 Hi, I'm Zor. Welcome to Unisor Education. Today's lecture continues the topic of motion and we will talk about trajectory of the motion. Conjecture, that's the theme of this lecture. Now, this lecture is part of the course called Physics for Teens. Physics 14 presented on Unisor.com website. Now, if you found this lecture somewhere else on YouTube, for instance, I do recommend you to go actually through the website because it contains notes and exams for some students if they want to take it. So, basically, it's a little bit more functionality than just a plain lecture. Okay, trajectory. First of all, what is trajectory? Now, we know that motion is actually change of the position of certain object which we actually model as a geometric point in three-dimensional space where we live. Okay, now, when the object moves from one position to another, well, it leaves actually the previous position empty and then it takes another one and then it leaves this position empty and then goes to somewhere else. So, as we observe the object, we can see its position at certain moment of time and then maybe another moment and then another moment. So, what is trajectory? Well, trajectory is basically exactly a set of all the points visited by this particular object during its motion. So, this is kind of an explanatory discussion. Now, a little bit more rigorous definition of what actually trajectory is. Well, in some way, it's similar to a definition of graph of the function. So, if you remember, the graph is just a bunch of points on two-dimensional plane with x-coordinate being an argument and y-coordinate being a function. Here we have exactly the same thing, but it's in a three-dimension. Now, what is the argument? The argument is time. Now, as we know, the position of our object or point is defined by three functions. That's what we have discussed in the previous lecture. So, these three functions which are x, y, and z-coordinates in some Cartesian system of coordinates established in our space. So, these three functions actually define the motion. Now, t is time, x, y, and z are coordinates. Now, usually the time is a variable, the argument actually, which belongs to certain interval, let's say from zero to some time limit capital T, whatever it is. So, now the definition of the trajectory is a set of all points with these three coordinates where t is any value from zero to some kind of a limit T. So, the collection, the set of all the points visited by our object during the time of motion and the time of motion is from the time equal to zero to time equal to capital T. So, the collection, the set of all these points represents actually the trajectory of the motion. Now, can we see this trajectory? Now, the graph of the function we can actually represent and we can see the graph, right? Well, basically in three-dimensional space, it's a little bit more difficult, but in certain cases we can actually see it quite, quite visibly. For instance, if our movement is not really in three-dimensional space, but in the plane, two-dimensional plane, and it represents it actually as a paper, and we use the tip of the pencil moving on this paper. Well, whatever the trace it leaves, this is a trajectory of the tip of the pencil. Now, we don't have three-dimensional paper, so to speak, and some kind of a pencil which we can see. However, in certain cases actually it is possible. For instance, there is a famous Wilson cloud chamber. It's actually a chamber with vapor. And whenever some kind of elementary particle is going through this chamber, it leaves within the vapor inside the chamber, it leaves some kind of a trace. And you can see, you can photograph this trace. That's how certain physical experiments with elementary particles were conducted. So in some cases we can see it even in three-dimensional space. But we don't really have to see or feel it. You can always imagine that there is some kind of a curve in the three-dimensional space which describes the movement of the tip of this pen, for instance. So I'm putting this something in the air and you can imagine that there is a trace in it. That's the trajectory. Now, I would like to point a very important property of this trajectory. Well, actually it's the property of these three functions. Now, these three functions must be continuous in mathematical sense. Why? Well, let's just imagine that one of those functions, let's say, x function is discontinuous, which means it goes within the x-axis at certain speed, let's say. And then all of a sudden it jumps to a different location. Let's say from a location x equals 2 at time equals whatever. Immediately after this is the next infinitesimally small interval of time. It jumps to location x is equal to 2, for instance. Well, it doesn't happen in practical life, right? Because that's actually the dream of science fiction. When we have some kind of a rocket or whatever else, you press the button and it immediately transports you to another planet. So that does not happen in real life. So in physics which we are learning, we are assuming that these functions are continuous. Alright, so I think the only thing which is left is just to have a couple of examples of trajectories which we will probably be dealing with. The simplest trajectory is always a straight line in a three-dimensional space. And in this particular case when we know that the object is moving along a straight line, now basically the system of Cartesian coordinates is at our control. What's very convenient is to choose, let's say, the x-axis to be along that line of movement. y and z would be perpendicular and that's why the coordinates, y and z-coordinates will always be 0, right? So if you have this Cartesian system of coordinates and we basically are talking about movement within the x-axis, because it's our choice, right? So we have chosen x-axis to be along the movement. And zero point, obviously we have chosen as the position where movement starts at t equals to zero, right? So what will be the good example of movement down the x-line? Well, this is the function which basically corresponds to this movement. So as t is increasing, x of t, which is x-coordinate, is also increasing. And basically it's increasing in such a way that for equal increments of time would be equal increments of lengths of the distance covered, so to speak. But obviously for any point on x, the y and z-coordinates will be 0. So these three functions describe the motion along the straight line, which is an x-axis, and not just any motion, but the motion which is actually quite even, which means with even increments of time you will have equal increments of distance. Now, obviously the straight line movement doesn't have to be along the straight line with such an equality in increments, right? We can have some kind of jiggery movement. So one of the examples of basically moving back and forth would be if x function is not something like this, but is changing sine or cosine, whatever, so it goes back and forth, back and forth all the time. Now, another example, for instance, of straight line movement would be something like this, 2 to the power of t. Well, this movement is actually accelerating because with bigger t we will have the increments moving a little bit faster. I mean, the distance covered would be faster than the small t's, right? So that would be an uneven, so to speak, not very constant. It would be accelerating, actually. We didn't talk about acceleration, yes, but you obviously understand that's what it is. As t is increasing, the distance would be increasing with more and more speed, so to speak. So these are all examples of straight line movement. Now, what other movements we will be talking about? Well, circular movement, that's another example. Now, whenever you have a circular movement, it's very convenient to choose the x, y plane of the system of co-ordinate within the plane of the movement, right? And the center of the circle around which our object is moving should be actually the origin of the coordinates. And the z-axis should be perpendicular to this axis, in which case z-coordinate will always be equal to zero, right? So the movement is within the x, y plane, so that's how it will move all the time. And the center of the circle is origin, and z is perpendicular to the plane of movement. Now, what can be an example of the movement of this particular type? Now, obviously, whenever you are, your x-coordinate and y-coordinate should always satisfy this equation, right? Because of the Pythagorean theorem. This is, for instance, y, and this is x. And obviously, it's always, sum of squares is always equals to r square. So as long as you have this and z is equal to zero, you will have a circular movement. Now, examples, for instance, well, x of t is equal to r cosine t, y of t is equal to r sine t, and z is equal to zero. This is an example of such a very evenly spaced, evenly, with the same speed, so to speak. Again, we did not define speed, but you understand that this is some kind of a circular motion with no jiggering around, etc. But obviously, we can have some other functions which satisfy this equation, which means it's still circular, but it can be this way and then this way and then this way or something like this. So that's another example of the trajectory. So we have a straight line, and as an example, we have a trajectory which is within the plane, for instance, circular. And the third example which I would like to offer you is a real three-dimensional trajectory. Now, my example is basically a very simple one. We all know how the corkscrew works, right? So you, let's say your bottle stands on the table, you put the corkscrew on the top, you screw it, and the tip of the cork is going down, down, down in this movement which is actually a spiral, right? So in this particular case, what makes sense is to have, well, let's say this is your bottle, this is your cork, and this is your screw which goes in. So let's consider this tip. Now, what's very convenient is, in this case, put z-axis down, down the center of the bottle, right? The axis of the bottle, and it will be the center of this spiral corkscrew. And this plane which is perpendicular should be actually the plane where you will have x and y. Now, what happens in this particular case is, on one hand, as we go down, the screw goes down, and the tip is changing its z-position in some way as it just basically going down, down, down. An example can be, obviously it depends on the speed how we rotate the corkscrew, but for example it can be something like this. So as t is increasing, well, a is positive, hopefully, we go down increasing the z-coordinate. Now, how the tip would be relative to x and y-coordinates? Well, it actually is circling. So if we project the tip of this corkscrew onto this plane which is aligned with the bottle, with the top of the bottle, it will actually do the circular movement, which means x can be equal, for example, r cosine t and y can be equal to r sine t. Where r is the radius of this spiral from its axis to the outer diameter. So this is a trajectory. Basically trajectory will be this spiral, or helix I think, it's called scientifically. So the trajectory will be this spiral, and these are equations which represent functions. Obviously this is x of t and this is y of t, which describe this particular trajectory. So these are a few examples of trajectories in our three-dimensional space. And obviously there are infinitely many different, very, very complicated trajectories of God knows what and God knows how, but obviously we will study only the most simple ones. Something like straight line, or a circular movement, or similar to these. We are not going to do some kind of a research in very difficult trajectories, etc. Our movements, our laws of physics will be related only to simple ones. But look, I mean, any complicated movement can be always represented as a combination of simple ones. So that's one of the ways around this. Alright, so that's it for today. Thank you very much and good luck.