 Hi, I'm Zor. Welcome to Unisor Education. Let's solve a few problems in kinematics, especially related to the frame of references. Now, this lecture is part of the Physics 14 course offered on Unisor.com. I suggest you to read the notes for this lecture on the Unisor.com and basically try to solve these problems yourself. Now, the notes have answers, so you can check yourself. But I will offer these problems with solutions. So, after you have spent some time solving these problems successfully or unsuccessfully, it doesn't matter. Well, then you can watch this lecture and check it against your own opinion. Alright, first problem. Let's say you have a bullet train goes 230 kilometers per hour in one direction and you have a regular train which goes 130 kilometers per hour in another direction. Here you have a passenger and this particular train has 30 cars 20 meters each. Considering the length of the car plus attachment to connect to another car, right? So, basically they have the total length would be 30 times 2600 meters total length of the train. Now, in this train there is a passenger who is looking in this direction perpendicularly to the track. Track is considered to be straight line parallel to two tracks are parallel to each other. The question is how long will it take for this passenger to see the train passing by? Alright, now we know the speed of each one of these trains relative to the ground. Whenever we are talking about one passenger here looking at this train, we are talking about a different reference frame, reference frame related to this passenger, right? Now, considering these are opposite directions, so one train goes into let's say positive direction, another is negative and obviously the relative velocity of this train relative to this passenger would be in this direction and the speed would be a sum of these, 360 kilometers per hour, right? So, the passenger will see himself as the origin of its own frame of reference and this train passing by with a speed of 360 kilometers per hour. Well, considering the length is 600 meters, all we have to do is to convert this thing into meters per second, let's say and then we will know basically the seconds which takes for this train to pass by. Well, 360 kilometers per hour is 360,000 meters divided by how many seconds are in an hour, which is 3,600 and the result would be 36, 100 meters per second. So, that's the speed and with the speed 100 meters per second, my 600 meters train will pass in 600 divided by 100. Lengths divided by speed, the time would be 6 seconds. So, that's the answer. Next. Next, we have a river which goes this way with unknown speed, X. Now, we have a boat which goes down the river stream in 9 hours, but whenever it goes back from B to A, it spends more time, 11 hours. Obviously, because in this case the flow of the water helps and in this prevents, right? So, if, now my problem is to find out what's the speed of the water flow. Now, obviously I'm considering that the river is straight and the flow is constant, so everything is in nice condition. Now, let's assume that this particular boat has certain engine which allows it to go into a steady water, let's say in a lake, with certain speed, its own self, so to speak, speed. Okay, so self speed would be S. This is the speed in the standing water. Only the engine of the boat works. Now, if it goes this way, by the way, I didn't specify 198 kilometers is the distance from A to B. Alright, now if this is the speed in the standing water and whenever it goes downstream, obviously the flow helps, so every hour let's say the boat itself goes along this distance S in one hour and then the flow of the river helps by X during this same hour, so the total combined distance covered in one hour would be S plus X, so that's the speed. S plus X kilometers per hour, let's say. So, that's the time it goes down the river. So, I know that 198 distance divided by speed, which is a combined speed of these boats, boats is equal to 9 hours. So, this is kilometers and we assume this is kilometers per hour, right? Now, whenever it goes in opposite direction, every hour the engine itself pulls it by S kilometers, and the flow of river pushes back by X, so the total speed would be S minus X and that would be 11 hours. All I have to do is to solve this system of two equations with two unknowns. Alright, so let's do it. 198 is equal to 9X plus 9S and 198 is equal to 11S minus 11X. So, we need X, right? So, to do it really easy we can multiply this by 11 and this by 9 and subtract and S would cancel out, right? So, 198 times 11 is equal to 11, it's 99X plus 99S. And here I multiply by 9, it's equal to again 99S minus 99X. Now, if I will subtract them, my 99S would cancel out. Here I will have 99X minus minus 99X, it's 198X. And here I will have 198 times 11 minus 198 times 9, it will be 198 times 2. Now I can cancel 198 and I have X is equal to 2, that's my speed of the flow of the river. Now, basically all these problems are kind of easy, it's all distance divided by time or distance divided to get the speed or distance divided by speed to get the time or speed multiplied by time is equal to distance. I mean, you can, distance is equal to speed multiplied by time so this formula can be basically used in all the different kinds whatever is necessary to get the answer. Now next, next is I have an escalator, escalator which goes up with certain speed. Well, let's consider the speed is X and it has certain lengths. So, if speed is number of steps per second because it's very conveniently in this particular case to measure the speed and the distance in number of steps. So, this is number of steps and this is number of steps let's say per second it moves. So, if I want to know the time when escalator pulls the person from the bottom to the top it would be N divided by X, right? Number of steps divided by number of steps per second it moves. But our person is not just stanging still it's going up and it's counting the steps. So, it goes up with certain speed let's say S and it counts the steps once and it got 24 steps. So, as he is going up with his own speed S it's like the boat in the standing water this is the passenger on the subway let's say which basically goes from one step to another regardless whether step is moving or not moving he has his own speed that's his own speed of moving up. Now, then he decided to double his speed and he counted 32 steps. Question is what's N? What's the number of steps here? Now, we can think about well we have too many unknowns here we have N, X and S and we have only basically two equations. Well, let's make these two equations maybe we can solve them. So, first equation so he as he goes up with certain speed of his own and he counts 24. Now, that's a good equation if I know the time. Well, this is the first. Now, what is the time he goes up counting these steps? Well, I have the distance which is N and I have a speed. Speed is his own speed plus the speed of the escalator. Right? Now, next equation the same distance as he covered with double his speed plus the speed of the escalator. That's the time now his speed is 2S now it's doubled and he counted 32. Okay, so now we have these two equations with three unknowns now we have to solve it for N. It's actually not as bad because we can always reduce it to two unknowns how by reducing by S here, what do we see here? S here, S here would be 1 and S X divided by S right? And here we will also reduce it by S so you will have S count cancels out here and this X divided by S. So now as you see we have only two unknowns one is N and other is X divided by S. So that's easier now we have two equations with two unknowns. Well, let's call X divided by S Y let's say we don't need it but not interested in this we are interested in N right? So we will have N is equal to 24 times 1 plus Y 24 times 1 plus Y and 2N is equal to 64 plus 32 Y. Okay, so this is let's open it up 24 plus 24 Y. All right, fine. So now we can very easily determine N because we have two independent equations. Now how to get rid of Y in this particular case? Well, very easily we will multiply this by 3 no by 4 by 4 and this by 3 we will have 96 Y and we will have 96 Y and then we will subtract. So we will have 4N equals to 96 plus 96 Y. 6N is equal to 192 plus 96 Y. Subtract from this subtract this this will go out this will be 96 192 minus 96 and this will be 2N. So the N is equal to 48. So that's the answer 48 steps our escalator has. All right, next. Next we'll talk about average speed. Now average speed is basically distance divided by time, right? All right, now we have a situation. We have two cars. One car has decided to go from point A to point B. Half a distance, half a distance with a speed U and another half a distance with a speed V. Question is what's the average speed? Well, if you think it's U plus V divided by 2 you're wrong. It's not. Again, total distance divided by total time. Now what is the total distance we know? Now what's the total time? Well, we divide distance in two halves, right? So we have one half of the distance with a speed U. So that's the time spent in to cover the half a distance. Then I have another piece of time which has the same distance G over 2 with a speed V and that's the time for the second one. So this is the total time. And if I will divide the total distance over this, I will get, what will I get? Obviously, D cancels out. I will have 2 divided by 1 over U plus 1 over V. So this is the formula. I mean, you can simplify it differently. You can put 2UV divided by U plus V. However, this is something which is called, this is something which is called harmonic mean of two numbers. So if you have two numbers, U and V, harmonic means is you invert each one of them, you average them divided by 2 and then you invert again. And that's what you will get. That's harmonic. Now, another car decided to do it differently. Another car decided to have the entire time of the trip divided by 2. So he spent certain amount of time with the speed U and then exactly the same amount of time with the speed V. Okay, let's say the total time is T, which is very unknown. But if you will multiply U by this half a time and then V by half a time, that will be my total distance. Now, the total time is still T. So if I divide my distance to the total time T, T cancels out and I will have U plus V over 2. And this is arithmetic average, kind of things which we are used to have. So arithmetic average is if you divide time in half. Harmonic average is if you divide distance in half. Okay, that's it. Next. Okay, you have a car and this is the wheel of the car. Now, the car wheel makes n revolutions per second. The diameter of the car is D. What's the speed of the car? Okay. Whenever you are talking about rotating very conveniently to deal not with the regular velocity, which is like car velocity which we would like to find out, but with angular velocity of the rotating wheel in this particular case. Now, rotating wheel has certain angular velocity and it's defined by this. Now, angular velocity is measured in radians per second. Now, if it's n revolution, each revolution is 2 pi radians. So it's 2 pi n is equal to omega. Omega is angular velocity, radians per second. Okay, now, if you have an angle, let's say you have time t. If time t, then my wheel turns by r times omega times t, right? If omega is angular speed, then omega times time would give me whatever the angle my wheel turned during the time t, right? So it turned, let's say this is angle phi. So angle phi is equal to omega times t. Now, during this time, my center of the wheel moved from this position to another position. Now, what is this distance? It's exactly the same distance as this, right? So the arc. So I need the length of the arc. So during this time, my wheel turned on the angle of w t. Now, my r times this angle is the length of the arc. So this is basically how far my center of the wheel, which means the entire car actually, moved during the time t. So the speed is this. So that's the speed of the car. Now, if I know the diameter, obviously, I have to say that my speed of the car is equal to r omega, which is d over 2 times 2 pi n, which is equal to pi dn. So that's the answer. So this is my linear speed of the car if the wheel makes n revolutions per second, and d is the diameter of the wheel. And by the way, in some cases, you can imagine that this device which gives you the speed of the car, it might actually calculate the speed exactly using this formula, because it knows the diameter of the tires, and it obviously can count how many revolutions a second the wheel actually makes. And by using this formula, you calculate the speed. Okay, and the last one, I have a funicular. Now, funicular is a device which is used to move people up the mountain. So you have on the top of the mountain, you have a wheel, and then you have two cars. Now, whenever a wheel moves this way, this car goes up and this one goes down. Whenever the wheel moves that way, this one will go down and this up, and that's how people are moving back and forth, back and forth, all right? Now, what's known about this is the diameter of the wheel is g, and a passenger who is sitting in one car looks at the speed another car passes by, and there is a speed v, which this passenger noticed. Now, my question is what's the angular velocity of the wheel which moves the whole system? All right, let's just think about it. If my relative speed of one car relative to another car is v, and we know that both cars are moving with the same speed in different directions, it means that the absolute speed of each car is v over 2. This one into this direction and this one into this direction, also v over 2. Only then we have the passenger here thinking that this car is actually moving past it with a speed v. So we have a linear speed. Now, this linear speed is actually the same as linear speed of the wheel. So we know that this every point on this wheel is moving whenever this guy sees another car, this thing is moving with certain speed which is linear speed. The point on the rim moves exactly the same way as this particular, as the linear speed, absolute speed of this car. Now, again, linear speed is equal to r omega, where omega is angular speed and the radius is r. So right now, well, we know the diameter, so it's diameter over 2 omega. And what do we know about this linear speed? Well, d omega divided by 2 is equal to linear speed of the car, from which omega is equal to v over d. That's the answer. So if you have this car passing the passenger here with the relative speed v, and the diameter is d, then the angular speed of the wheel is omega, so basically these are six relatively simple problems, which are only part of the whole set of problems which I would like to spend some time. Maybe there will be another set of problems. And that's my plans for the next lecture. So I do suggest you to go through these problems again just by yourself. Go to the website Unisor.com. Go to Physics for Teens, Mechanics, Kinematics and Frames of Reference. And there you will have this problem one chapter. And try to solve these problems yourself again, by yourself. And see if you have the same answers which I have provided in the notes for this lecture. All right, that's it for today. Thank you very much and good luck.