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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 have once in physics class ...
... but they are also common problems you would see in your everyday life ...
... because you're trying to figure out how far you have gone ...
... or how fast you are going or how long it might take you to get to some place.
So first I have: If Shantanu was able to travel 5 kilometer north in 1 hour ...
... in his car, what was his average velocity?
So let's just review a little bit about what we know about vectors and scalars.
So they are giving us that he was able to travel 5 kilometer to the north
So they gave us a magnitude ...
... that's the 5 kilometer ... that's the size of how far he moved.
And they also gave a direction.
So he moved a distance of 5 km. Distance is the scalar ...
... but if they 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 is many ways 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 that we put little arrows on top of vector quantities.
Normally they are bolded ... you can have a type face ...
... and they have an arrow on the top of it.
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.
The arrow isn't a 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 postition.
So you can say its displacement ...
... and the letter for displacement is "s" ...
... and that its a vector quantity.
So that is displacement.
And you might be wondering: ...
... why dont 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 thats why we use "s" for displacement.
If someone has a better explanation of that ...
... feel free to comment on this video ...
... and 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 words, so you 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 some time.
So these two ...
... you could call them formulas ...
... or you can call them definitions ...
... although I would think they are 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 directions ...
... so you're dealing with vector quantities.
This is where you're not so conscientious about directon ...
... and so you use distance, which is scalar ...
... and you use rate or speed, which is scalar.
And 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 ...
... 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 ... is ...
... his displacement was 5 kilometers to the north ...
So his displacement ... The displacement was 5 kilometers ...
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.
Sometimes ...
... - 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 expicitly means "change in".
It looks like a very fancy mathematics when you see that.
But a triangle in front of something litterally 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 over 1 ...
... let me just write it 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 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 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 ....
... we don't care about the direction now ...
... it's 5 kilometers ...
... and he does it in 1 hour.
His change in time is 1 hour.
So this is same thing as 5 kilometers per hour.
So once again:
We're only giving the magnitude here.
This is a scalar quantitiy.
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 the term of meters per second.
Here I gave you kilometers ...
... or kilometers depending on how you want to pronounce it ...
... kilometers per hour.
What if someone want 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 does not to hurt to work on that right now.
So if we wanted to do this to meters per second.
How would we do it?
Well, a 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 will 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 5 ...
... and then the only unit you have in ...
... oh, I should say 5000 ...
... so you have 5 times 1000 ....
... so this is ... so let me write this ...
... 5 times ... I'll do it in the same color ....
... 5 times 1000, so I just multiplied the numbers.
When you multiply something you can switch around the order.
Multiplication is communitativ ...
... I always have trouble pronouncing that in 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 5000 meters per hour.
And you might say:
Hey Sal, you know, ... I could have ... you know, ...
... I know that 5 kilometers is the same thing as 5000 meters.
I could do that in my head.
And you probably could.
But this canceling out dimension ...
... or what it is often called "dimensional analysis" right here ...
... can get useful once you started doing really, really complicated things ...
... with less intuitiv units than something like this.
But you should always do an intuitiv gut check, right here.
You know that, if you do 5 kilometers in an hour that's a ton of meters, right?
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 intuitiv gut check.
If something is traveling a certian amount an hour, ...
... it should travel a much smaller amount in a second ...
... or, you know, 1/3600 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 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 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 seconds per minute ....
... times 60 minutes per second ...
... the minutes canc... oh Sorry! ... times 60 minutes per hour, I should say ...
... the minutes cancel out ....
...60 times 60 is 3600 ...
...seconds per hour.
Or if you flip it ...
... so you can say this is 3600 seconds for every 1 hour.
Or if you flip them, you would get ...
... 1 over 3600 hours - or hour - per second.
Or hours per second, depending on how you want to do it.
So one hour is the same thing as 3600 seconds.
And so now this hour cancels out with that hour ...
... and then you multiply or appropiatly divide the numbers right here.
And you get ... this is equal to 5000 over 3600 ...
... 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 can do this by hand but just because this video is already getting a little bit long ...
... let me get my trusty calculator out ...
... just for the sake of time ...
... 5000 divided by 3600 ...
... which should be really the same thing as 50 divided by 36 ...
... that is 1.3 ... i'll just round it over here ... 1.39.
This is equal to 1.39 meters per second.
So Shantanu was traveling quite slow in his car.
And well, we knew that just by looking at this.
5 kilometers per hour, that's pretty much ... just like the car roll pretty slowly.
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first of all I appreciate your work I have a question: what if we have to times and to speeds with different direction? how can we get the average velocity? thanks..
opps!! you forgot the direction of the avg. velocity :) V avg. = 1.39 m/s north
nice
we should say displacement as 's' because distance is known as 'd'
why can't we say 1.39 as 1.4 m/s when calculated? and round off and 1 m/s?