 Hi, I'm Zor. Welcome to Unisor Education. Today we will talk about relative motion of one object relative to another object. In all the cases, we will be talking about uniform motion, which means that velocity is a constant vector and acceleration is equal to zero. Now, this lecture is part of the course of Physics 14's presented on Unisor.com. Together with the video lecture, the site contains numerous notes. Actually, every lecture has notes. It has also exams for those people who are willing to challenge themselves. The site is completely free, no advertisement. Also, on this site you can find the course called Mass 14's, which is definitely a prerequisite to Physics 14. You really have to know mass, especially vectors and calculus. Let's talk about relative motion. We are talking about motion of one object relative to another. Let's consider that first of all we have certain original frame of reference. Let's say for our purposes it's the frame of reference which is fixed on the surface and we consider this frame of reference to be inertial. Now, we have two objects. We have object, let's say A and object B. And they are moving with uniform velocity, so there is some kind of a vector here and vector there. Now, what we are interested in is motion of one object relative to another, let's say B relative to A. Now, for this purpose, we are considering another system of coordinate which has an origin where my A object is located, let's say at time t is equal to zero. And then this frame of reference is moving with axes parallel to original, let's say the ground-related frame of reference, initial frame of reference. So it's moving towards the vector of speed, vector of velocity actually of this particular object. So this frame of reference is always related to the moving object A. Now, obviously at any moment of time, position of A is described by some kind of function, vector function of the time. That's the position of object A, this is index A. And the position of object B, so this is this vector and this is this vector. This is PB of t and this is PA of t. So PB is position of the B object. Now, obviously these are vectors which are changing and that's why I have a t parameter here. Now, the first derivative of these are velocity vectors. And velocity vectors we have agreed are constant, the uniform motion of both of them, right? So this particular derivative is equal to VA and I do not put dependence on time because it's a constant vector and this is VB also constant vector. Now, what do we know from the vector algebra? If I will take the vector from A to B. Now, A, as you remember, is an origin of the new system of coordinates, new frame of reference which is moving with A with axis parallel to the original frame. So let's call this vector PAB and this is obviously again dependent on t, on the time. So this is vector from A to B and now knowing the rules of addition of vectors I can say that vector from P, from original system 00000 point to A plus the vector from A to B is equal to the vector from the origin to B, right? So I can always say that PA of t plus PAB of t is equal to P of B of t. So to get to the point B from the origin of the original system I can always go to A and from A to B. That's what basically it says. It's a simple vector arithmetic. Now, if I will differentiate this thing I will get VA plus VAB is equal to VB. This is a constant and this is a constant. That's why this is also supposed to be a constant. So the vector which connects from A to B as a position if I will take the first derivative of this vector it will be a constant velocity of B. Now, let's just think about it relative to what? I mean obviously position vector from A to B is basically position of object B in the system of co-ordinate related to point A moving with A and correspondingly this equation with the speed. This is actually a speed or rather velocity vector of object B relative to object A. And these are exactly what we want to find out. We want to find out what's the location of B in the system of co-ordinate related with associated with object, with moving object A, right? So this is exactly how we can find out. So from the first one we can see that PAB is always equal to PB minus PA and these are all functions of time. And the velocity is correspondingly difference between velocities. So let's just concentrate on this and this. This is actually the reflection of Galileo's relativity principle. Now this relativity principle basically tells us how to calculate relative speed of one object relative to another. Well in this particular case we are talking about uniform motion and most problems which we will be dealing with probably will be of that kind in kinematics. So that gives you basically the tools how to calculate relative position if you know absolute positions and speeds or velocities rather. And vice versa if you know for instance one of them and relative to another then we can find the original. So this is basically the meaning of the Galileo's relative motion principle. Now what's interesting is in Einstein's theory of relativity this type of addition or subtraction of velocities is not exactly the same. I mean there is certain nuance which actually is related to shortening of the distance and the time as the speed is increasing etc. We are not talking about this I'm just telling you that this is a classical approach which goes back to Newton. Now it doesn't mean that this is wrong. No to a certain degree of precision this is absolutely correct. But if you go into the speeds which are relatively comparable to the speed of light and the precision of your instruments is really very very high you will see that this is not exactly the law of nature and the law of nature has certain I would say corrections. Very small corrections mind you but nevertheless they are. So this is the Galileo's relativity principle. Now let's just apply this particular principle let me put this on the top and I will apply it to a few practical problems. So let me put it again. V a plus V a b is equal to V b or V a b is equal to V b minus V a. These are our laws of adding the velocities and this is the vectors obviously all. Now the first problem is very easy and I think it's like a must problem for anybody who is studying kinematics. You have two trains going let's say in opposite direction along parallel straight line rails one let's say goes to the speed of 100 kilometers per hour to the opposite direction 90 kilometers per hour. Now what does it mean? What did I not say and I assume? Well I assume that there is a system of coordinates related to the ground and assumed to be inertial system and these are speeds relative to this particular system of coordinates. Now obviously it's very convenient to choose this system of coordinates in such a way that this is my x axis let's say the direction is towards this train which goes with 100 kilometers per hour but the line the x axis actually is coinciding with the lines of the motion of the train. Then my y and z coordinates are equal to zero and I'm not going to mention them at all. So we are actually right now we are talking about one dimensional system not three dimensional but one dimensional and this is the dimension. Now so this is my v a 100 kilometers per hour and what's my vb? Well not exactly 90 it's minus 90 right because the direction is this way right so it must be negative it's against the x increasing. Now what if I want to know how this particular train moves relative to this one? Now what's the practical situation? Well for instance there is a passenger in this particular train looking at the window and he is basically measuring the speed of the passing train. So what's the speed of this train relative to this passenger? Again what I did not really say but I assumed I assumed that this particular passenger is the origin of a new system of coordinates new frame of reference which is moving this way with this speed and that's why this is v a and I'm measuring speed of this object which has absolute speed relative to the Earth's 90 kilometers per hour in that direction but I would like to measure the speed of this thing in the system of coordinates related to this passenger. So what do I need? I need to determine v a b that's what I need to determine. Well again it's the difference between this is b by the way it's minus 90 minus 100 it's minus 190 kilometers per hour. Now first of all why is it minus? Well because relative to this passenger this train goes against the x axis increasing right? So that's why it's minus and as far as the 190 again it's kind of obvious that whenever trains are moving against each other then the speed must actually be combined the speed of this guy approaching this train is the combination of both. So it's kind of intuitively obvious but I would like it not to be based on the intuition but based on consideration of the frames of reference. One frame of reference is relative to the Earth's another frame of reference is relative to the passenger who writes in this particular train and both frames of reference are inertial because this is a uniform motion right? And that's how you can find out relative speed of this train relative to this one. Now what if the train goes this way? Well then the speed will be 90 this arithmetic would be minus 10. Now why is it minus? Because again this guy is slower they are moving towards the same direction but this guy is slower this guy is faster which means this guy will go this way and this will always be behind right? So and it goes behind more and more so it's like increasing this distance in this particular direction and that's why it's minus. But it's only 10 because in this particular case the velocities are subtracted because we are moving in the same direction one is a little slower but nevertheless there is a difference and since there is a difference this is. Now if this guy was faster than this then it's also the difference in absolute value but the sign would be positive because relative to this one this who moves let's say 120 kilometers per hour then it would be moving relative to this one with the speed only 20. Same thing as you are driving the car and you're driving the car let's say with 90 kilometers per hour or 60 miles per hour whatever and then somebody else is moving a little bit faster than you then it moves really forward right? But if it moves slower than you it seems to me that this guy relative to you moving backwards right? More and more. Okay next problem. Next I have a few other examples which basically are exactly the same as far as their meaning is but nevertheless it's good illustration kind of thing. So one illustration is let's say we have a platform which is moving at certain speed forward let's say three meters per second. Now on this platform there is a person who is moving again towards the same direction and his speed is let's say one meter per second. Now I would like to know how fast this person moves relative to another person who is standing still on the ground. What do I have in this case? Well again obviously I presume that the X axis of my original ground related system of coordinate is towards this movement so this is my X coordinate. Now my platform would be my A object and the person would be B object. So I know that VA is equal to three miles per hour per second three meters per second. Now whenever I'm saying that the person is moving on the platform with this speed one meter per second it means that this is relative to the platform that's what it means and from here I can do my absolute speed of this particular person relative to my original ground related system of coordinates as in this case sum 3 plus 1 which is 4 meters per second. So again my first original system of coordinate is related to the earth to the ground. Now my moving object A is the platform and this is another system of coordinate another frame of reference and basically I'm moving relative to it that's why it's VAB is equal to. Now and the third example would be about rivers and boats. Now let's assume we have a river with straight banks and it goes with certain the flow goes at three miles three meters per second okay that's the flow of river and we are assuming that this is a uniform motion all the water uniformly moves along the straight line along the obviously X axis that's our now let's assume we also have a boat now the boat moves in the water it can move in the standing water like in the lake or it can move in the river but in any case whatever we are saying about the speed of the of the boat this speed is relative to the water where it is located so let's say we have a boat and its speed is 10 meters per second in the water now if it moves down the river the flow of river helps the boat because the whole water mass is moving and the boat is moving within that mass of water with the speed in which case we will have what VA is equal to three meters per second and this is relative to the water right so this is VAB is equal to 10 so if I would like to know how fast the boat moves relative to the bank of the river let's say I have two points A and B and I would like to know how long it will take to get from A to B so I know the distance but this distance is along the ground so I have to I have to calculate everything along the ground so obviously in this case the speed of the boat relative to the ground VB would be 3 plus 10 which is 13 now if my boat moves in this direction from B to A obviously I have to subtract then now why do I have to subtract well because my VAB would be negative in this case so VAB in this case would be negative 10 meters per second so I still move the water this way that's 3 and then I have to instead of plus 10 I have to do minus 10 because the speed is now negative since it's against the X growth and that will be minus 7 again minus because the boat will move against the growing of the X coordinate and then and 7 is its absolute value okay basically that's all different problems I would like to use as examples but the most important is this type of addition or subtraction whenever so it's either VA plus VAB is equal to VB or VAB is equal to VB minus VA same thing in both cases so these are a few examples which illustrate this particular thing okay that's it for today thank you very much and good luck