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Main content
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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 once 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 someplace so first I have if Shantanu was able to travel five kilometers north in one 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 five kilometers to the north so they gave us a magnitude that's the five kilometers that's the size of how far he moved and they also give a direction so he moved a distance of five kilometers distance is the scalar but if you give the direction to you get the displacement so this right here is a vector quantity he was displaced five kilometers to the north and he did it in one 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 it but this tells you that not only do I care about the value of this thing I care about or I care about the size of this thing I also care about its direction that's what the arrow the hour isn't necessary erection just tells you that it is a vector quantity so the velocity of something is its change in its change in position including the direction of its change in position so you could say it's displacement 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 didn't 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 DS don't get confused and that's why we use s for displacement if someone has a better explanation of that feel free to feel free to to comment on this video and then I'll add another video explaining that better explanation so velocity is your displacement over time is your displacement over time if I wanted to write an analogous thing for the scalar quantities I could write that speed I could write and I'll write out the word so we don't get confused with displacement or maybe I will 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 the distance that you travel over over some time so these two you could call them formulas or you can 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 use rate or speed which is scalar here use the displacement and you use velocity now with that out of the way let's figure out what his average velocity was and this keyword 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 or 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 his velocity is his displacement was five kilometers to the north so his displacement the displacement was five kilometers five kilometers all right just a big capital well let me just write it out five kilometers north over his the amount of time it took him over the amount of time and let me make it clear this is change in time change in time sometimes this is also 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 very fancy mathematics when you see that but a triangle in front of thing something literally means change in change in so this is change in time so it goes five kilometers north and it took him one 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 if you just look at the numerical part of it it is 5 over 1 let me just write out 5 over 1 kilometers and you can treat the units the same way you would treat the quantities in a fraction 5 over 1 kilometers Coulomb eaters per hour kilometers per hour and then to the north 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 per hour to the north to the north to the north so that's his that's his average velocity 5 kilometers per hour now 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 does your average is 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 it's five kilometers and he does it in one hour his change in time is one one hour so this is the same thing as five five kilometres per hour so once again we're only giving the magnitude here this is a scalar quantity if you want the vector you get you have to do the north as well now you might be saying hey you know in the previous video we talked about things in terms of meters per second here I gave you kilometres or kilometers depending on how you want to pronounce it km/h what if someone wanted in m/s or what if I just wanted to understand how many how many meters he travels in a second and there just becomes a unit conversion problem and I I figure it doesn't hurt to to work on that right now so if we wanted to do this to m/s 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 five kilometers per hour and we want to we want to convert it to 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 1000 meters for every one kilometer 1000 meters for every one 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 five five and then the only unit you have in the five I should say five thousand so you have five times a thousand so this is let me write this five times I'll doing the same color five times 1000 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 an associative and then in the unit's in the numerator you have meters and in the denominator you have hours meters per hour and so this is this is equal to this is equal to five thousand five thousand meters per hour and you might say hey Sal you know I could have you know I know that five kilometers is the same thing as five thousand meters I could do that in my head and you probably could but this cancelling out de mint or what's often called dimensional analysis right here can get useful once you start doing really really complicated things with less intuitive units than something like this but this should you should always do an intuitive gut check right here you know that if you do five kilometers in an hour that's a ton of meters right so it should get you should get a larger number if you're talking about meters per hour and now when we want to go to second 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 you know 136 hundredths 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 m/s but let's actually do it with the dimensional analysis so we want to cancel out the hours and we want to left be left with seconds in the denominator so the best way to cancel this hours in the denominators by having hours in the new-new in the denominator in the numerator so you have hours per hours per seconds so how many hours are there per second or another way to think about it one hour think about the larger unit one hour is how many seconds well you have 60 60 seconds per minute times 60 minutes per second the minutes cancel sorry times 60 minutes per hour I should say times 60 minutes per hour the minutes cancel out 60 times 60 is 3,600 seconds per hour 3,600 seconds per hour or if you flip it if you so you could say this is 3600 seconds for every one hour or if you flip them you would get one over 3,600 hours or hour per second or hours per second depending how you want to do it so one hour is the same thing as 3,600 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 over 3,600 meters 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 by well we could I could do this by hand but just because this video is already getting a little bit long let me get my trusty calculator out if I get my trusty calculator out just for the sake of time five thousand 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 1.3 9 so one point this is equal to 1.3 9 meters per second 1.3 9 meters let me meet what 1.3 9 meters per meters per second so Shantanu was travelling 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 roll pretty pretty slowly