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Same rate with different units


Video transcript

So I have a car here, and let's say that in 3 hours, this car is able to travel 150 kilometers. So what I want to think about in this video is, what are some reasonable ways to express the rate at which this car traveled for those 3 hours? And I encourage you to pause this video and think about it for yourself. One, you could calculate the rate. But also think about different units that you could use to express that rate, and which ones would be useful-- which ones what would be reasonable, and which ones would be unreasonable. So let's just remind ourselves what rate even is. So you could think about distance as being equal to rate times time. Or you could imagine if you divide both sides by time, you could imagine that distance divided by time is equal to rate. So they've given us a distance, they've given us a time. So we could just divide the distance divided by the time to figure out the rates. And I'm going to keep the units here. Because it's really important to recognize that the units, to some degree, they can also be manipulated algebraically. Now they aren't variables, but they follow the same rules, I guess I should say, as a variable would. So for example, if I said, look rate is distance divided by time. So I could say that my rate in this situation is going to be 150 kilometers divided by 3 hours. So if we just look at the numeric part of this, what's 150 divided by 3? Well that's going to be 50. So this is going to be equal to 50. But we can keep the units the way that they are, right over here. This is 50 kilometers per hour. This is what I meant by saying look, we're dividing this quantity, expressed in kilometers, by this quantity, expressed in hours. We can divide the numeric part, 150 divided by 3. But then we could leave at the units in that relationship that they were before. So you can kind of algebraically keep them that way. And you'll see in a second we're going to manipulate them a lot more, using what's often called the dimensional analysis. But anyway, this is a reasonable way to express a rate. 50 kilometers per hour. I can imagine this. I can imagine that in 1 hour, you're going to go 50 kilometers. Let's think about other ways that we could represent that. So 50 kilometers per hour-- and this is where we're really going to do some dimensional analysis with our units. So 50 kilometers per hour. Let's say we want to express it in terms of kilometers per second. So how could we write 50 kilometers per hour, in terms of kilometers per second? Well it's always good, actually, as a first approximation, to just think about it. If you went this far in an hour, then the number of kilometers you go in a second, is that going to be less, or more? Well a second's a much, much shorter period of time. There's 3,600 seconds in an hour. So you're going to go 1/3,600 of this distance. But let's think about how we would actually work out with the units. Well, we want to get rid of this hours in the denominator. And the plural, obviously the grammar doesn't hold up with the algebra, but this could be hour or hours. So we could think about well, 1 hour-- I'll write an hour in the numerator that's going to cancel with this hour in the denominator. But we want it in terms of seconds. So 1 hour is equal to how many seconds? Well 1 hour is equal to 3,600 seconds. This is what I meant by saying that using dimensional analysis, which is what I'm doing right now, we can essentially manipulate these units, as we would traditionally do with a variable. So we have hours divided by hours. And so when we do the multiplication, we can multiply the numeric parts. So we have 50 times 1, divided by 3,600. Let me write that. 50 times 1 over 3,600. And then our units left are kilometers per second. Or I could say seconds. So we can play around with the plural and singular parts of it, but I'll just write it as kilometers per second. And so this is 50/3,600. And this fits our intuition. In a second, you're going to go 1/3,600 as far as you would go in an hour. But let's actually think about what this is equal to. 50/3,600-- so this is going to be the same thing, as-- Let me just simplify it over here. So 50/3,600 is the same thing as 5/360, which is the same thing as-- let me write it this way-- 10/720. And I did that way because that makes it clear that that's the same thing as 1/72. So you could write this as, you're going, this is equal to 1/72 of a kilometer per second. Now I would claim that this is not so reasonable of units for this example right over here. 1/72 of a kilometer every second? That doesn't help me too much. I guess I'll know that in 72 seconds I will have gone a kilometer. But this is something that's kind of strange for me to conceptualize. If I wanted to get my calculator out, 1/72, 1 divided by 72-- if someone said, hey, I'm going 0.0139 kilometers in a second, that doesn't seem to make a lot of conceptual sense to me. So I would say that this, right over here, is a very reasonable way to express our rate. This one seems like more of an unreasonable way. But we could salvage this. Because we're going 1/72 of a kilometer per second. Now this is a small number, but how could we make it much larger? Well, what if we thought in terms of, not kilometers per second, but if we thought in terms of meters per second. A kilometer's 1,000 meters. So if we think about this in terms of meters per second, we're going to get a larger number here, in fact larger by a factor of 1,000. So let's think about that. Let's try to convert this kilometers to meters. So how would we do that? Well, once again, if we have kilometers in the denominator this kilometer will cancel with that kilometer. And we want a meter in the numerator. So we want to think about how many meters are there per kilometer. Well there's 1,000 meters per kilometer. Kilometers cancel out, and we are left with 1,000 times 1-- I'll just write that as 1,000-- over 72. And now we're left with in the numerator, we're left with the unit meters per second. And I know I keep writing second in different ways. Oftentimes actually you'll see people write second like that. So actually let me just go with that. So s is second, is seconds, is sec, just like that. So is this fairly reasonable? Well actually, this feels pretty good. Let's get our calculator out and figure out what that is. So 1,000 divided by 72 gives us 13, if I round, that's about 13.9. So this is approximately equal to 13.9 meters per second. Which I can visualize. I can imagine how far 13.9 meters is, and of doing that distance in one second. So this actually also seems like a reasonable way to express the rate. So I could say hey, this thing's going 50 kilometers per hour. I can imagine it going roughly 13.9 meters per second. So this is reasonable as well. But to say it's going 1/72 of a kilometer per second doesn't really seem to make sense. And also, if I try to think about it in terms of meters per hour, that also would be strange. Actually, I encourage you to calculate it. Try to convert this right over here to meters per hour. Then we would say well, we could use the same thing here. That's going to be 1,000 meters for every 1 kilometer. Kilometers cancel out. And I'm going to be left with 50 times 1,000 is 50,000 meters per hour. So I have trouble imagining-- well, obviously if I convert to kilometers in my head I could imagine it-- but this is kind of a crazy large number. 50,000 meters per hour. So at least in my eyes, using kilometers per hour to describe this rate seems useful. Describing this rate as meters per second seems useful. But describing it as kilometers per second, or meters per hour, seem a little bit unusual.