so it doesn't move, it doesn't accelerate, so everything is equal to zero. And this is the simplest kind of case which we will probably never be dealing with this kind of movement. It's really trivial. Now, the second also absolutely trivial is when the body is also at rest, but not at the beginning, not at the origin of coordinates, but somewhere else. Let's say the initial position, which does not change with time, is three, ten, and minus six. So, we are at point in the three-dimensional world with three coordinates, three, ten, and six, and minus six, and the object doesn't really move from this point at all. What happens with the first derivative, with velocity? Well, again, the constant, the derivative from the constant is zero, so we also have exactly the same thing. And obviously, if the body is not moving, the velocity vector of velocity, its speed, obviously it's all zero. And if it's not moving, if it's standing still, obviously there are no accelerations. The second derivative is also equal to zero. And obviously the magnitude, the speed is equal to zero, and acceleration as a constant, as a scalar, is also zero. Again, very trivial case. Now, we will slightly change this thing. My third example is three plus six t. y is equal to ten minus eight t. And z is equal to minus six plus ten t. Now, what does it mean? Well, at t equals to zero, we are in exactly the same position as in the previous example, three, ten, and minus six. But now, as t increases, as the time goes by, my object is moving. Now, these are all linear equations of time, so it's very easy to prove that the resulting movement will be a long straight line. Some straight line. It doesn't really matter which one, but some straight line. Now, what's interesting is how fast I will move. So, what's my velocity? Well, the vector of velocity is the vector of derivatives, right? Now, the derivative of three plus six t, well, that's the sum of two functions. So, derivative of the sum is equal to sum of derivatives. Derivative of three is a constant which means it's zero. Derivative of six t is six. Derivative of ten minus eight t is minus eight. And derivative of minus six plus ten t is ten. Okay, so that's my velocity. Now, look at this. Velocity is a vector. Now, what's interesting is this vector doesn't change with time. What does it mean? It means that our linear speed along that line is exactly the same and it does not change. So, the vector of velocity which has a magnitude and direction, since it does not change, there is no dependency on the time parameter, right? It doesn't change. It means that the direction and the magnitude of this vector are always the same. So, direction obviously is defined by three coordinates. Now, it's magnitude. It's very easy to calculate. So, the magnitude is equal to square root of six square plus minus eight square plus ten square, which is thirty-six, sixty-four, a hundred plus a hundred, two hundred, so it's ten square root of two. So, that's the magnitude and that's the speed. So, this velocity vector defines direction, obviously along that straight line defined by these parameterized equations. T time is a parameter and the magnitude is this one. So, the direction is within this straight line and the linear speed along this straight line is this. Now, finally, acceleration. Now, acceleration, these are constants, so derivative from the constant is zero. So, there is no acceleration, which means that the speed is constant. It doesn't do it faster. It doesn't do it slower, because with a certain constant speed along the line defined by these equations, our body is moving and moving and moving without any change in the rate of movement. So, that's my third example. And the fourth example, let me just wipe it out. I want to draw a picture. Okay, so this is a tower of Pisa. So, I'm sure everybody knows that there is a town in Italy called Pisa and there is a tower in there, which is basically slanted a little bit and Galileo was using it as an experiment. He was dropping balls down to the earth and he was actually counting whatever the time covered, maybe the speed of the ball, it's not really easy to do the speed, but anyway, he was doing the experiments with things which he dropped from the tower of Pisa. And he found out that whatever object he drops, the time until it reaches the ground is exactly the same. And he was kind of surprised a little bit because he was thinking that heavier objects should really fall faster. But that's not the case. Anyway, so this is the tower of Pisa and let's consider that the height vertical is capital H. So, we are dropping certain object from the top of the tower of Pisa down to the ground. Now, first of all, I have to talk about coordinate system. Now, here is coordinate system. This is the origin. These are on the ground X and Y and this is Z axis. All the way up to the top of the tower of Pisa, which means that our original position, so only the Z component is not equal to zero. The X and Y component of the object which we are going to drop from the top of the tower, so the Z component is H, X and Y are zero. That's pretty much understood. Now, I would like to tell you how exactly this particular object will be falling down. Well, it will fall down along this vertical line, which means X and Y coordinates will not change. It will be still zero. So, this is zero and this is zero for any T. It doesn't change. But the Z coordinate is changing with the time. Now, initially, at T equal to zero, it's H and then it becomes smaller and smaller. That's why there is a minus sign. And what's interesting is it's proportional to the square root of the time interval passed from the beginning, where G is some kind of a constant. We are not talking about what's the meaning of this constant and I put the coefficient two just for convenience because this is a constant anyway. I put K but I decided to put G over two to make my further calculations just slightly easier. So, there is some kind of a constant and this is given. I mean, I'm giving it to you by definition that this is the law of falling from the height H because the T is equal to zero, obviously Z of zero is equal to H as it should. Now, at some time, let's just think about what time it will reach the ground. Well, it will reach the ground when Z of T will be equal to zero, right? So, the moment capital T, when it reaches the ground, it means that zero is equal to H minus G T square over two from which T is equal to 2H divided by G square root, right? So, this is the time if my body is falling according to this formula, then this is the time when it will reach the ground but that's not exactly my point right now. This is the side issue. My point is let's find velocity at each moment of time and acceleration. Okay. So, we will disregard this. We don't really need it. Okay. Velocity is the first derivative from the coordinate functions, right? So, what is this vector? Well, the X coordinate is constant zero. So, the first component of velocity, the derivative from the X will be zero. Y will be exactly the same thing. Now, the derivative of this, well, this is the constant minus remains. Now, T square is derivative is 2T, right? This is just a coefficient. So, it will be G times 2T divided by 2. So, it will be minus GT. Well, that's why I put divided by 2 to get a little bit nicer velocity. Okay? Otherwise, I would have to put coefficient 2 here. So, this is my velocity as a vector. So, let's just analyze it. First of all, you see zero and zero, which means my body would not move outside of the vertical line. It's only within the Z axis it will move. No movement outside of the Z towards X or increasing or decreasing X or Y. It will be zero all the time. Now, along the Z, it's minus. What does it mean that it's minus? Well, that's because we are decreasing, right? We were on the H and we are going down to zero. So, we are decreasing. Since we are decreasing, function is decreasing and decreasing function have negative derivative, right? So, that's why it's minus GT. So, what's interesting is that my velocity is not constant now. I mean, it's constant by direction, but it's not constant by magnitude because the magnitude of this thing is square root of zero square plus zero square plus minus GT square, which is GT. So, magnitude is increasing as T is increasing. So, as the time goes by, my body moves faster and faster and we all know that in the beginning basically the velocity is equal to zero, right? We just dropped it. We didn't push it down. We just dropped it. And if we dropped it, the initial speed is equal to zero. So, if T is equal to zero, my initial velocity is equal to zero. Now, if the time goes by and as you see the magnitude of the velocity is increasing, so the velocity will be the maximum at the very end, right? So, if you remember the very end, T was what? Square root of two H over G, right? I was just driving this. So, at moment T, my velocity will be G times T, which is equal to square root of two HG, right? In the top and square root of G in the bottom, so it will be square root of G in the numerator. So, that would be my final velocity when it reaches the ground. But anyway, this is my vector of velocity. This is its magnitude. And what's my vector of acceleration? That's the derivative of this. Zero, zero. Derivative of minus GT is minus G. So, my acceleration is constant. So, my speed is not constant. My velocity vector is not constant. But the acceleration is constant. So, whenever the body goes down, it accelerates, which means the instantaneous increase of the speed is constant. Increase of the speed is not constant. That's the velocity. But instantaneous increase of instantaneous increase of the position is constant. That's what's very important. And it's obviously related to gravity. Our planet attracts everything. And the result of this attraction is basically the force which brings down this particular body. What's interesting about the whole story here is that the acceleration is constant. Well, that's basically, that concludes my few examples of knowing the coordinate function, whatever the coordinate function is. We can basically find out everything. Now, in this particular case, we found out the velocity. We found out the acceleration. We found out the moment of time when the movement actually is ending, when it reaches the ground. And we even found the final speed at that particular moment. When this particular object hits the ground, that's the speed of hitting the ground. Everything is derived from equations of motion. Here. This, this, and this. As long as you know how the object moves, you can calculate all these little things which are needed to basically analyze the movement. All right. That's basically it for today. Thank you very much and good luck.