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Calculating average velocity or speed

Although speed and velocity are often words used interchangeably, in physics, they are distinct concepts. Velocity (v) is a vector quantity that measures displacement (or change in position, Δs) over the change in time (Δt), represented by the equation v = Δs/Δt. Speed (or rate, r) is a scalar quantity that measures the distance traveled (d) over the change in time (Δt), represented by the equation r = d/Δt. Created by Sal Khan.

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Video transcript

Now that we know a little bit about vectors and scalars, let's try to apply what we know about them for some pretty common problems you'd, one, see in a physics class, but they're also common problems you'd see in everyday life, because you're trying to figure out how far you've gone, or how fast you're going, or how long it might take you to get some place. So first I have, if Shantanu was able to travel 5 kilometers north in 1 hour in his car, what was his average velocity? So one, let's just review a little bit about what we know about vectors and scalars. So they're giving us that he was able to travel 5 kilometers to the north. So they gave us a magnitude, that's the 5 kilometers. That's the size of how far he moved. And they also give a direction. So he moved a distance of 5 kilometers. Distance is the scalar. But if you give the direction too, you get the displacement. So this right here is a vector quantity. He was displaced 5 kilometers to the north. And he did it in 1 hour in his car. What was his average velocity? So velocity, and there's many ways that you might see it defined, but velocity, once again, is a vector quantity. And the way that we differentiate between vector and scalar quantities is we put little arrows on top of vector quantities. Normally they are bolded, if you can have a typeface, and they have an arrow on top of them. But this tells you that not only do I care about the value of this thing, or I care about the size of this thing, I also care about its direction. That's what the arrow. The arrow isn't necessarily its direction, it just tells you that it is a vector quantity. So the velocity of something is its change in position, including the direction of its change in position. So you could say its displacement, and the letter for displacement is S. And that is a vector quantity, so that is displacement. And you might be wondering, why don't they use D for displacement? That seems like a much more natural first letter. And my best sense of that is, once you start doing calculus, you start using D for something very different. You use it for the derivative operator, and that's so that the D's don't get confused. And that's why we use S for displacement. If someone has a better explanation of that, feel free to comment on this video, and then I'll add another video explaining that better explanation. So velocity is your displacement over time. If I wanted to write an analogous thing for the scalar quantities, I could write that speed, and I'll write out the word so we don't get confused with displacement. Or maybe I'll write "rate." Rate is another way that sometimes people write speed. So this is the vector version, if you care about direction. If you don't care about direction, you would have your rate. So this is rate, or speed, is equal to the distance that you travel over some time. So these two, you could call them formulas, or you could call them definitions, although I would think that they're pretty intuitive for you. How fast something is going, you say, how far did it go over some period of time. These are essentially saying the same thing. This is when you care about direction, so you're dealing with vector quantities. This is where you're not so conscientious about direction. And so you use distance, which is scalar, and you use rate or speed, which is scalar. Here you use displacement, and you use velocity. Now with that out of the way, let's figure out what his average velocity was. And this key word, average, is interesting. Because it's possible that his velocity was changing over that whole time period. But for the sake of simplicity, we're going to assume that it was kind of a constant velocity. What we are calculating is going to be his average velocity. But don't worry about it, you can just assume that it wasn't changing over that time period. So his velocity is, his displacement was 5 kilometers to the north-- I'll write just a big capital. Well, let me just write it out, 5 kilometers north-- over the amount of time it took him. And let me make it clear. This is change in time. This is also a change in time. Sometimes you'll just see a t written there. Sometimes you'll see someone actually put this little triangle, the character delta, in front of it, which explicitly means "change in." It looks like a very fancy mathematics when you see that, but a triangle in front of something literally means "change in." So this is change in time. So he goes 5 kilometers north, and it took him 1 hour. So the change in time was 1 hour. So let me write that over here. So over 1 hour. So this is equal to, if you just look at the numerical part of it, it is 5/1-- let me just write it out, 5/1-- kilometers, and you can treat the units the same way you would treat the quantities in a fraction. 5/1 kilometers per hour, and then to the north. Or you could say this is the same thing as 5 kilometers per hour north. So this is 5 kilometers per hour to the north. So that's his average velocity, 5 kilometers per hour. And you have to be careful, you have to say "to the north" if you want velocity. If someone just said "5 kilometers per hour," they're giving you a speed, or rate, or a scalar quantity. You have to give the direction for it to be a vector quantity. You could do the same thing if someone just said, what was his average speed over that time? You could have said, well, his average speed, or his rate, would be the distance he travels. The distance, we don't care about the direction now, is 5 kilometers, and he does it in 1 hour. His change in time is 1 hour. So this is the same thing as 5 kilometers per hour. So once again, we're only giving the magnitude here. This is a scalar quantity. If you want the vector, you have to do the north as well. Now, you might be saying, hey, in the previous video, we talked about things in terms of meters per second. Here, I give you kilometers, or "kil-om-eters," depending on how you want to pronounce it, kilometers per hour. What if someone wanted it in meters per second, or what if I just wanted to understand how many meters he travels in a second? And there, it just becomes a unit conversion problem. And I figure it doesn't hurt to work on that right now. So if we wanted to do this to meters per second, how would we do it? Well, the first step is to think about how many meters we are traveling in an hour. So let's take that 5 kilometers per hour, and we want to convert it to meters. So I put meters in the numerator, and I put kilometers in the denominator. And the reason why I do that is because the kilometers are going to cancel out with the kilometers. And how many meters are there per kilometer? Well, there's 1,000 meters for every 1 kilometer. And I set this up right here so that the kilometers cancel out. So these two characters cancel out. And if you multiply, you get 5,000. So you have 5 times 1,000. So let me write this-- I'll do it in the same color-- 5 times 1,000. So I just multiplied the numbers. When you multiply something, you can switch around the order. Multiplication is commutative-- I always have trouble pronouncing that-- and associative. And then in the units, in the numerator, you have meters, and in the denominator, you have hours. Meters per hour. And so this is equal to 5,000 meters per hour. And you might say, hey, Sal, I know that 5 kilometers is the same thing as 5,000 meters. I could do that in my head. And you probably could. But this canceling out dimensions, or what's often called dimensional analysis, can get useful once you start doing really, really complicated things with less intuitive units than something like this. But you should always do an intuitive gut check right here. You know that if you do 5 kilometers in an hour, that's a ton of meters. So you should get a larger number if you're talking about meters per hour. And now when we want to go to seconds, let's do an intuitive gut check. If something is traveling a certain amount in an hour, it should travel a much smaller amount in a second, or 1/3,600 of an hour, because that's how many seconds there are in an hour. So that's your gut check. We should get a smaller number than this when we want to say meters per second. But let's actually do it with the dimensional analysis. So we want to cancel out the hours, and we want to be left with seconds in the denominator. So the best way to cancel this hours in the denominator is by having hours in the numerator. So you have hours per second. So how many hours are there per second? Or another way to think about it, 1 hour, think about the larger unit, 1 hour is how many seconds? Well, you have 60 seconds per minute times 60 minutes per hour. The minutes cancel out. 60 times 60 is 3,600 seconds per hour. So you could say this is 3,600 seconds for every 1 hour, or if you flip them, you would get 1/3,600 hour per second, or hours per second, depending on how you want to do it. So 1 hour is the same thing as 3,600 seconds. And so now this hour cancels out with that hour, and then you multiply, or appropriately divide, the numbers right here. And you get this is equal to 5,000 over 3,600 meters per-- all you have left in the denominator here is second. Meters per second. And if we divide both the numerator and the denominator-- I could do this by hand, but just because this video's already getting a little bit long, let me get my trusty calculator out. I get my trusty calculator out just for the sake of time. 5,000 divided by 3,600, which would be really the same thing as 50 divided by 36, that is 1.3-- I'll just round it over here-- 1.39. So this is equal to 1.39 meters per second. So Shantanu was traveling quite slow in his car. Well, we knew that just by looking at this. 5 kilometers per hour, that's pretty much just letting the car roll pretty slowly.