Mr.p: Good morning.
Today we are going to learn about
vectors and scalars.
Here we go!
Bo: Hey guys.
Billy: Hey, Bo.
Bobby: Hi, Bo.
♫ (lyrics) Flipping Physics ♫
mr.p: As an introduction to vectors and scalars,
I want to walk through a rather simple example.
In order to do so,
I need something that moves at
a rather slow, constant velocity.
And I actually had a really hard time
finding something that would do this,
so I made one.
I made a race car.
I made Slow Velocity Racer.
(engine revs)
(engine screeches)
(clicking)
Bobby: Really?
Bo: Uhh.
mr.p: Now we need to know how fast
Slow Velocity Racer moves.
So I had her race right next to a meter stick.
Remind me Billy, how long is a meter stick?
Billy: Uh, one meter?
mr.p: Yep, somehow people sometimes forget
that a meter stick is one meter long,
which is also 1000 millimeters.
And you can see that Slow Velocity Racer
takes 23.8 seconds to travel 1000 millimeters.
So we can use the equation
speed equals distance over time
to figure out the magnitude of her velocity.
Now we can substitute in 1000 millimeters
for her distance traveled and the time it took her
to travel that distance which was 23.8 seconds.
And what do we get for the speed of our race car?
Billy: Uh, 42.0168 or 42.0 millimeters per second.
mr.p: Actually, for the purposes of today's examples
we're just gonna use two significant digits.
So now we know that Slow Velocity Racer
can move at approximately 42 millimeters per second.
We could put Slow Velocity Racer
on the track at the top that is moving
at 42 millimeters per second to the left,
or we could put her on the track on the bottom
that is moving at 42 millimeters per second
to the right.
Considering Slow Velocity Racer wants to move
as quickly as possible,
Bobby, on which track should she race?
Bobby: In order to move as quickly as possible,
she should race on the bottom track.
mr.p: Yes Bobby, and why should she choose
the bottom track?
Bobby: Because then she'll be moving at
84 millimeters per second.
mr.p: Absolutely correct, Bobby.
The speed of the car plus the speed of the track,
or 42 millimeters per second
plus 42 millimeters per second,
is equal to 84 millimeters per second.
In other words, if Slow Velocity Racer
races on the bottom track,
she will be moving at 84 millimeters per second.
Bo, how fast will she be moving
if she races between the two tracks?
Bo: 42 millimeters per second.
mr.p: Correct, Bo.
The speed of the race car plus the speed of the track
or 42 millimeters per second plus zero,
is equal to 42 millimeters per second.
In other words, she'll just be moving
at 42 millimeters per second
if she races between the two tracks.
And Billy, how fast will she be moving
if she races on the top track?
Billy: Well, she actually wouldn't end up moving.
mr.p: Correct.
In that case we get her total speed
by subtracting the speed of the track
from the speed of the race car.
In other words, 42 millimeters per second
minus 42 millimeters per second
which is equal to zero.
So if she races on the top track,
she actually isn't moving at all.
And I do recommend that everybody try this
at some point on an escalator or moving walkway.
It is quite fun.
(jazz piano)
Bobby: What?
Bo: That's weird.
mr.p: What you've all done here instinctively
is called tip-to-tail vector addition.
It's also sometimes called tail-to-tip vector addition
but it's the same thing.
So we have a representative vector right here
and you can see this is the tail of the vector,
and this is the tip of the vector.
So for our bottom track,
we had both objects,
the track and the race car were moving to the right,
so both vectors are gonna be drawn to the right.
And you can see we've replaced the tip
of one vector on the tail of the other vector,
and the net vector,
or the resultant vector is twice as long,
or 84 millimeters per second.
For the top track,
the track itself was moving to the left
so we have the vector representation for the track
moving to the left.
The arrow goes to the left.
But the race car was moving to the right
therefore that vector is going to be to the right.
And you can see we put the tip of one vector
on the tail of the other vector,
and these two end up
cancelling one another out completely.
Therefore the net speed for the top track
of Slow Velocity Racer is equal to zero.
That is the basic concept
of tip-to-tail vector addition
and we're gonna spend more time talking about
that in our next lesson.
Bo: You know, that's actually pretty obvious.
mr.p: That is exactly my point, Bo.
Vector addition is at its root level,
quite simple and something you probably
already inherently know how to do.
I will of course in the future make it
more difficult and confusing,
however please remember that it is
based off of this intuitive concept.
Bobby: Why would you wanna make it more difficult?
Billy: And confusing.
Bo: Because that's his job.
mr.p: Yeah well, we do have to make progress
in this class.
So let's define vectors.
A vector is a quantity having both
magnitude and direction.
Bo, remind me what is the definition of magnitude?
Bo: It's like the stuff.
Billy: What?
Bo: The stuff in the quantity.
Billy: That's not helpful.
Bobby: It's like the amount of the quantity.
Billy: That doesn't really help me, either.
mr.p: OK, you know what?
We'll come back to that question.
I think let's do this first,
I think it will help.
A scalar is a quantity that has magnitude
only it does not have direction.
And we've actually already in this class
have been working with vectors and scalars,
but we just haven't identified these variables
as vectors or scalars yet.
And I think if we do so now,
it will actually help us to define the term magnitude.
So class, let's start with vectors.
Could you please give me some examples
of variables that are vectors?
Billy: Displacement, velocity.
Bobby: Acceleration.
Bo: Force.
Bobby: Torque.
Bo: Momentum.
mr.p: Yes.
Displacement, velocity, acceleration
are all vectors that we've worked with,
and force, torque and momentum
are actually vectors we haven't yet worked with
but we will.
And now could I please have some variables
that are scalars?
Bobby: Distance.
Bo: Speed.
Bobby: Time.
Bo: Volume.
Billy: Density and money.
Bo: Money?
mr.p: Yeah, distance, speed, time, volume,
density and money.
Those are all scalars.
As far as money, think of it this way:
does five dollars east make any sense?
Bo: No.
mr.p: Then money is a scalar.
Basically anything where adding direction
doesn't make sense is a scalar.
For example, the number of hairs on my head.
(jazz piano)
The number of beads in this box.
The number of markers in my hands.
The number of pieces of plastic
in this Hoberman sphere.
The number of crayons in this box.
The number of pages in this book
written by my cousin.
The number of pentagons on this ball.
The number of feathers in this bag.
The number of swords my daughter owns.
The number of air molecules in this ball.
(cymbals)
The number of peanuts I can get
to land in my mouth.
(peanuts hitting floor)
Bo: I think we get it.
mr.p: Good, let's get back to this term magnitude
and let's use displacement as an example
to help us define magnitude.
If you recall from a previous lesson,
if I start right here...
And I end right here,
my displacement is approximately two meters east.
What then is the magnitude of my displacement?
Bo?
Bo: Oh, it's just two meters without the direction.
mr.p: Correct.
Bobby: So it is the amount of the quantity.
mr.p: Pretty much, Bobby.
The magnitude is the numerical amount
of the quantity.
And the velocity of Slow Velocity Racer itself
was 42 millimeters per second east.
Therefore Billy, what was the magnitude
of the velocity of our race car?
Billy: Ah!
The magnitude of the velocity is just
the numerical amount of the velocity
without the direction.
Ah, so just 42 millimeters per second.
mr.p: Yes.
Now be aware that scalars can be negative.
Initially some people think
that scalars can't be negative,
however please realize that scalars
can be negative.
An example to prove my point,
we already know money is a scalar,
therefore can you have negative money?
Billy: Absolutely!
Bo owes me the two dollars he borrowed from me
in third grade because he wanted to buy a ring
for Barbie because he thought
it would impress her. (laughs)
Bobby: Oh, he was wrong.
Bo: I can't believe you guys can remember that.
Bobby: Ah, that was good.
mr.p: So yes, Bo has negative money.
Quick review: tip-to-tail vector addition
is something that you probably already
inherently know how to do.
We illustrated that using the Slow Velocity Racer.
A vector is a quantity
that has both magnitude and direction.
A scalar is a quantity that has magnitude only
and does not have direction.
Thank you very much for learning with me today.
I enjoyed learning with you.
(laughs) One landed in my shirt, too. (coughs)