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Current time:0:00Total duration:9:55

Same rate with different units


Video transcript

so I have a car here and let's say that in three hours this car is able to travel 150 150 kilometers so what I want to think about in this video is what is what are some reasonable ways to express the rate at which this car traveled for those three 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 would be reasonable and which ones would be unreasonable so let's just remind ourselves what rate even is so you could imagine you could think about you could imagine 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 the distance divided by time distance divided by time is equal to rate 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 unit's 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 say look rate is distance divided by time so I could say that my rate my rate in this situation is going to be 150 kilometers 150 kilometers divided by 3 hours 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 unit's the way that they are right over here this is 50 kilometers km/h 50 km/h our 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 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 can we're going to manipulate them a lot more do it using what's often called 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 one 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 kilometers per hour let's say we want to express it in terms of I don't know 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 is the first approximation to just think about it if you went this far in an hour how far in the number of kilometers you go in a second is that going to be less or more well seconds a much much shorter period of time there's 3,600 seconds in an hour so you're going to go 136 hundredths of this distance but let's think about how it would actually work out with the unit's well we want to get rid of this hours in the denominator and so and you know the plural obviously the grammar doesn't hold up with the algebra but this could be hour or hours so we could think about what 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 3,600 seconds and this is what I meant by saying that we can think about we can mention analysis which is what I'm doing right now we can essentially manipulate we can manipulate these units as we would traditionally do with variable so we have hours / hours and so when we take and then when we do the multiplication we can multiply the numeric parts so we have 50 times 1 divided by 3600 let me write that 50 times 150 times 1 over 3600 and then our units left are kilometers 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's kilometers per second and so this is 50 over 3600 and this is this fits our intuition in a second you're going to go 1 3600 as far as you would go in an hour but let's actually think about what this is equal to 50 over 300 3600 so this is going to be the same thing as this is e let me just simplify it over here so 50 over 3600 is the same thing as 5 over 360 which is the same thing as let me write it this way 10 over 720 and I did that way because that makes it clear that that's the same thing as 1 over 72 so you could write this as you're going this is equal to this is equal to 170 second 170 second of a kilometer of a kilometer per second kilometer per second now I would claim that this is not so reasonable of units for this example right over here 170 second 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 one over 70 to 1 divided by 72 you know someone said hey I'm going 0.013 9 kilometers in a second that doesn't seem to make a lot of conceptual sense to me so I would say I would say that this right over here is a very reasonable way 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 172nd of a kilometer per second now this is a small number but how can 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 is a thousand meters so if we think about this in terms of meters per second we're going to get a much much we're going to get a larger number here in fact a larger number larger by a factor of a thousand 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 if we have a kilometer in the denominator this kilometer will cancel with that kilometer and we want a metre 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 1000 times one I'll just write that as a thousand over 72 over 72 and now we're left with in the numerator we're left with the unit meters meters per second meters per second and I'll 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 especially if we let's get our calculator around and figure out what that is so 1,000 divided by 72 gives us 13 if I round that's about thirteen point nine so this is approximately equal to thirteen point nine meters per second thirteen point nine meters per second which I can visualize I can have add I can imagine how far thirteen point nine meters is and of doing that distance in one second so this actually also seems this 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 thirteen point nine roughly thirteen point nine meters second so this is reasonable as well but to say it's going 172nd of a kilometer per second doesn't really seem to make sense and also if I tried to think about in terms of meters per hour that also would be strange actually I encourage you to calculate it try to try to convert this right over here to meters per hour then we would say well that's going to be we could use the same thing here that's going to be 1,000 meters for every one kilometer kilometers cancel out and I'm going to be left with 50 times 1,000 is 50,000 50,000 meters per hour meters per hour so I have trouble imagining well obviously if I convert it 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 saying using kilometers per hour describe this rate seems useful to use describing this rate as meters per second seems useful but describing it as kilometers per second or meters per hour seem a little bit a little bit unusual