 Hi, I'm Zor. Welcome to Unizor education. Today we will talk about speed of the motion and I will talk about the difference between speed and velocity. Both words are actually used in our common language but there is a big difference between them in physics. Now this lecture is part of the course which is called Physics 14 presented on Unizor.com. I suggest you to watch this lecture from this website because it not only has the reference to the lecture itself, the video but also notes and exams and there are other courses on this website, for instance mathematics, math for teens and also US law for teens and the site is completely free and there are no advertisements so I do suggest you to watch the lecture from the site. Alright, so we'll talk about speed and velocity. Now what's very important is for this particular lecture and probably for the rest of the course is your certain level of mathematics, of mathematical knowledge is assumed. Now if you have come to this course after you basically learned math in a pretty good level or for instance you have completed math for teens course on the same website then you are prepared. Now, but this is the first lecture where your real mathematical knowledge will be used and I would like actually to warn you that the course itself will pretty much be dependent on your good mathematical background. Now the lectures which precede this one about time, space, general concepts of motion, trajectory, they did not really use any mathematics but this one and probably most of other lectures in this course will be dependent on your mathematical background. Now in this particular case I would like to point out that vectors and derivatives are definitely required pieces of math knowledge for this lecture and again vectors and derivatives are fundamental for old classical physics and there are some other things but for now for instance if you are not very comfortable I do suggest you just go to theunisword.com and refresh these topics. You have to understand what vector is, how to operate with vectors and you have to know what the derivative is and how to get the derivatives, what's the properties of derivatives, etc. The course relies on this knowledge. Okay, now let's talk about speed. Well everybody knows that speed is distance over time, right? Well it's kind of true but it's not really a scientific definition, not rigorous enough. What kind of time are we using? Is it an hour or is it a second? Why am I actually pointing this? It's because when somebody is driving a car, the car is not actually driven with constant velocity or constant speed. The speed is changing, the rate of change obviously is different so it's very important to be a little bit more precise whenever we're talking about the definition of the speed. Now let me point to the definition of average speed first. Now the average speed, let's consider so far a direction only within x-axis. So you have your three-dimensional space but the movement is only within this x-axis. The y-coordinate and z-coordinates are always equal to zero. Now the definition of the average speed is really very simple. You have two moments of time and you define average speed from moment t1 to moment t2 to along the x-axis as difference between positions divided by difference of time. So this is basically what distance over time means. This is the distance covered. For instance at point, at time point t1 the moving object was here and point t2 is here so you have the difference between them that gives you the distance and then you divide by time. So that's easy. So we start with the concept of average speed. Now average speed is defined quite well. Now let's talk about what is an average speed for a specific kind of a movement. Now let me choose a particular movement except t is equal to a times t. It's called a uniform movement where a is some number, t is time. Now what happens if instead of x of t I'm using this exactly function. I will have a times t2 minus a times t1 divided by t2 minus t1. A goes outside of the parenthesis and t2 minus t1 can be cancelled so the result will be a. Well that's interesting that average speed in this particular type of motion which by the way it's called uniform motion along the straight line. So whenever we have a uniform motion along the straight line average speed on any time interval will be exactly the same. So it's independent of time interval and that's why it's called uniform obviously because whenever you are uniformly moving along the straight line in this particular case along the x-axis and this is basically the function which describes your motion then your average speed will be exactly the same on this time interval or on this time interval and any time interval will be the same and that's why it's called uniform. Okay that's just an example. I mean if you will take some other function a g square for instance obviously will not have the same independence of interval. On some other intervals you will have other average speed. Alright so let's just keep it in mind and let's again think about how can we define the speed a little bit more precisely not the average speed. I would like to be able to define so called instantaneous speed. Speed at any moment of time because at this moment of time the speed might be different from that moment of time. So what do we do to define the speed? Well actually it's very easy. For instance you have certain moment of time t. Let's call this moment of time is t1 and then another moment of time which is slightly after that where delta t is an increment of time and this will be our t2 and I will make the same calculation on this interval of time from t1 to t2 from from moment t to moment t plus delta t. Now if my increment delta t is small then I have a very small time interval where the speed can actually change right and the smaller the delta t is the closer this interval will be to initial point t where we would like to measure our instantaneous speed. So in this particular case let me call it this way. So it will be from t to t plus delta t and it will be x of t plus delta t minus x of t divided by t plus delta t minus t which is equal to x of delta sorry t plus delta t minus x of t divided by delta t. Now we are talking about people with mathematical background who know what limits are obviously and in this particular case obviously my continuation of my definition is that the instantaneous speed at moment t is equal to the limit of that thing. So s of t is equal to limit of that thing as delta t goes to zero. Now whenever I have a limit as delta t goes to zero now what is this? Well this is the definition of the first derivative of function x of t and the derivative I just marked with this prime symbol and that's the answer what is an instantaneous speed at moment t. An instantaneous speed at moment t by definition is equal to the first derivative of the co-ordinate function x of t by t by time at this moment t. So that's the definition. So the definition of the instantaneous speed is derivative of the co-ordinate function of time. And that's why I said that you really have to know what's the derivative and how it's defined etc. Alright so we basically have finished with one-dimensional case whenever we are talking about a particular object moving along the straight line which is an x-axis. Now you remember that in the lecture related to space I suggested that we not only can view a position as a point but also as a vector which originates from the origin of the co-ordinate towards that point where the object is. And the vector considerations are more convenient because then you can talk about displacement from one point to another also as vector and it is a vector because any displacement obviously has two characteristics it has a length and this has a direction right which is a vector. So before going any further into three-dimensional space let me just talk about the vector character of this. It's also a vector if you just think about it. Now it's obviously the movement is obviously within the same straight line within the x-axis but it also has a direction because if my function is such that incrementing of time increments the distance from zero which means our function is increasing as the time the distance from zero is increasing as the time increasing then it's positive and the direction of this movement will be at that particular moment obviously would be positive. Now if my direction is let's say towards this direction of the x-axis but that actually means that we are diminishing our numbers then this will be negative and as a result my derivative will be negative. So I don't have a lot of directions but at least I have two positive and negative directions so if my movement is along the straight line I'm also talking about vector character of the speed because it can be either towards the positive direction of the x-axis or negative there are only two directions so the sign of this actually is characteristic of the direction. Now we are ready to go to three-dimensional case. In three-dimensional case we basically have exactly the same situation just repeated for each axis separately so we don't have this we have certain movement in three dimensions so we have point A here at moment t which has coordinates x of t, y of t, z of t and then it moves somehow to point B. Point B is the position at moment t plus delta t right? y of t plus delta t and z of t plus delta t. Now we are interested in instantaneous speed at point at time point t which means right at the position where our object was at point A right? So how can we do this? Well we have to make this delta t as small as possible so my B actually is closer and closer to point A. Now how can I measure both the magnitude and direction of the speed at point A? Well it's exactly the same thing let's just measure it one coordinate by another so each coordinate is measured separately and I can do exactly the same thing I'm using the vectors now so this is a vector into the A this is a vector into the B right? So let's say this is A and this is B. I will take the difference between these vectors and I will try to squeeze my delta t as much as possible so the difference between these two vectors obviously is also a vector right? So the vector will be with coordinates x of t plus delta t minus x of t y of t plus delta t minus y of t that's the difference between two vectors between vector A and vector B right? And z of t plus delta t minus z of t. So this is my vector of displacement from point A to point B. Now as my delta t goes to zero this vector is obviously changing. Now how can I characterize the change we are talking about average speed of change right? So I have to divide by delta t that's my average speed from t to t plus delta t and if I will take the limit of this as delta t goes to zero I will have a resulting vector which is the derivative of x derivative of y and derivative of z and this is I call the speed at moment t a vector right? Exactly the same as in one dimensional case. So it's the definition of instantaneous speed at moment t and at point A in this particular case. And now about the words actually. In physics this vector is called not a speed but velocity and that's the real scientific name for it. I didn't use it before just I just didn't want to go into more complications than necessary. So let's just forget about the word speed I was using before it's really called velocity. So velocity is a vector so what is the speed? Well the speed is the magnitude of this vector. Now do you remember what the magnitude of the vector is? Well if the vector is given by its coordinates like in this particular case. So this is velocity and I will use now V with the bar on the top. Now the speed is absolute value of this velocity vector. It's magnitude which is actually a square root of sum of the squares of the coordinates right? Square plus y square plus z square. Y? Well that's a piece of warren theorem right? If you have some vector what it's what's its length what's its magnitude? Well first we calculate its projection on the xy and this is y and this is x and this is x so this is square root of x square plus y square right? And then if you have this right triangle and this is z it will be correspondingly x square plus y square plus z square that's simple thing. So this is the speed and this is the velocity. Velocity is a vector, speed is a scalar. It's the magnitude of this vector. Well that's it for today. That's all I wanted to talk about. I have introduced the concept of speed and velocity for a moving object with x of t, y of t and z of t coordinates of this of the motion of this particular object. Now what's really very important is the following. If you remember when I was talking about these coordinate functions before I was talking about their continuity. Why? Because we don't want these scientific jumps in no time at all from one point to another which doesn't happen in the real life right? Now in this case I think that if you wanted to know the speed and velocity of the object which is moving in our three-dimensional space, we need a little bit more requirement for these functions. We need differentiability because we have to take the derivative. So differentiability is something which we will probably assume always. We completely or almost completely exclude any kind of movement when the speed for instance is instantaneously changing or the position first goes very very slowly and then boom it goes very very very fast speed forward. So we are avoiding in our course we will avoid situations when these functions are not really smooth in terms of differentiability. And that concludes my lecture for today. Thank you very much and good luck.