4. 2D Kinematics
Intro to Relative Velocity
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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're 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 you're solving for. But I'm gonna show you, there's really just a couple of rules and steps that we can follow, so we can 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 c which we write in the notation vac is equal to the velocity of a relative to b plus the velocity of b relative to c. 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 that we're gonna do down here, we're gonna be talking about a car and a truck. 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 subscripts of the terms on the right side of the equation, so these are the subscripts that are the closest to each each other are gonna be the same. So notice how I have b and b. And then the outer subscripts on the right side, meaning the subscripts that are farthest away from each other on the on the right side of the equation are going to equal the subscripts of the term on the left side. So a and c is we have a and c on the left side as well. So really these are 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 meters per second East relative to the ground, and then we've got a truck that's ahead of you also moving at 60 meters per second. And what's the velocity of the truck relative to your car? So the first kind the first step that you're gonna do in solving all these problems is 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. We're 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 ground over here and we've also got the truck. So that's the first step. So now we identify all the objects in the references. The next we have to do is write each of the given velocities with the notation. Remember that the subscript notation means you're always gonna measure or you're always gonna write the velocity of one thing relative to another. So when it says the velocity of the car, what it's actually saying is the velocity of the car relative to the ground. So this isn't just v c, it's v c g, and in the similar way this is VTG. So what do we ask 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 be VT measured by or relative to c. That's actually what we're asked for in the problem. So VTC is going to be our target variable and we know what VCG is, that's just 45. And we know VTG is equal to 60. So now that brings us to the 3rd step, which is we have to write the relative velocity equation according to the rules for inner and outer subscripts 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 what we'd have to do is we're gonna have to set up the terms so that the outer subscripts, basically the ones that are farthest away from each other on the right hand side, are the same as the subscripts on the left side. So what I need is I need a term that has v as the first or t as the first subscript, and then I need the other term to have c as the last subscript, right? So that's gonna obey that rule. So if you look to my terms here, I've got one term that has a T in the front and that's VT g. Right? So that's the 3rd 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 inner subscripts to be the same. So that's the way to set up the equation. So if we look through our variables here, this v t c is what I'm looking for. This v t g is the 60 meters per second and I need VGC. So I've already got these. I've already got this variable covered here. Now if we notice, this variable that we're given, the 45, isn't vgc. It's VCG. It's the same letters, but they're just flipped backwards. So how do we deal with this? Well, if you're ever given a velocity with the correct subscripts, but just in the opposite order, then you can flip or reverse the subscripts. And whenever you do that, whenever you flip the subscripts, then 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 v b a. So what that means in our example here is that if we want v g c and we're 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 or sorry, now that we have this variable and 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 v t c is equal to 15 meters 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 meters 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 meters 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 meters per second faster than you. That's really what that 15 meters per second means. Alright guys, that's it for this one. Let me know if you have any questions.
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