Today we're going to start Chapter 3, but before we get into
chapter three, let's have one last question about the chapter two material.
Let's go over to this concept test.
This is the same concept test, we had at the end of last time.
I've a truck travelling a constant velocity, 50 kilometers,
an hour, approaches a car stopped at a red
light, the car is approaching stopped at the red
lightfrom the The truck's approaching the car from behind.
When the truck is 100 meters from the back of the
car, the light turns green and the car immediately begins to accelerate.
At 2 meters per second squared.
Which of these graphs represents this situation?
Ok times up, let's talk about this one. So the car starts
from rest, and begins to accelerate. And
I can say the initial position of the car is x equals 0.
>> So the constant acceleration formula for the position of the car is x naught,
which is 0, plus v naught t, which is 0, plus 1 half a t squared.
If I graph this, what's the shape called?
>> Parabola.
>> Parabola, yeah. It's a parabola.
So, so here's the car
On the other hand, the position of the truckt
is the initial position of the truck, x naught
t, plus the velocity of the truck, v naught
t times time, and what's the acceleration of the truck?
0.
So this is it for the truck This is the graph of a straight line, so at t
equals zero, when the car takes off, is the
truck behind the car, or in front of the car?
Behind. Is that positive x, or negative x?
Negative x. So the, the truck starts out here,
and, so here's the truck.
Your job, is to figure out when the
distance between them, x car minus x truck.
Is minimum.
Well I can figure that out if I know the time when they're closest.
They're closest together about here it looks like.
What's special about that closest distance?
The velocities are the same.
The truck, and the car are closest together
when their velocities are equal. That gives you a condition on the time.
Can you see why this is?
Initially, the truck's going faster than the car,
so the truck is gaining on the car.
But after awhile, the car will get going faster than the truck.
And so the car will pull away from the truck.
So first, the distance is closing and then the distance is widening.
That crossover Is when the velocities,
the speeds, are exactly equal.
That homework problem was hard so I wanted to give you
a little start, and also reward you for coming to lecture.
Yeah, so the right answer is b.
So in chapter two we talked about motion in one dimension.
In chapter three we're going to talk about motion in two dimensions.
But before we talk about two dimensional motion,
we have to do a little math interlude.
We have to talk about vectors. And understand some vector math.
A vector is a mathematical object that consists of a size, a
bigness, a magnitude. Plus, a direction.
So far we've encountered two different vector quantities.
Velocity is a vector, the magnitude of the velocity is called the speed.
The direction of a velocity is the direction in which the object's moving.
Another vector quantity we've seen is the acceleration.
Acceleration has a size, a bigness, it can
be big acceleration or small acceleration, it has magnitude.
And acceleration has a direction.
Recall that the direction of the acceleration, is the direction
of that thing we call delta v, the change in velocity.
I can represent a vector Graphically, with an arrow.
The length of the arrow tells me the magnitude, and the
direction of the arrow tells me the direction of the vector.
If I want to use a symbol
to represent a vector, I'll emphasize that it's a
vector by putting a little arrow over the symbol.
If I want to indicate the magnitude of a vector What
I'll do is put absolute signs around the vector like that.
Another way to represent the magnitude of a vector is
to just draw the symbol without the arrow over head.
So if I lift the arrow off I mean the magnitude.
If I put
the arrow on I mean the arrow.
So this length right here is the magnitude.
Of the arrow.
Now, I just want to emphasize that magnitude of the positive number always.
Now, in two dimensions, suppose I'm working in the x, y plane.
Suppose I draw a vector like so Okay. I can indicate that
vector, I can specify it completely by giving you 2 numbers.
I can give you the magnitude, and the
angle that tells me which way it's pointing.
So, vector A.
In 2D, in two dimensions, I need
two different numbers to specify completely the vector.
I can give you the magnitude, A, and the angle, theta.
Or,
I can give you what are called the components, of the vector.
If I imagine.
Casting a shadow onto the x axis,
I'll get a number that we call a sub x, the x component of the vector.
And if I cast a shadow onto the y axis, I'll get
a number called a sub y, the y component of the vector.
We'll talk about components later on.
But the point is, there's more than one way to specify a vector using numbers.
But if you're in 2D, you've got to have two different numbers.
In 1D, 1D was sort of a special case.
We could specify the direction of a vector in
one dimension by just doing a plus or minus sign.
So a single-signed number was all I needed to specify a vector in 1D.
But in 2D, I gotta have two numbers.
And in 3D, I'll need three numbers, because
I'll need a component in the z direction.
Now, I want to talk about vector addition.
Vectors are not numbers, so vectors don't add the way
the numbers do. There's a special rule for adding vectors.
And I'm going to talk about graphical vector addition.
This is how to add vectors by drawing pictures.
Drawing, drawing graphs.
Suppose I have two vectors.
Suppose I have a vector a that points to
the upper right like that, that's a big long vector.
Then I have a small vector b. That points straight up.
A vector is a magnitude and a direction. It's not a location.
I can draw the vector wherever I think is convenient.
Let me show you a really cool
[UNKNOWN]
simulation called a vector addition.
So, I've got these little vectors in a, in a bucket.
I'll grab a vector and pull it out here.
I can change the length and the direction of the vector.
Now once I've got the vector length, and the vector direction
set It doesn't matter where I put it, it's the same vector.
So if two vectors have the same length and the same direction,
they're the same vector.
These two vectors, are equal no matter where I put them.
Location is not part of the specification of a vector.
So when I'm thinking about these vectors I can draw them on the page wherever I like.
What do I mean by the vector A plus B?
When I add two vectors, vector A plus vector
B, I get a third vector that I'll call C.
And here's the rule for
getting that vector C.
I think of the two vectors A and B like steps.
From a starting point to a destination.
I start anywhere, anywhere I like. And I think of A as the first step.
So I'm going to draw that vector, it's to the upper-right, here's the vector A.
After the first step, I'm right there.
Then I think of b as the second step. After I take the second step,
I'm right there.
Okay, it's like steps on the way to the treasure.
Step a, step b.
After the two steps I'm done. There's my finish point.
And the rule is the vector c, is the
vector that points directly from the start to the finish.
So there's the vector c.
The way I've drawn this diagram is called tiptotail.
We call it tip-to-tail because I've put the tip of A against the tail
of B.
There's another way to draw the same diagram, I can
put the two vectors A and B tail to tail.
See if there's the vector A and there's the vector B.
If I now complete the parallelogram, the foursided objects, the foursided object
with parallel sides, then the diagonal of the parallelogram is the vector C.
I hope you can see that this diagram, and this diagram are the same.
I've just moved the vectors around.
Now, I hope It's clear, that vector addition is.
What's it called when I'm allowed to switch the order?
A plus B is called B plus A.
A, A plus B, is the same as B plus A I claim.
What's that called?
Commutative law! Thank you.
Vector addition is commutative.
I can show that on the phet sim real quick.
So if I've got two vectors, 'kay.
Here it's just two random vectors, okay.
I've got this little thing called Show Sum, let me click that.
This green arrow appears.
I claim that this green arrow is the vector
sum of this arrow and this arrow. Well, let me put tip to tail.
And sure enough, look at that.
Vector a plus vector b is this green arrow, the sum vector.
But the order doesn't matter.
I could put this vector first and this one second, and I'll still add up.
So vector addition is commutative. A plus b is b plus a.
I can add any number of vectors.
I can take a vector a. And
I can add it to a vector b,
and I can add that to a vector c, and I can add
that to a vector d, and it's like throwing toothpicks down, you know?
So I'm going to show some, I claim that this green
arrow is the vector sum of all of these red arrows.
Well, the only way to tell that is to line 'em up, tip to tail
Oh it's going to work. Look at that.
Vector A step A plus step B plus step C plus step D.
All them all up to
[INAUDIBLE]
, you get the vector sum.
So, sorry let me just finish the diagram here.
This diagram here is called the parallelogram rule.
If you're adding two vectors, you can
either do parallelogram rule, or tip to tail.
But if you're adding a whole bunch of vectors, you've got
to use tip to tail, that's the only way to do it.
Let's have a concept test.
Two vectors of unequal magnitude, I've got a vector
A whose magnitude is not equal to the magnitude of
vector B.
Can 2 vectors of unequal magnitude add up to zero?
I've made a techical mistake in this equation here.
When I add 2 vectors together, I have to get a 3rd vector.
When I say 2 vectors add up to zero, I mean
they add up to a special vector called the zero vector.
The 0 vector is the vector of 0 magnitude.
So, to be technically correct, I should've put a little arrow over
this 0 to represent the 0 vector.
Anyway, can two vectors of unequal magnitude add to 0?
Yes or no. Confer and vote, please.
Okay, I'm hearing some good discussion.
But I'm also starting to hear some off-topic discussion.
So let's, let's finish up the vote. Well if I want two vectors to add to
0, I hope it's clear that the second step has to exactly undo the first step.
So if I have two vectors, and I want them to vector sum to 0.
The only way to
get the sum to 0 is the have the second vector to be equal and opposite
to the first vector It's gotta exactly cancel.
There we are.
Two vectors equal an opposite. This little, this little sum
vector just disappears, like the smile of the cheshire cat, just.
Gone, so it's got to be equal and opposite in order to cancel.
So if they're not equal magnitude they
can't possibly cancel, so the answer is no.
92%, pretty good.
Okay, this next question, I'm going to do something a little unusual.
We're going to vote on it twice.
The first vote is going to be a silent vote, no discussion.
And the second vote will be with discussion, so.
Please don't discuss with your neighbor. Let's just look at the question and vote.
Consider this vector diagram which of
these equations correctly describes this diagram.
Vector A plus vector B plus vector C plus vector D is 0 or A plus B plus C
is D. Or A+B is C+D, or A+D is B+C or
none of these is correct. Please do not confure with your neighbors.
Put on your own we will confure in a minute but not right now.
If you are clueless just guess. Times up in 3, 2, 1 Get them all in.
Okay, now, discuss with your neighbors before voting.
We're going to vote on this a second time. Okay.
All right.
It looks like everyone's, no, not everyone's voted.
Vote a second time here.
I've got the vote open for a second time. Time's up.
Three, two, one.
Get them all in.
There's always people who, like, wait 'til the last second.
>> Okay, time's up.
>> Okay, I've redrawn the diagram here, using
two different colors, so it's kind of suggestive.
I noticed that vector A and vector B are tip to tail.
Well, since they're nicely lined up for me tip to tail, I can see from
the diagram what the vector sum is. I'll call the vector sum.
I'll call it s for sum, a plus d, let's see if I start there, a is
the first step, d is the second step,
so there's the vector sum right there, there's s.
But I notice that b and c are also lined up tip to
tail and so vector b plus vector c is the same vector s.
So, this diagram is showing the equation A, vector
A plus vector D is vector B plus vector C.
So what's the right answer?
Ooh, 94%, you guys are catching on, okay. How'd you do on the first try, 68%.
Group think works, I like it. Okay?
Talk to your neighbors!
You know, if you're going to be an engineer, you're never ever going to
solve a problem on your own, you're always going to be working on a team.
So get use to taking advantage of the minds around you.
Before we go further I need to define some other operations involving vectors.
I have to define what I mean by a negative of a vector.
If I have a vector A, I'll represent with an arrow,
the vector negative A, minus A, means the
vector with the same magnitude but opposite direction.
So if I just flip the direction of the arrow,
keeping the parallel and magnitude the same, that's minus A.
The next operation I have to define is the operation of a number
[SOUND]
. Multiplied times a vector.
Suppose I have a number.
I'll call it b.
And let me be specific suppose the number is 3.
So b is 3.
And then I have a vector. I'll call it vector a.
I want to know, what is the vector 3 times a or b times a mean?
It means this.
If I start with the vector a, and I stretch.
The vector
by a factor of 3, then I'll get a vector
that's three times as long, but in the same direction.
I'll try to indicate, so here's the vector 3A, which is b times A.
Ok, so multiplying a number times a vector means
just stretch the vector by the factor of that number.
Incidentally, There are 2 different ways to define
the multiplication of a vector by a vector.
It's called the scaler product or cross product,
but we'll get to those later in the course.
Okay, and finally, I have to define what I mean by vector subtraction.
If I have a vector. A.
And I subtract the vector b. I gotta know what the heck that means.
It means this.
It mean, take the vector a, and add to it the negative of b.
We've defined what it means to take the negative of, of a vector.
And we've defined what it means to add vectors.
So, so this relationship defines subtracting vectors.
Let me give you an example.
Suppose I have a vector A, that points upper right like that.
And I have a vector B that points to the right like that.
A minus B. How do I do that?
Well first I have to construct the negative of B.
There's the negative of B. Now I'm going to add
the vector A to the vector the negative of B.
And I'll do that sub tail.
I start here. Here's my vector A.
We're going to add to that tip to tail the vector minus B, and after those two steps,
A plus a minus B, I wind up here. Here's the vector A minus B.
Time for some concept tests. Three vectors, A, B, and C
are shown. I've shown them in the X-Y plane.
Which vector.
Is the vector A plus B minus C.
So here the choices or maybe a none of these.
Confer before voting, please. I see that most of you have voted.
So may I have the remaining votes please.
Time's up.
Let's see I've reproduced the question on this
transparency so I can draw right on the transparency.
I'm just going to do graphical tip to toe addition.
So if I start there.
Here's the vector A.
The vector B is two down and two to the left.
So here's the vector B. Here's the B.
The vector, minus c, is the opposite of the vector
c, so there's the vector minus c, three to the right.
Okay so I'm going to add that vector minus c here, tip to tail.
Okay so there's the vector minus c.
Okay so I wound up there, so, the vector a plus b minus c is the vector that takes me
from the start Directly to the finish, so there's the
vector C, that's one to the right and four down.
So what's the right answer?
D is three to the right and four down, so D is close, but no cigar.
It's E, none of these.
Oh, I didn't
stop the vote in time.
[LAUGH]
[LAUGH].
Yeah,
[LAUGH].
Yeah,
[LAUGH].
Before I stopped the vote, I could see that only about 40% of you
had voted correctly, and the others didn't notice that the barn door was open.
Okay, so.
The point here is that, even though we're drawing pictures, it's not qualitative.
It's exact.
You can draw the pictures exactly correct, or you can draw it
a little incorrect and a little incorrect is the same as wrong.
So be careful.
Let's try another one here.
We're about to start talking about components,
and before we talk about components let's have a little review of trigonometry.
Take a look at this diagram.
This solid line has length a, and it
makes an angle theta with the negative y direction.
Whats the length of this dotted line right here?
The length of this dotted line is, well I'll let you read the expressions.
They involve Co sign, sign and tangent, the discussion
has died away so I'm stopping the voting now.
Times up in 3, 2, 1. OK.
So, if I complete the rectangle, that this diagonal has length A,
I want this side, I notice that this side here is the same as this side over here.
>>
[SOUND]
And that's the side of this right triangle that is
adjacent to the angle and there's this pneumonic that some people use.
There's this Indian Princess name, what's it called?
>>Sokatowa. >> Sohcahtoa, thank you.
I don't use Sohcahtoa because I can't remember how to spell Sohcahtoa.
So, I have a different way of thinking about it.
Incidentally, what's the answer.
To the length here, what is it, its, the answer is, sign or cosign or what.
So its a cos theta.
I'm with you. Let's see how you did here.
You're engineers, so you all know trigonometry.
Ooh, ouch, 20% of you are unclear on sine and cosine.
That's perfectly Ok, to be unclear today in
lecture, but get it under control before the exam.
I can't remember Sohcahtoa, so I have a different way, a
way that my physics TA told me about 35 years ago.
So I notice that this side, this side I'm
looking for, It's the side that's touching the angle.
It's the side that's adjacent to the angle.
It's the side that's intimate with the angle.
It's the side that's cozy with the angle. It's the cozy side.
So it's A cozy side.
A Cosine Theta. Get it?
[LAUGH]
And how do you remember this one? It's the other one.
A.
[LAUGH]
So it's either A cozy side theta or it's the other one, A sin theta.
Okay next I have to talk about these things
I alluded to before called the components of a vector.
The components of a vector are signed numbers.
They're numbers with plus or minus signs.
[SOUND].
And the way I define the components of a vector is like this.
Let's consider a vector b in the xy plane.
Let me draw it like this, alright? There's the vector b.
Now, if I imagine casting a shadow onto the x axis.
I imagine there's a light source directly
overhead, casting a shadow onto the x axis.
I call that the, I call the shadow, the projection
of the vector onto the X axis.
The length of that shadow is how big BX is and BX has a plus or minus sign.
Depending on which way the shadow points.
In the case, the shadow is pointing in the negative X direction.
So this vector has a negative X component. So BX is Less than zero.
To get the y component, I project, I cast a shadow,
onto the y direction. So here's my b-y.
And I can see that the b-y shadow
is pointing in the plus y direction, so this b-y has a Is a positive number.
So this vector has a positive y component and a negative x component.
If you look at the factsim there's a wonderful way to visualize this.
Let me pull out a vector here, and there's something called
component display.
These purple arrows represent the
components, they've got different styles here.
So here's the style where I'm showing the projections.
So, this purple arrow represents the x component of the
vector, this one represents the y component of the vector.
Notice that components don't change when I move the vector around.
Components of a vector specify the vector completely, but
location is not part of the specification of a vector.
Now that we have components, there's a way I can
do vector addition using a calculation, rather than drawing a graph.
It's called vector addition by components. If I have a vector A and I add to it a
vector B and that gives me the vector C, I claim
that a way to get the x and y components of the the sum vector, C,
is by doing this. Ax plus Bx I claim will give me
Cx. And Ay plus By, I claim, will give me Cy.
And the proof of this is what I call proof by diagram.
If you draw the correct diagram, it's obvious.
Let's look at the
[UNKNOWN]
to draw the diagram.
If I take a vector A and add to it a vector B.
Well, to do vector addition I add them tip to tale.
Let me just make them a little bit longer, okay.
And then we display the components.
So, this purple arrow is the x-com
proponent of this red arrow, I'm going to call
this one A and this one B. So, can you see that Ax+Bx is this
length right here, this big long length. Well, let me now show you the sum.
There's the sum with it's components represented by those green arrow.
So if you look carefully, A-x, this length right
here, added to B-x, is going to give me C-x,
the x component of the sum.
Likewise, A-y stacked with B-y gives me C-y.
Stare at this diagram in a cool dark room if that doesn't make sense.
Let's have a question about adding vectors by adding vectors component.
Did we have this question?
Oh, we didn't, we didn't, okay. So, sorry I skip one.
Let's go back to this one please.
What's the correct formula for the ay, the y component of this vector a.
I'll let you read the choices.
I see that everyone voted or almost everyone.
Okay. So,
I want the y component of this vector.
The y component of a vector is the projection of the vector onto the y axis.
So graphically, I think of the y component as that shadow right there.
Now, without doing any math, no calculator,
just look at the diagram and think.
Is that Y component a positive number or a negative number?
Negative yeah, cause the shadow
is pointing in the negative Y direction. So it's negative.
Is this the cozy side or the other side? Cozy side, so it's A cosine theta.
Notice that A, A without the arrow over it means the magnitude.
So that's always positive.
Cosine theta, because I've chosen an angle
less than 90, cosine theta is always positive.
The only way to get cosine or sine to be negative is
to have angles that are bigger than 90.
So, that minus sign is the only minus sign around, so that
minus sign right out front is showing me the sine of ay.
How'd you do?
78%, not bad, I wish that was a little higher.
A lot of people today. A has the wrong sign.
You get the sign by thinking about the diagram, the plus or minus sign.
You get the a cosine or a sin
theta by using whatever technique you're comfortable with.
[INAUDIBLE]
if that works for you, stick with it. Cosyside that's what work for me.
Alright, the last thing I need to talk about in today's lecture.
Is something called unit vector. There's a special set of vectors called
the unit vectors. They're represented either by a hat, j
hat, k hat or x hat,
y hat, z hat. What do I mean by the vector i hat?
So in the XY plain I hat which is the
same as x hat just different textbooks use different notation.
I hat is a vector that has magnitude 1.
That's why we call it the unit vector, its magnitude is unity.
The little hat over a vector is a notation that indicates its a unit vector.
Any time you see a hat you know its got magnitude one.
The i hat vector, or x hat vector is
the unit vector that points in the positive x director.
So here's I hat.
The Y hat vector which is the same as the J hat
vector is the unit vector that points in the positive Y direction.
So here is the J
hat vector. I can represent any vector, vector A.
By giving its components, and using these unit vectors.
Now, vector a, for instance, I can write it as ax i hat plus ay j hat.
Let me give you an example to maybe make this a little more clear.
Suppose ax is 3,
so I have 3 i hat, and ay is negative 2. Minus 2 j hat.
Let me represent that vector graphically here.
So, I'll start by drawing the vector 3 times i hat.
i hat is the vector pointing in the plus x direction.
When I multiply by 3, that gives me a vector that.
3 long.
Okay so here is the vector, 3 i hat. The vector minus 2 j hat,
well j hat points in the plus, y direction, minus j hat would point in
the minus y direction. When I multiply, by 2 I stretch
it a factor of 2 So here is the vector,
-2j^. Now, when I add these
two vectors using the parallelogram rule, I get this vector, and
there's the vector A. It's 3i^-2j^.
Vector added.
Let me finish up with a concept test,
that we'll discuss at the beginning of next time.
Oh, shoot. Sorry.
I keep forgetting I always have one more concept test.
The vectors A, B, and C are shown.
The vectors sum of these three vectors.
Vector S is vector A plus vector B plus vector C.
What is the value of S sub x, the x component of this vector S?
Confer and vote, please.
Now, we have two minutes left, so can I have the rest of the votes.
Time's up. Okay.
Well, if vector S is vector A, plus vector B
plus vector C, then I can do the component thing.
That means, Sx is going to be Ax plus Bx
plus Cx. Let's see.
Can someone read ax from the graph for me. What's ax?
It's a positive 3, thank you. Bx is minus 3, cx is minus
3, so 3 minus 3, minus 3 is minus 3 for s sub x.
So, how many of you can do. Arithmetic, let's find out.
[LAUGH]
Oh, very good, 76% of you can do arithmetic, excellent.
[LAUGH]
Yeah it's easy to get confused, be careful.
Okay.
Got less than a minute left so I'm
going to send you off with one last concept test.
We'll pick this up at the beginning of next time.
I've got five different vector equasions here.
How many of these equations make no sense?
How many of these equations are just nonsense?
0, namely all of them make sense or 1, 2, 3 or 4, soon as you've voted, you can
take off. I'm out of time, I'll see you on Monday.