Hey guys. So you may come across some problems which have to calculate the velocity of one object that moves relative to or with respect to something else. So I want to introduce you in this video, the idea of relative motion or relative velocity And really, what we're gonna see is that it kind of just comes down to simple addition and subtraction of velocities. So let's check it out. What's this whole idea about? Well, whenever we measure velocity, we are measuring it relative to some reference points, which we call a frame of reference. So, for example, you're standing on the side of a road. You have a fancy speed gun or a radar gun that cops used to measure how fast you're driving and you're measuring the you know, the speed of cars passing by. So your reference frame, your frame of reference is the earth. That's the thing that you're measuring the velocities with respect, Thio. And so, in fact, most of the time and problems, your frame of reference is gonna be the ground or the earth unless they otherwise tell you. So let's just jump right into an example so I can show you how this works. So imagine that you're an observer, right? And you're standing on the outside of a moving platform. Are moving walkway like the kind that you've seen airports. You have one of these fancy speed guns with you, and you're gonna measure the velocity of three different people on this moving walkway. So person a standing still Person B's walking to the right. The person sees walking to the left. So what would our speed gun measure their velocities to be? Well, the whole idea here is that if you look at T equals zero, all of these people are kind of lined up along the same position. But then one second later, the walkway has kind of moved everybody to the rights. But because some of them are walking forward, some of them are standing still and some of them are moving backwards. Then they're gonna be a different distances or different positions one second later. So what happens with person? A. What happens is the walkway is just gonna move them some distance over, you know, some amount of time in one second. And so because person A is just standing still on the moving walkway that is moving them along at 3 m per second. And if we pointed our speed gun at person A. It would also show them moving at 3 m per second. So here's why. Here's a way you can visualize this. So imagine my pen here is you the observer. My hand is the moving walkway and person A is my red pen. So as the moving walkway basically moves this person along at 3 m per second, they are together and because they are together, they have the same velocity. In fact, whenever you have a stationary or not moving object that is inside or within or on top of another moving object and they basically have the same velocity, they share the same velocity. And so, in other words, another way we can say that is that the moving walkway and the person A has the same relative velocity to all other reference frames. So to you, the observer, the person A and the moving walkway have the same velocity relative to any other reference frame in another way of also putting that is that they have the same velocity in their velocities relative to each other. is equal to zero. So let's move on to person B now. So person B is moving along in 2 m per seconds. So what happens with them? Well, in the same amount of time, the same one second person be is because Because they're walking forwards. We're gonna travel higher distance or a greater distance in the same amount of time. Now we know that the velocity is equal to the change in position or the displacement over time. So if they're traveling a farther distance in the same amount of time, then that means that the velocity must be higher. So if we put it, our speed gonna be, what would we measure? Well, because B is moving at 2 m per second. Sorry, because they're walking forwards at 2 m per second. Then there, 2 m per second in addition to the 3 m per second on the walkway is moving them along. Then when we point our speed gun at them, they're not gonna be moving at three. But they're gonna be moving at three plus two, which is 5 m per second. And so what happened now happens with person, see? Well, person sees moving 2 m per second in the opposite direction. So in the same amount of time, they're going to cover a less distance than the other two persons A and B, and so their velocity must be lower than five and three. And so the idea is basically just the reverse. Their velocity relative to the moving walkway picks up a negative sign. And now what happens is if you were to point your speed gun at person, see, it wouldn't measure five. It wouldn't measure three. It would measure three minus two because the velocities point in opposite directions and they kind of cancel each other out, and you would see the moving at 1 m per second. So in, you know, it's a summarize. Everything really relative velocity is just the addition or subtraction of velocities. And it all depends on which directions that those relative velocities Aaron that's it for this one. Guys, let me know if you have any questions
Solving Relative Velocity Problems in 1D
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Hey, guys. So in the last video, we were introduced to the idea of relative motion or relative velocity. And what we saw is that to solve relative velocity problems, you really just adding or subtracting velocities in different reference frames. Now you're gonna need to know exactly how to solve relative velocity problems in one dimension. So I'm gonna show you how to do that in this video, and it really just comes down to setting up what's called the relative velocity equation. Now setting up this equation is the hardest part of solving these kinds of problems because it's tough to figure out exactly which variable use solving for But I'm gonna show you. There's really just a couple of rules and steps so we can follow so we could always set up the equation the right way and get the right answer. Let's check it out So this relative velocity equation is written in the following formats. So the velocity of a relative to see which we right in the notation V A. C is equal to the velocity of a relative to be plus the velocity of be relative to see now different books and professors are gonna write this using different letters. They might even have it, like, you know, with some negative signs in a slightly different order. But this is the order that we're gonna use in our videos because we think it's the easiest for you to set up and understand. So as I mentioned before, we might actually see, you know, different letters, and it really kind of just depends on the problem. So, for example, in the in the example we're gonna do down here, we're gonna be talking about a car and a truck so you might use the letters C and T instead of A and B. So instead of memorizing the letters, what I want you to do is memorize the rules because those are much more important. And the rules are actually fairly straightforward. So your inner sub scripts of the terms on the right side of the equation. So these are the sub scripts that are the closest to each other are gonna be the same. So notice how have b and B and then the outer sub scripts on the right side, meaning the sub scripts that are farthest away from each other on the on the right side of the equation are going to equal the sub scripts of the term on the left side. So A and C is we have a and see on the left side as well. So, really, these air, just the two rules. If you set this equation up using these two rules, you're always gonna get the right answer. So let's see how this works using this example down here. So we're in a car moving at 45 m per second east relative to the ground, and then you've got a truck that's ahead of you also moving at 60 m per second. And what's the velocity of the truck relative to your car? So the first kind the first step you're gonna do in solving all these problems, you're gonna draw a diagram and you're going to identify all of the objects and references. So, for example, we've got the road over here like this were driving. So you're driving in your car. So we've got one object here. That's the car, and the velocity of your car is equal to 45. Now. We also have the truck that's ahead of you and it's also traveling to the East. V truck is equal to 60. But there's a third thing here because remember that both of these velocities here are measured relative to the ground. So that's another reference. It's not really an object. It's more of a reference. So you've got the ground over here and we've also got the truck, So that's the first step. So now we have identified all the objects in the references. The next we have to dio is right each of the given velocities with subscript notation. Remember that the subscript notation means you always gonna measure or you're always gonna right the velocity of one thing relative to another. So when it says the velocity of the car, what is actually saying is that the velocity of the car relative to the ground? So this isn't just VC, it's V C G. And in the similar way, this is vtg So what we asked for in the problem. So we're asked for the velocity of the truck relative to your car. So if we use the same notation, it's gonna VT measured by our relative to see that's actually what we're asked for in the problem. So Vtc is going to be our target variable And we know what V c g is. That's just 45 and we know vtg is equal to 60. So now that brings us to the third step, which is we have to write the relative velocity equation according to the rules for inner and outer subscript over here. So we're looking for VTC. So we're gonna write that on the left side of the equation. So VTC is gonna be equal to and now we have to do is gonna have to set up the terms. So the outer sub scripts, basically the ones that are farthest away from each other on the right hand side are the same as the subscript on the left side. So what I need is I need a term that has V as the first T is the first subscript and then I need the other term to have see as the last subscript right, so that's gonna obey that rule. So if you looked at my terms here, I've got one term that has a t in the front, and that's V T G. Right? So that's the third object or third reference that we need is the ground and then the other term here is has to be a G in order for the inter sub scripts to be the same. So that's the way to set up the equation. So if you look through our variables here, this is vtc is what I'm looking for. This vtg is the 60 m per second and I need v g c. So I've already got these. I've already got this variable covered here. Now if we notice this variable that were given the 45 isn't bgc. It's V c g. It's the same letters but they're just flipped backwards. So how do we deal with this? Well, if you ever given a velocity with the correct sub scripts but just in the opposite order, then you can flip or reverse the sub scripts. And whenever you do that, whenever you flip the sub scripts and you're basically just flipping the sign of the number from negative to positive or vice versa. So in General v. A B is gonna be the negative of the B A. So what that means in our example here is that if we want v G C and were given V c G. Then we can always just reverse the order of the subscript. So v g c is just gonna be negative. 45. And now that we have this letter over here are sorry, now that we have this variable in this number now we can actually plug it into our equation and solve. So our vtc is just gonna be the 60 the truck relative to the ground, plus negative 45. And so what happens is vtc is equal to 15 m per second. So that's the answer. That's the velocity of the truck relative to you. So let's summarize and actually put yourself in this scenario here. So you're driving along in your car at 45 m per second. That's relative to the ground, which means if somebody were on the ground with their speed gun measuring your velocity, it would read 45 m per second. But if you had your own speed gun and you were measuring the truck ahead of you, it wouldn't read 60. It would read 15 m per second faster than you. That's really what that 15 m per second means. Alright, guys, that's it for this one. Let me know if you have any questions
A boat on a river is traveling from a pier to a point 500 m upstream (against the river’s current). The current flows at 4 m/s. If the boat makes the trip in 250 s, what is the speed of the boat relative to the water?
Flying in the Wind
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Hey, guys, let's take a look at this problem here. So we got these two cities without any wind. This airliner could make the trip between them and some amount of time, but then the wind starts blowing, so there's 225 kilometer wind that blows from west to east. We're trying to figure out how long the trip is from A to B is gonna take now that the wind is blowing. So we're gonna solve this because we have some relative motions and relative to the air stuff like that, using our relative velocity steps. But there is a little bit of a before and after case that's going on here, so we're gonna draw these diagrams. But I'm actually gonna draw to one for the before situation and one for me after, So let's get to it so that before situation that happens is in this first part of the problem where we're told that without any wind, the trip is, you know, the airliner could make the trip in some amount of time, so the before situation is with no wind, and then the after is gonna be when the wind starts blowing. So this is gonna be some from the wind. So let's check this out here. So we got these two cities, right? So we've got a and B, we know be lies directly west of our east of a on DSO. The Delta X, we're told, is equal to 27 75 kilometers and we know the time that it takes to travel between these two cities is 3.3 hours. So what happens is if the plane is just traveling through the air like this is the plane like this, then we can actually just straight up figure out what the velocity is just by doing Delta X over Delta T. So it's just 27 75/3 270.3 hours. And so the velocity of the plane through the air is 8.89 point 41 is 841 kilometers per hour. So that's the speed of the plane, right? Without any wind. But now what happens is the wind is gonna start blowing. And so now there's gonna be a couple of velocities to consider. There's gonna be some relative velocities, so let's draw the same exact diagram. So we've got again City A and city be and we know the Delta X over Delta T or Sorry, we know the Delta X. We know that this displacement is 27 75. What's different now is because there's some wind that's potentially helping or changing the velocity of the plane. This Delta T is gonna be different. So that's what we're actually asked for in the problem. How long does this trip now take with some wind blowing. So I'm gonna call this Delta T initial. That's the 3.3 hours. But what I'm really looking for is Delta T final. What is the change? What is the amount of time that it takes now that the wind is blowing? So this is really what I'm interested in. Okay, So what's going on here? So now we're just gonna go ahead and stick to the steps for solving relative motion problems. So you've got this diagram here. Now we just identify the objects in the references. So we've got the plane that is traveling, So we've got this plane over here That's one of our objects were told that the plane in the second part of the problem has the same speed relative to the air, which means that the air is actually itself a reference or an object. So we're gonna call the air the object here, and we're told that the air also has a speed. The velocity of that wind that's blowing from west to east is 225. And so, you know, it's positive because it points in the same direction, right? So velocity of the air. So the next question is, what is the velocity of that air measured relative to? Well, so are third reference is gonna be the ground. So that's the third reference here. The grounds we got the plane is traveling through the air, but the air is traveling relative to the ground. So those there are three objects and references. So now we're gonna write each of their velocities with the given subscript notations. So we're told that the velocity of the plane relative to the air is going to be the same as before. So that means that this variable over here was V P. A. And so that's 841. And the air is real, is measured relative to the ground. That's the wind speed or whatever. And so that's V A G. So we've got our two variables here. So you got V EPA, which we know is 841. We've got V A G, which we know is 225. So remember, we're gonna need a third one in order to start writing a relative velocity equations. What's that? Third variable. Well, the only other thing that we can use is the fact that it covers some distance which is on the ground basically the ground distance between A and B in some amount of time. So what this means is that the velocity of the plane to the air and the velocity of the air relative to the ground can come and combine, and they'll produce a velocity of the plane relative to the ground. That's V P. G. So both of these things here combined form V P. G. That's the third variable. So this is actually what we're looking for here, R v p g. Or at least we're not giving it. So think about this, right. If we're trying to figure out the total amount of time that it takes for the airplane to travel this distance here, then we're gonna need to know it's plain to ground speed. So if I can figure out what this V p. G is by using my relative velocity equations, then I can actually just go ahead and solve for Delta T. So I know this is gonna be Delta X over T final. So therefore, my Delta T final, it's going to be Delta X over the velocity of the plane relative to the ground. So again, if I can figure out this V p. G, then I can actually go ahead and figure out what this this time is. So now that brings us to the third step. We're just we're going to set up the relative velocity equation, so v p g, if this is what we're looking for and now we're just gonna stick this to our rules, we want the outer subscript to B, P and G. That means I want the first script subscript of the first term on the on the right to BP, and then I want the last subscript two BG. And then the letter that goes inside of the inside subscript is just gonna be the only letter that we haven't used yet, which is a so this is gonna be a and A. So now the intercept scripts of the same the outer ones of the same as the first term. So now we're just gonna go through and figure out if we have all of our terms if we need to flip anything, stuff like that. So, um bbd that's what I'm looking for. This is gonna be V P. A. This is the 841 plus V a G. So I actually have all of those numbers here, so I could just go ahead and straight up, plug them in, so V p g is just equal to 841 plus the 225. And if you work this how you get 10 66. So if you think about this, the velocity of the plane relative to the ground is just the velocity of the plane relative to the air, plus the velocity that the wind is basically helping it move along the ground. So these things kind of add together to form a faster velocity, right? So It's kind of like if you were running and you know you have wind at your back, that kind of helps you and pushes you along and, you know, going a little faster. So now that we have this velocity here now, I could just plug it back into our equation and solve for Delta T. So a Delta T final is gonna be the same distance 27 75 between the two cities, divided now by 10. 66. And if you work this out, you're gonna get 2.6 hours. So that's how long it takes. That makes some sense. If the wind is helping you along, then whereas before it took you 3.30 hours travel the distance between them. Now it only takes 26 because the wind is kind of helping you push you a little faster. So we're answer. Choice is B and that's it for this one. Guys, let me know if you have any questions.
Additional resources for Intro to Relative Velocity