- [Voiceover] 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
seen in 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
too, 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 them,
but this tells you that not only do I care about the value
of this thing, but I care about, or I care about
the size of this thing,
I also care about its direction,
and that's what the arrow,
the arrow isn't necessarily its direction,
it just tells you that it's 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 its 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 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 derivitive operator,
and that's so that the D's don't get confused,
and that's why we use S for displacement.
If someone has a better explanation of that,
feel free 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 you don't
get confused with displacement,
or maybe I'll 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 could 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 you use rate or speed which is scalar.
You 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 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,
his velocity is,
his displacement was five kilometers to the north,
so his displacement,
the displacement was five kilometers,
five kilometers, I'll write just a big capital,
well let me just write it out.
Five kilometers north
over 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 something
literally means "change in".
Change in.
So this is change in time.
So he goes five kilometers north,
and it took him one hour,
so the change in time was one hour.
So let me write that over here.
So over one hour.
So this is equal to, if you just look at the
numerical part of it, it is five over one.
Let me just write it out,
five over one kilometers.
You can treat the units the same way you would treat
the quantities in a fraction.
Five over one kilometers,
kilometers per
hour.
Kilometers per hour,
and then, to the north.
To the north.
Or, you could say this is the same thing as
five kilometers per hour north.
So, this is
five kilometers per hour,
per hour, to the north.
To the north.
So that's his average velocity,
five kilometers per hour, and you have to be careful,
you have to say to the north if you want velocity.
If someone just said five kilometers per hour,
they're giving you a speed or a 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 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,
is five kilometers,
and he does it in one hour.
His change in time is one hour,
so this is the same thing as
five kilometers
per hour.
So once again, we're only giving the magnitude here.
This is a scalar quantity.
If you want the vector, 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 kilo-meters
or kilometers, depending on how you want to pronounce it,
kilometers per hour.
What if someone wanted it in meters per second,
or what if I just wanted to understand how many
meters he travels in a second?
And there just becomes a unit conversion problem
and I figure it doesn't hurt to work on that right now.
So if we wanted to do this to meters per second,
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 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 1,000 meters for every one kilometers.
1,000 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,
I should say 5,000,
so you have five times 1,000,
so let me write this,
five times, I'll do it in the same color,
five times 1,000,
so I just multiply the numbers.
When you multiply something, you can switch around
the order.
Multiplication is commutative.
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,
this is equal to 5,000 meters per hour.
And you might say, hey Sal, I could have,
I know that five kilometers is the same thing
as 5,000 meters.
I could do that in my head,
and you probably could, but this
cancelling out dimension,
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,
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 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, 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 the 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, oh 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,
so you could say this is 3,600 seconds for every one hour,
or if you flip 'em, you would get
one over 3,600 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 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,
this is equal to
1.39 meters per second.
1.39 meters per,
meters per second.
So Shantanu was traveling quite slow in his car.
Well, we knew that just by looking at this.
Five kilometers per hour,
that's pretty much saying the car rolled pretty slowly.