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# Intro to dimensional analysis

CCSS.Math:

## Video transcript

we've seen multiple times in our life that distance couldn't be viewed as rate times time and what I want to do in this video is use this fairly simple formula right over here this fairly simple equation to understand that units can really be viewed as algebraic objects that you can kind of treat them like variables as we work through a formula or equation which could be really really helpful to make sure that our that we're getting results in units that actually make sense so for example if someone were to give you a rate if they were to say a rate of let's say five meters per second and they were to give you a time a time of 10 seconds then we can pretty in a pretty straightforward way apply this formula we say well distance is equal to our rate five meters per second times our time times our time which is ten seconds and what's neat here is we can treat the units as I just said like algebraic constructs kind of like variables so this would be equal to well multiplication doesn't matter what order we multiply in so we can change the order this is the same thing as 5 times 10 five times 10 times meters per second times meters per second times seconds times seconds and if we were to treat our units as these kind of algebraic objects so excited look we have seconds divided by seconds or you have a seconds in the denominator multiplied by seconds in the numerator those are going to cancel out and 5 times 10 of course is 5 times 10 of course is 50 so we would be left with 50 and the units that we're left with are the meters 50 meters so that's pretty neat the unit's worked out when we treated the unit's out like algebraic objects they worked out so that our end units for distance were in meters which is a unit of distance now you're saying okay that's that's cute and everything but this seems like a little bit of too much overhead to worry about when I'm just doing a simple formula like this but what I want to show you is that even with this simple with a simple formula like distance is equal to rate times time what I just did could actually be quite useful and this thing that I'm doing is actually called dimensional analysis and it's useful for something as simple as distance equals rate times time but as you go into physics and chemistry and engineering you'll see much much much more I would say hairy formulas and when you do the dimensional analysis it makes sure that your that the math is working out right and make sure that you're getting the right units but even with this let's try a slightly more complicated example let's say that our rate is let's say let's keep our rate at 5 meters per second but let's say that someone gave us the time instead of giving it in seconds they give it in hours so they say the time is equal to 1 hour so now let's try to apply this formula so we're going to get distance is equal to 5 meters per second 5 meters per second times time which is 1 hour times 1 hour well what's that going to give us well the 5 times the 1 so we multiply the 5 times the 1 that's just going to give us 5 but then we have to remember we have to cheat the unit's algebraically we're going to do our dimensional analysis so it's 5 so we have meters per second times hours times hours or you could say 5 meter hours per second well this doesn't look like a this isn't a set of units that we know that that makes sense to us this doesn't feel like our traditional units of distance so we want to cancel this out in some way and it might jump out of you oh if we can get rid of this hours if we can express it in terms of seconds then that would cancel here and we'd be left with just the meters which is a unit of distance that we're familiar with so how do we do that well we want to multiply this thing by something that has hours in the denominator and seconds in the numerator times essentially seconds per hour well how many seconds are there per hour well there are 3600 let me do the sunna I'll do it in this color there are 3,600 seconds per hour or you could even say that there are 3600 seconds for every one hour so when you now when you multiply these hours will cancel with these hours these seconds will cancel with those seconds and we are left with we are left with five times 3,600 what is that that's five times 3000 would be fifteen thousand five times 600 is another three thousand so that is it's equal to eighteen thousand and the only units that were left with we just have the meters there eighteen oh eighteen thousand eighteen thousand eighteen thousand meters but and so this is we're done we've now expressed our distance in terms of units that we recognize if you go five meters per second for one hour you will go 18,000 meters but let's just use our little dimensional analysis muscles a little bit more what if what if we didn't want the answer in meters but we wanted the answer in kilometers what could we do well we could take that 18,000 meters 18,000 meters and if we could multiply it by something that has meters in the denominator meters in the denominator and kilometers in the numerator then these meters would cancel out and we'd be left with the kilometers so what can we multiply it so we're not really changing the value well we want to multiply it by essentially one so we want to write equivalent things in the numerator in the denominator so one kilometer is equivalent to equivalent to 1000 meters so one way to think about it we're just multiplying this thing by one one kilometer over a thousand meters well one kilometer is a thousand meters so this thing is equivalent to one but what's neat is when you multiply we have meters cancelling with meters and so you're left with 18,000 divided by 1,000 is equal to 18 and then the only units were left with is the kilometers and we are done we have re expressed our distance instead of in meters in terms of kilometers