- Good morning.
Let's begin our review
of the entire AP Physics C: Mechanics Curriculum,
and we will start with kinematics.
♫ Flipping
♫ Physics
- Class, dimensions are our--
- [Students] Friends!
- In other words, we have to remember
the units for everything
and we have to identify every number answer with units.
- Unless it doesn't have units!
- Like Mu.
- Yeah.
- Correct.
It also means we have to be able to do conversions.
Bobby, please convert the density of the gas krypton,
at standard temperature and pressure
of 3.75 grams per centimeter cubed,
to kilograms per meter cubed.
- Okay, multiply by
100 centimeters over one meter,
and cube it to cancel out centimeters cubed,
and multiply by one kilogram over 1000 grams
to cancel out grams,
and we now have 0.375 kilograms per meter cubed.
- And now vectors versus scalars.
- [Bo] Mr. P?
- Yes, Bo?
- That answer is not right.
It should be 3,750 kilograms per meter cubed.
I think Bobby forgot to cube the numbers
as well as the units.
- Yeah, that's right.
- Bo, thank you, that is absolutely correct.
Students will sometimes forget to apply an exponent
which is outside the parentheses
to everything which is inside the parentheses.
I see it all the time.
Okay.
Class, a vector has both--
- [Students] Magnitude and direction.
- And a scalar has?
- [Students] Magnitude only.
- This difference might seem minor, however,
it comes up on every AP Physics exam.
One example is that momentum is a vector
and students will sometimes forget
to include both the x and y directions
when solving a conservation of momentum problem.
Billy, what is the equation for velocity?
- Are you asking for average or instantaneous velocity?
- [mr.p] That is an awesome question, Billy.
Please give me both.
- Okay.
Average velocity is change in position over change in time,
and instantaneous velocity is the derivative of position
as a function of time.
- And Bo, what is the difference between
average and instantaneous?
- Average means over a time period,
and instantaneous means at a specific point in time.
- [mr.p] And what does the derivative represent?
- The derivative gives the slope of a line.
- We can also write out the equations for average
and instantaneous acceleration.
Average acceleration equals
the change in velocity over change in time
and instantaneous acceleration equals
the derivative of velocity with respect to time.
Now, let's talk about Uniformly Accelerated Motion, or UAM.
We can use the UAM equations
when the acceleration of the object is uniform or constant.
In other words, when the acceleration equals a number,
not an equation, which changes as a function of
time or position.
Bo, what are the five uniformly
accelerated motion variables?
- Acceleration, obviously.
Velocity final, velocity initial, displacement,
and change in time.
- There are four UAM equations.
Now I know these four--
- Mr. P!
Those are not the equations we're going to get
on the equation sheet on the AP exam.
- Now I know these are not the UAM equations
which are on the equation sheet
which you're going to get on the AP exam.
So I wanna make sure you understand the differences
between what I've written here
and what's going to be on that equation sheet.
So, the first of the equations which I've written here
is written as such on your equation sheet.
Velocity in the x direction equals
the velocity in the x direction naught,
plus acceleration in the x direction times time.
Billy, what does this naught,
or this zero, as a subscript mean?
- Velocity naught, or velocity zero,
means the velocity at time t equals zero
or the initial velocity, correct?
- Correct.
So, v sub x naught
means the velocity initial in the x direction.
I, instead, write v sub i for velocity initial,
and I add subscripts whenever necessary.
And there's also this difference between
the change in time and time.
Change in time equals time final minus time initial
and using t for time final and letting time initial equal
to zero, means that change in time equals t.
But again, remember this assumes that
the time initial is zero.
Bobby, what about this v sub x,
on the left hand side of the equation,
what does that mean?
- That is the velocity final of the object
in the x direction, right?
- Correct.
So you can see, this first equation on the equation sheet
describes the final velocity of an object
as a function of time, assuming that the time initial
is equal to zero.
And here are the other two UAM equations
on the equation sheet.
These last two UAM equations have displacement,
replaced with position final, or x,
minus position initial, or x naught,
and I don't know why my last UAM equation
is not on the AP equation sheet.
Okay, class, using my UAM equations,
how many UAM variables are there?
- [Students] Five.
- And how many UAM equations?
- [Students] Four.
- And if you know, how many of the UAM variables?
- [Students] Three.
- You can find the other--
- [Students] Two.
- Which leaves you with one--
- [Students] Happy physics student.
- Which is always fun.
And, I will point this out.
Velocity is the derivative of position with respect to time
and if we use the UAM equation from your equation sheet
of position as a function of time
and we take the derivative of that with respect to time,
we get the first of the UAM equations.
(laughing) Which is pretty cool.
Okay, let's move on to the concept of free fall.
An object is in free fall if the only force acting on it
is the force of gravity.
In other words, there cannot be any air resistance.
I like to call this being
in the vacuum that you can breathe.
When this occurs, the acceleration in the y direction
equals the negative of the acceleration due to gravity
or the negative of little g,
and the acceleration due to gravity here
on planet Earth is positive 9.81 meters per second squared.
Which means, when an object is in free fall,
the acceleration in the y direction of that object
equals negative 9.81 meters per second squared,
or 9.81 meters per second squared down.
This means free fall is an example
of uniformly accelerated motion
where the acceleration is equal to
a constant negative 9.81 meters per second squared.
And you can use all the UAM equations.
Bo, please, walk me through all the graphs
for position, velocity, and acceleration
as a function of time
for an object that is in free fall
dropped from a height of 2.0 meters.
- Is there any air resistance?
- [mr.p] Unless it is otherwise stated in AP Physics,
we are always in the vacuum you can breathe,
so no air resistance.
- Okay.
Let's start with the acceleration graph
because that one is the easiest.
We know the acceleration is a constant, it's constant
at a value of negative 9.81 meters per second squared, so,
acceleration as a function of time will be a horizontal line
at negative 9.81 meters per second squared.
The initial velocity of the ball is zero, and the slope of
a velocity versus time graph is acceleration,
so, the velocity versus time graph will be a line
which starts at zero and has a constant slope
of negative 9.81 meters per second squared.
The position as a function of time graph
will start at 2.0 meters,
and because the slope of a position versus
time graph is velocity,
the initial slope of the position versus time graph
will be zero.
As time goes forward,
the velocity becomes more and more negative,
so the position as a function of time graph
will have a negative slope which increases
in magnitude over time.
So it will be concave down until the position final is zero.
- Bo, thank you, that was very impressive.
Let's move on to component vectors.
Component vectors are the vectors
in the x, y, and possibly z directions,
which when added together result in the original vector.
Billy, please determine the component vectors
of the acceleration vector,
2.5 meters per second squared
at an angle of 26 degrees east of north.
- Well, the x direction uses cosine,
so the acceleration in the x direction
is 2.5 times the cosine of 26--
- No, it's not.
You have to draw the vectors out and use SOH CAH TOA.
- Mr. P sets up his examples this way
to make sure we show our work.
- Aw, yeah, so, once we draw the vectors,
we can use sine theta equals opposite over hypotenuse,
which equals acceleration in the x direction
over acceleration,
multiplied by acceleration,
to get acceleration in the x direction,
equals acceleration times sine theta
or 2.5 times the sine of 26 degrees,
which equals 1.09593,
or, with two significant digits,
1.1 meter per second squared.
We can do the same thing with
cosine theta equals adjacent over hypotenuse,
and we end up with the acceleration in the y direction
equals acceleration times the cosine of theta,
or with numbers, 2.24699,
or with two significant digits,
2.2 meters per second squared.
- Which brings us to the unit vectors i, j, and k.
Which are vectors with a magnitude of one
in the x, y, and z directions, respectively.
In other words, Bobby, what is, in unit vector form,
an acceleration of 2.5 meters per second squared
at an angle of 26 degrees east of north?
- Well, we just determined the components of that vector
in the x and y directions,
so the acceleration is 1.1i
plus 2.2j meters per second squared.
- What is that weird hat symbol over the i and j?
- That is the symbol for unit vector,
and yes it is often called a hat.
- Oh.
- So these two expressions describe the same acceleration.
Plus, unit vectors make vector addition much easier.
For example, let's say we are adding together
three displacement vectors for an object.
Without unit vectors,
we would need to break every one of these vectors
into its components,
in the x direction and the y direction.
Then we would need to add all the x values together,
and then all the y values together,
and then determine the magnitude and direction
of the resultant vector.
However, because all of these vectors are
in unit vector form, all we need to do is
add all the vectors in the x or i direction together
and all the vectors in the y or j direction together.
Which is a lot easier.
And notice also, this gives us the r vector
to identify the position of an object.
Because it's not just the position
in the x or y direction of an object,
the r position vector can give you the position
of the object in two or even three dimensional space.
Next, relative velocity.
Relative velocity is vector addition.
Let's say we need to fly a plane due north,
relative to the Earth,
in wind which is blowing eastward.
What direction do we fly the plane?
We know the velocity of the plane with respect to the Earth
equals the velocity of the plane with respect to the wind
plus the velocity of the wind with respect to the Earth,
and we need to draw a tip to tail vector addition diagram.
And you can see, the plane needs to be flown
in a northwesterly direction.
Let's move on to projectile motion,
where an object is flying through
the vacuum that you can breathe
in both the x and y directions.
Bo, what is our general strategy
for solving projectile motion problems?
- We start by listing our known values,
but we split them by direction,
listing what we know in the x direction
and what we know in the y direction separately
because an object in projectile motion only experiences
the force of gravity which is in the y direction.
The object has no net force in the x direction
and therefore no acceleration in the x direction.
That means the velocity of the projectile
in the x direction is constant
and we can use velocity in the x direction
equals displacement in the x direction over change in time.
In the y direction the object experiences free fall
so the acceleration in the y direction
equals negative 9.81 meters per second squared
and we can use the UAM equations.
And the variable which connects the two directions is time,
because time is the only variable which is a scalar.
And we usually have to break the initial velocity
into its components.
- That completes my review of kinematics.
Next, you can enjoy my review of dynamics,
which is Newton's three laws, forces, equilibrium,
and so forth.
Or you can visit my AP Physics C review web page.
Thank you very much for learning with me today.
I enjoyed learning with you.