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Solving Relative Velocity Problems in 1D

Patrick Ford
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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