AP Physics 1 review of 2D motion and vectors | Physics | Khan Academy

- [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.