- [Instructor] What do vector components mean?
Vector components are a way of breaking any vector
into two perpendicular pieces.
For convenience we typically choose these pieces
to lie along the X and Y directions.
In other words to find the vertical component
of this total vector knowing this angle,
since this vertical component is opposite to this angle,
we could write the vertical component as the magnitude
of the total vector times sine of that angle.
And since this horizontal component
is adjacent to that angle,
we could write the horizontal component
as the magnitude of the total vector
times cosine of that angle.
And if instead we were given this angle
and we wanted to determine
the vertical component of the total vector,
since this vertical component is now adjacent to this angle
we'd write the vertical component as the magnitude
of the total vector times cosine of this angle.
And since the horizontal component
is now opposite to this angle
we'd write the horizontal component
as the magnitude of the total vector
times sine of this angle.
So remember, to find the opposite side you use sine,
and to find the adjacent side, you use cosine.
So what would an example problem
involving vector components look like?
Say you had this question and you wanted to determine
the X and Y components of this velocity vector.
Since the horizontal component is adjacent
to the angle that we're given,
we're gonna write the horizontal component
as the magnitude of the total vector 20 meters per second
times cosine of the angle,
which gives us 10 meters per second.
And since the vertical component is opposite to this angle,
we can write the magnitude of the vertical component
as 20 meters per second times sine of the angle
which gives us 17.3 meters per second.
But since this vertical component is directed downward,
technically this vertical component
would be negative 17.3 meters per second.
So using sine and cosine will give you the magnitude
of the components, but you have to add
the negative signs accordingly.
So if the vector points right,
the horizontal component will be positive.
If the vector points left,
the horizontal component will be negative.
If the vector points up,
the vertical component will be positive.
And if the vector points down,
the vertical component will be negative.
What does tail to tip or head to tail vector addition mean?
This is a graphical way to add
or subtract vectors from each other.
And the way it works is by taking the tail
of the next vector and placing it at the tip
or the head of the previous vector.
And once you're done doing this for all your vectors,
you draw the total vector from the first tail
to the last head.
In other words, if you were adding up vectors A, B, and C,
I would place the tail of vector B to the head of vector A,
and then I'd place the tail of vector C
to the head of vector B.
And once I'm done I would draw the total vector
going from the first tail to the last head.
And that total vector would represent the vector sum
of all three vectors.
And if you had to subtract the vector,
you could still use vector addition.
Simply add the negative of that vector.
In other words, if you had some vector B
and you wanted to subtract vector A,
instead of thinking of it as subtracting vector A,
think of it as adding negative vector A.
And the way you find vector negative A
is by taking vector A and simply placing the arrow head
on the other end of the vector.
So what would an example involving tail to tip
vector addition look like?
Imagine we have this question,
and we have these four vectors,
and we were asked to determine what direction
is the sum off all of those vectors?
So we'll use tail to tip vector addition.
I'll take vector A preserving its direction.
I'm not allowed to rotate it or change its size.
And I'll add to that vector B.
The way I do that is putting the tail of vector B
to the tip of vector A, and we add vector C.
And the way we do that is put the tail of vector C
to the tip of vector B.
And finally we'll add the vector D
by putting the tail of vector D to the tip of vector C.
And now that we've drawn all of our vectors,
our total vector will go from the tail of the first vector,
to the tip or head of the last vector,
which means this is the direction and magnitude
of the total vector, A plus B plus C plus D.
Another more mathematical way of adding vectors
is by simply adding up their individual components.
So to find the total vector A plus B,
instead of graphing them tail to tip,
you can find the horizontal
and vertical components separately
by adding up the individual components.
In other words, to get the total horizontal component
of the total vector A plus B,
I could just add up the horizontal component of A,
which is negative 20, and the horizontal component of B
which is negative five to get negative 25.
And to find the total vertical component
of the total vector A plus B,
I can simply add up the vertical component of A
which is negative 15 to the vertical component of B
which is 10, to get negative five.
This technique lets you quickly determine
what the individual components are of that total vector.
And again, if you need to subtract a vector
you can still add the components,
except the components of a negative vector
all get multiplied by negative one.
In other words, if vector A has components negative 20
and negative 15, then vector negative A
would have components positive 20 and positive 15.
So what would an example
of vector component addition look like?
Let's say you had this question.
You had vectors A and B with these components,
and you wanted to know the components
of the total vector A plus B.
So to find the horizontal component
of the total vector A plus B,
I can add up the individual components of A and B.
So the horizontal component of A is four,
plus the horizontal component of B is negative one,
gives me a total horizontal component of positive three.
And I could do the same thing for the vertical component.
I could add up the component of A plus the component of B,
which would be five plus negative four,
gives me positive one.
So since my horizontal component
of the total vector is positive,
I know it points to the right three units.
And since the vertical component of the total vector
is positive, I know it points up one unit.
That means my total vector A plus B
points up and to the right.
One unit up, and three units to the right.
How do you deal with 2D kinematics problems?
2D kinemtatics are projectile problems
describe objects flying through the air at angles.
For these objects, if there's nothing acting on them
besides gravity, their vertical acceleration
is gonna be negative 9.8.
And they will have no horizontal acceleration
since gravity doesn't pull sideways.
Also, the X and Y components behave independently.
That means you'll use different equations
to solve for vertical components
than you will for horizontal components.
Since the vertical acceleration is constant
you can use the kinematic formulas
to solve for quantities in the vertical direction.
But you can only plug in vertical quantities
into these equations.
Similarly, since the acceleration is zero in the X direction
you can simply use distance as a rate times time
to relate the quantities in the X direction,
but you should only plug in horizontal components
into this equation.
In other words, as a projectile is flying through the air,
since there's no horizontal acceleration,
the horizontal component of the velocity
is gonna remain the same for the entire trip.
Which means the rate at which this projectile
is moving the X direction never changes.
But since there is acceleration in the vertical direction,
the vertical component of the velocity
will get smaller and smaller until it reaches the top,
and that means the total speed of the projectile
is gonna decrease as well as you approach the top.
And then at the top there is zero
vertical component of velocity
since the projectile's not moving up or down at that moment.
And then on the way down the vertical component
of the velocity gets more and more negative,
which increases the speed of the projectile.
Keep in mind during this entire trip
the vertical acceleration is the same, negative 9.8
on the way up, at the top, and on the way down.
The vertical acceleration never changes.
So what would an example problem
involving 2D kinematics look like?
Let's say a meatball rolls horizontally
off of a dinner table of height H, with a speed V.
And we want to know how far horizontally does the meatball
travel before striking the floor.
So the first thing we should do is draw a diagram.
So the height of the table is H.
The initial speed of the meatball is V,
and we want to determine how far horizontally it makes it
from the edge of the table.
But note this problem's symbolic.
We're not given any numbers so we're gonna have to give
our answer in terms of given quantities
and fundamental constants.
The given quantities are things like H and V,
and fundamental constants are things like little G.
So this quantity we want to find
is the displacement in the X direction,
which is gonna be the speed in the X direction
times the time of flight.
We know the speed in the X direction,
it's gonna remain constant.
So this V is gonna be the horizontal speed
for the entire trip.
So we can plug in V for the speed, but I don't know
what the time of flight is gonna be.
To get the time of flight we'll do another equation
for the vertical direction.
The vertical displacement is not gonna be H,
it's gonna be negative H since this meatball fell downward.
And the initial velocity in the Y direction
is not gonna be V, it's gonna be zero,
since this meatball had no vertical velocity
right when it left the table.
It only had horizontal velocity.
The acceleration is negative 9.8,
but we're gonna write that in terms of fundamental constant,
so we'll write that as a negative G.
This lets us solve for T.
We get the square root of two H over G
which we can now bring over to here
to get the horizontal displacement of this meatball
before it hits the ground.
Something else you'll definitely have to know
for the AP exam is how to graph data to a linear fit.
And what I mean by that is that when you graph data
it doesn't always come out linear.
And when you don't get a linear graph
it's hard to find the slope of that curved graph.
However you can force your data to be linear
if you write down the expression that gives the relationship
between your data in the form of a straight line.
So the form of a straight line is Y equals M X plus B.
Y would be the vertical axis.
X would be the horizontal axis.
M is the slope, and B would be the Y-intercept.
In other words, if you had the expression
P equals 1/2 D squared,
if you just plot P versus D you're gonna get a parabola,
and that means you got problems
'cause finding the slope of a parabola is hard.
But if you instead choose to plot P versus D squared,
you will get a straight line
because now you've required P to be your vertical axis,
you've required D squared to be your horizontal axis,
and the slope that's multiplying what you called X
is just a constant, and that means
your slope's gonna be constant.
So this lets us predict what the slope would be
if I plot P versus D squared since the slope
is always what's multiplying my horizontal axis,
the slope in this case should just be 1/2.
So in other words, if you force your expression
to take the form of a straight line,
now only will you get a linear fit,
but you can predict what the slope is
by looking at everything that's multiplying
what you called X.
But a lot of people find this confusing and strange,
so what would an example problem
where you have to graph data to a linear fit look like?
Say you were given this question.
You repeatedly roll a sphere off a table
with varying speeds V,
and then you measure how far they travel delta X
before they strike the floor.
If the table has a fixed and known height H
what could we plot to determine an experimental value
for the magnitude of the acceleration due to gravity?
So we repeatedly change the speed
and measured how far the ball went.
That lets us know these are the quantities that are varying,
so these are gonna be involved in the X and Y axes.
But to figure out what to plot we need to find
some sort of relationship between these two variables
so that we could put that relationship
in the form of a straight line.
Now in the 2D kinematics section right before this,
we derived a formula for how far a ball goes
rolling off a table in terms of the height of the table
and the acceleration due to gravity.
So this is the expression that relates
how fast it was going to how far it went.
And we need to put this in the form of a straight line
so it's easiest to just make this left-hand side
since it's already solved for Y.
So our Y quantity on our vertical axis would just be delta X
and that's okay 'cause that's one of the quantities
that we're varying over here.
Similarly, the other quantity that's varying is V,
so I'll just call V X.
That means the horizontal axis is gonna be V.
I was able to do that since this was just V.
If this had been V squared, I would've had to have plotted
V squared on the horizontal axis.
And if this was square root of V,
I would have to plot square root of V on the X axis.
But since it was just V, I can get away with just plotting V
on my horizontal axis and now we can figure out
what our slope would represent.
We've got Y equals M times X.
Everything that multiplies what we called X
is gonna be our slope.
The way it's written here
the M is multiplied on the right-hand side,
but it doesn't matter 'cause M times X
is the same as X times M.
That means this entire term, root two H over G
is the slope of this graph.
In other words, we're gonna get a linear fit,
and the number we find for the slope
is gonna equal the square root of two H over G.
And since there's no added B term,
there's no Y-intercept to worry about over here.
So how do we actually determine
the experimental value for G?
We take our data, we plot them,
we draw a best fit line through the data.
We would use points on our line
to determine the slope of this line
by taking the rise over the run.
That rise over run would be equal to the slope,
and we know that that number's gonna equal
the square root of two H over G.
So if we know this number for the slope,
and the table has a known and fixed value H,
the only unknown is G which we can now solve for.