AP Physics 1 Unit 1 Review - Kinematics - Position - Velocity - Acceleration - Vectors - Equations

welcome back everyone this is going to be unit one of AP Physics 1 if you haven't watched my introductory preliminary video I highly recommend you do that contains some prerequisites some knowledge some attitudes we should have

as we go into this course and I I won't be reviewing what I went over there now time to learn AP Physics one from the guy who took and self-studied every single AP exam so uh with unit one the

title of this unit is called kinematics okay and the principle of kinematics revolves around three quantities that is position velocity and

acceleration all right and we're going to do a very deep dive into precisely what each of those are uh we're going to be dealing with position on one dimension which means that uh we're

going to deal with position on something approximating a number line okay one dimension means you're moving along one axis I can either move forward or

backward we're not dealing with two Dimensions yet we're not going to deal with left and right just yet we're going to get into that in just a minute but I I want to simplify it down so that we can understand fundamentals first okay

so in any uh kinematics problem or in any problem in physics we need to Define where zero is okay so uh in all of

physics you have the discretion to Define where your axis is where your measurements are going to be so for example I could Define my position right

here as zero okay and that would mean that uh that side of the Whiteboard is approximately 1 M forward it's at the

position one cuz position is measured in meters as I covered in the introductory video that would be a position one now alternatively I could Define the axis to

say that this side of my whiteboard is going to be uh zero okay so if this side of my whiteboard is zero and I stand over here that would mean that relative

to my origin relative to my zero value I would be at position -1 I am 1 M behind my origin which would if I draw it out

on the onedimensional axis if I put Zer here 1 m to the right or 1 M forward would be at positive 1 and 1 meter backward would be at

position-1 and you're going to see that that is important because sometimes it's going to be advantageous for you to Define zero at a certain point rather than others position can be measured

with two quantities it can be measured either in terms of distance or displacement so when we change our position if I walk from position zero

right here and and I walk to 1 M forwards now I'm at position one I have traveled 1 M when we talk about distance

and displacement distance here would be a scalar if we remember that from the introduction and displacement would be a vector all right

so let's uh exemplify how those are different if I were to travel from my position zero to my position one over here here 1 M forward and then I were to

travel back to my original position position zero the distance I would have traveled I would have traveled 1 M and then 1 M

so I would have traveled at total of 2 m my distance in that case would have been 2 m my displacement however given that it's a vector if I were to travel

forward 1 M and then I were to travel backward 1 M you remember how we add vectors we add vectors tip to tail so forward 1 M backward one meter I start

here and I end here okay therefore the two vectors cancel each other out right tip to tail if I go forward

and then I add tip to tail and it takes me right back it means that my displacement is zero my displacement is zero because I ended exactly where where I began at position zero now you can

also have a velocity let's rewrite velocity down here velocity can be written as velocity or it can be written as speed velocity in this case being the

vector and speed in this case being the scalar okay so these two words are not interchangeable in the same way that these two words are not interchangeable I can

travel at 1 m/s this way I have a velocity of 1 m/s forward again the since this is a vector

it would be specified with the direction 1 m/s to the right or 1 m/s forward speed being a scaler is just that it doesn't have a direction you would

Define it as 1 m/s the derived unit for velocity or speed would be unit of me/ second okay division bar means that is

it is a rate of change okay your velocity or your speed defines the rate at which your position is changing okay if you have a velocity if you have a

speed that are nonzero numbers it means that your position is changing if I have a velocity of 1 m/s that means I am going 1 M every second

in the positive direction because it's a vector if I have a velocity of -1 m/s it means I am moving backwards 1 m/s I

moving in the negative Direction in one dimensional space you can Define direction as positive being to the right on the number line or negative being to the left on the number line so the

relationship between velocity and position is that velocity is the rate at which position changes okay and in that relationship position is the area under

a velocity time curve now I went over the velocity time curve in the previous uh video but now we can understand what it

means okay the area under a velocity time curve would be the displacement during that time notice how if I take the area under the curve of a

vector I get a vector I get the displacement let's understand what's going on here in the graph my velocity is increasing as time passes

that means at a Time Zero I'm not moving and as time starts to increase I slowly start to go faster and faster and faster

and faster which means as time goes on the distance that I am traveling every second increases because my speed increases as time goes on uh think of it

like a race car when the light is red the race car is not moving and the moment the guy swishes the flag and the light goes green the race car

immediately starts speeding up and it takes time for the car to speed up so the car is going to immediately after the light it's going to start moving slowly and then is progressively going

to get faster and faster and faster until it hits like 100 mph after velocity we have the quantity acceleration acceleration is the rate at which your velocity changes okay so if

we look at a velocity verse time graph uh if you remember from algebra we'd be able to calculate the slope of this graph all right and you remember that

slope is the rate of change of a graph so in order to find acceleration we can find the rate of change of this graph okay so we take one point let's

take the origin which is 0 0 and this point which we're going to call 10 10 and we're going to realize that the slope m equal 1 okay given that

acceleration is the rate of change of velocity and the slope of a velocity time graph is the rate of change of velocity that we can say that our acceleration in this

case is one okay and we can also say that the acceleration is constant meaning the acceleration is not changing because our

slope is the same everywhere on the line we have a constant acceleration acceleration is the rate of change of velocity we can then draw another graph

which would be acceleration versus time and acceleration would be in units of m/s squared if we take the area

underneath an acceleration versus time graph then that area would give us our velocity and the area under a velocity verse time graph would give us our

displacement now acceleration is a vector there is uh no scalar quantity for acceleration that we will need to be aware of in this course if you take the

area under the curve of a vector you're going to get a vector if you take the area under the curve of a vector you're going to get a vector uh let's try and extrapolate this area under the curve

business to interpret a more complex graph so if if I were to give you graph of acceleration versus time of course

m/s squared seconds this is my graph of acceleration versus time okay let's try and find the area under the curve okay

now this is an interesting concept when we look at area under curves an area that is above the xaxis is a positive area and an area that is

beneath the x-axis is a negative area and you can tell just by looking at the graph that I drew for you here the area here is appr approximately

equivalent to the area here so if here I have positive a and here I have a negative a that means if I add the two

together I'm going to get zero so that means that if let's say this is time equal 0 seconds and this is time equal 20 seconds that means after 20 seconds means at

this end after 20 seconds have elapsed the area under my curve is zero so what does that mean it means my velocity at time equal 20 seconds is

zero all right now why do we need to specify 20 seconds because the area under the curve changes when you change your time interval so for example I

could Define my interval from 0 to time = 15 seconds and so now this is my area and now my area would be about a

positive 12 a and that would be my velocity after 15 seconds if your velocity is positive it means your position is increasing

meaning you have a velocity in the positive direction means you are uh traveling further forward in the positive direction with respect to your position and the same principle applies

to acceleration if your acceleration is positive it means that your velocity is increasing all right that's what we mean when we Define the relationship of a

rate of change given that acceleration is the rate of change of velocity if acceleration is positive velocity is increasing now common misconception that

we need to be aware of just because velocity is increasing doesn't mean it's positive so if I were to draw a graph of velocity that looks like

this where this is our x-axis the velocity is increasing for the entire time interval graph however the velocity is negative on this interval and

positive on that interval you are not allowed to say that just because your velocity is increasing means it's positive that's not true if it's

increasing all that's telling you is that your acceleration is positive it does not tell you if your velocity is positive uh from there let's move on to calculating average values so uh the

normal way that you would calculate an average is you would take all of the different components of a series like let's say you wanted to find the average amount of

apples in each hoop you would uh add up all all the apples you have and divide by the number of Hoops and that's how you would find the average number of apples in each hoop but you can't quite

uh take that approach here because uh if you're looking at a graph there is an infinite amount of times at which you could evaluate the graph you got time

equals 1.0 1.01 1.01 1.01 and so on and so forth to Infinity so the way we find average

values of our different quantities here position velocity acceler ation is we uh go by area so let me again give you a graph of

let's say velocity versus time so if my graph looked like this it was horizontal what does that tell me it tells me that

my velocity is not changing my velocity is constant at some uh value y if I were to find the area of this region

from 0 to let's say 8 seconds uh the area would be our x value times our y value okay which would be equivalent to saying base time height which you

probably more familiar with because this the x value is our base and the Y value is our height if I want to find the average value of my

function I need to figure out what Y is so I take area and I divide it by X area over my x value gives me my y

value okay now let's see how that is applied to non-constant graphs like this one that is a triangle here uh the value

of my velocity is not constant but I want to find its average value the average value would be like saying what value of y is this graph most

um centered around like what that's not the center Center is lower down right around there but where is my graph centered at okay and the way we find

that find that by this equality the area of our shape whatever it may be equals our x value multiplied by the average y

value this is always true the area under our curve is equal to the x value multiplied by the average yvalue in this case our yvalue was constant so our y

value was our average yalue in this case our y value is not constant so we would need to find the area of this shape area equals base *

height over two and then we would uh use this equation ax equals our average y value and we would simply divide by X so remember how

I said B * H is equivalent to x * y our area is equal to x * y / 2 ax = y / 2 therefore this y

representing our final y value up here for the case of a triangle y final / 2 = y average and you can do that for

any shape parallelogram trapezoid any shape you see on your Gra you would find the total area under the

curve you would find the total area under the curve and you would divide that by your x axis your x value your bounds in this case that was 0 to

8 so a / 8 in this case would have given you the average y value if this was 0 to 10 our a / 10 would have given us our

average y value that's how we find average value of any function you're given the area under the curve divided X gives you average velocity because uh this sometimes kids confuse this with

change in y over change in X which is slope slope is average acceleration if we take the slope of the

Velocity time graph and the slope of our position time graph is our average velocity okay okay so you can find it two ways if you wanted to find average

velocity you get area under the velocity curve you divide it by X that gives you average velocity or you find the slope of the position time curve and that

gives you average velocity slope of velocity gives you average acceleration slope of position often denoted as X is

equal to average velocity [Music] okay so and the reason position is often denoted as X is what I was hinting at earlier uh position can be expressed on

multiple axes so in this case there's only one uh direction there's only one dimension we can express position in that's the X Direction you can either go

forward in the X Direction which is positive or you can go backward in the X Direction which is negative on a coordinate plane that has an X and A Y

AIS you can go forward in the X direction and you can go forward in the y direction and vice versa okay so you can have a position defined

in the X direction or you can have a position defined in the y direction and a lot of the problems we're going to be looking at in kinematics are going to feature two axes

okay same principles exact same principles I've been describing to you until now except we express it on multiple axes which means I can have a displacement of positive one in the X

Direction and a displacement of positive one in the y direction okay they can be considered exclusively from one another meaning if

I only want to consider my displacement in the y direction it is positive one if I only want to consider my displacement in the X Direction it is still postive 1 now my displacement in the Y Direction

has become positive2 but my displacement in the X direction is still positive one the uh Central sort of idea topic that

we're going to be covering here in kinematics are the kinematic equations that you find on your equation sheet in the introductory video I had advised you to print out your equation sheet I

sincerely hope you did so because we will be referencing it a lot and I am going to assume that you have it with you when you are watching said videos okay so your kinematic equations are

going to be the equations at the very top of your equation sheet on the left side in the mechanic section let's write them out and let's start to tackle what

exactly they mean you're at the very top we have the Velocity in the X Direction equals the initial velocity in the X Direction devoted denoted by the

knot plus the acceleration in the X Direction time time okay so going back to our noine solve

let's pay attention to what values participate in this equation uh V subx that would be our

final velocity after some time has passed VX knot would be our initial velocity

and ax would be our acceleration now important note for all the kinematic equations the first three equations on your equation sheet big

note they assume a constant

acceleration what does that mean it's means if I were to graph acceleration verse time it would be a horizontal line it means my acceleration never

changes if you are dealing with a non-constant acceleration if you are dealing with an acceleration that is changing you cannot use these

equations however the equations do not work with a changing acceleration however there's a way to circumvent that

if you have a changing acceleration let's say your acceleration looks like this graph um you are allowed to use these equations if and only if instead of a

subx you sub in you substitute in the average acceleration only then do these equations begin to work okay and the

average acceleration can be used in all cases because in the case of constant acceleration Your Average acceleration equals your acceleration for all points but in the case of non-constant

acceleration uh you can only plug in the average acceleration into your uh into your kinematic equations okay and uh again T equals

time all right that's our first kinematic equation on the equation sheet so when would we use this kinematic equation we'd use it when we know three

but we don't know one it's that simple okay excuse me next equation would be

our position x uh that's again if you were to look on the right side of your equation sheet as as to see what each

letter is mapped to X is denoted as position specifically X is used to denote position on the X AIS in the the X

Direction so let's write out the equation your position in the X Direction your position in the x axis is equal to your initial position on in the

X Direction plus your initial velocity in the X Direction time time plus 12 your acceleration in the X Direction

time time squ in this equation we have a final position we have an initial position now it's important to note that

if we take our X final our final position uh in the X Direction minus our initial position in the X direction that gives us the change in

position okay that's what the Delta symbol means change in position okay so if I want to calculate displacement displac is a change in

position I can say that Delta X which is equivalent to displacement I can rewrite the equation as such I can subtract X knot from both

sides of the equation and I'm left with our my displacement which is equivalent to x - x = v the Velocity in the X

Direction initially plus 1 12 the acceleration in the X Direction time time [Music] squ okay so as I was alluding to

before this only looks at uh the kinematic variables in the X Direction alternatively you could rewrite it the same exact equation but in the y

direction which would be y equals your initial position on the Y AIS plus your velocity uh in the Y your

initial velocity in the y direction times time plus 1/2 your acceleration in the y direction time time squ okay so that is an important factor to

consider these deal with your uh acceleration velocity position on an axis okay so if uh let's say from your

perspective towards the the white board is the positive X Direction meaning if I'm moving backwards I'm moving in the negative X Direction all right and the

right where my hand is extending right now is going to be the positive y direction okay meaning if I am moving like this my position in the X direction

is not changing at all only my position in the y direction okay if I were to move diagonally like this then that would mean that both my X position and

my y position are changing so I would need to calculate both of these forms of the equation in order to solve for a variable at my final position

okay and now let's look at the final uh kinematic equation that we are given here which would be your velocity in a given Direction could be X

or Y y as I've just uh stated your velocity in the X Direction squar equals your initial velocity in the X Direction squared plus 2 * your acceleration in

the X Direction multiplied by x - x which I just stated up here is the same as your

displacement is the same as your Delta X so you could rewrite this as your velocity in a particular direction equals your initial velocity in that direction plus 2 * the acceleration in

that direction time Delta X or your change in position or your displacement I know that was a lot to take in right now let's work through uh one of the

most common problems we see on kinematics together so you guys can get a hang of uh how we are going to apply this unit so in order to tackle uh this

problem we're going to need to take one more look at vectors all right so the interesting thing about a vector is it can be what is so-called resolved into

components what the heck does that mean let me explain so uh like I said before a vector is Illustrated as an arrow okay now uh that Arrow could be moving

on what is not one of the coordinate axes so if this is my x axis and this is my y AIS my Vector is moving in both the X

Direction and the Y Direction all right which means it is said to have both X and Y components okay so with any Vector

such as this one we can construct a right triangle between this vector and the x-axis or we can construct a right triangle between the vector and the Y AIS I prefer it to look like this but

it's ultimately up to personal preference like uh I said in my introductory video vectors add tip to tail okay which means a vector that

looks like this is the sum of a vector along the X Direction and a vector in the y direction such

that if this is Vector a and this is Vector B this is Vector C so the purpose of resolving a vector into components is to find a and b

given C if let's say I am given that this Arrow represents velocity a will represent my velocity in the X Direction and B will represent my velocity in the

y direction such that if I add the two vectors I get my original given velocity Vector this is a simple right triangle problem that you likely covered in your

algebra class okay if I am given the original velocity Vector again the magnitude of the vector is equal to the length of the arrow so

if I'm traveling at 25 m/s it means the length of the arrow is 25 okay so I am given the length of the hypotenuse

C and I'm trying to find the length of one leg X and the length of the other leg y all right and I would be given an angle in this case

Theta all right we are going to whip out our sin cosine tan functions remember SOA s of theta equals our opposite over

our hypotenuse cosine of theta equals our adjacent over our hypotenuse tan of theta equals opposite over adjacent

okay so if I were to look at sign for example our hypotenuse is C that is known

our opposite in this case opposite the angle is y such that c * sin Theta =

Y and our hypotenuse again is C our adjacent to the angle is X such that c * cosine Theta = X so this shows you how

that given uh the magnitude of the vector C and given the angle of the vector Theta you would be able to find X and Y you would be able to find the

component X vector and the component y vector and of course to get from X and Y back to C you would simply do Pythagorean theorem A2 + B2 =

c² okay let's get to our problem if we have a ball and we launch it up at a speed of

20 m/s at an angle Thea of 30° relative to the

[Music] horizontal how far will the ball go so in this case I already discerned out all

of the values that you're given I told you the initial velocity I told you the angle feta and I told you what we're looking for we're looking how far will the ball go we're looking for our

display placement Delta X now here's uh where defining your AIS defining your zero you're defining your

origin becomes important okay if I'm looking for displacement Delta X recall that displacement equals our final position minus our initial

position if our initial position if we were to Define that as position zero if we were Define that as the origin then our initial position becomes zero

therefore we can simplify Delta X as to equaling our final position we can simplify displacement as being just our final position first part is we're going

to need to take this vector and we are going to need to resolve it we're going to need to break it up into our X and Y components because what am I looking for

I'm looking for displacement in the X Direction so I'm going to consult my kinematic equations recall that your kinematic equations are written on your

reference sheet top left hand corner first three equations I'm looking for what can give me displacement here this is the equation that can give me displacement so what do I need I need

velocity in the X Direction I need time and I need acceleration in the X Direction my acceleration in the X

Direction there is nothing that's going to be changing in my velocity so this is going to be a constant acceleration all right there's no friction and the problem is going to tell you that you

can ignore the effect of wind resistance okay so if there's no friction there's no wind resistance it means we have nothing propelling us forward in the X

direction we have just our initial velocity in the case of the y direction however there is something accelerating you downward it's called the acceleration due to gravity the

acceleration due to gravity is denoted G it is a constant and it is on your reference table equals 9.81 to make everyone lives easier on

any AP exam you take you are allowed to round g210 you are allowed to make that round and that's going to make your life so much easier okay in fact all of the

correct answers on the multiple choice are derived assuming that gal 10 so you will in fact have it easier if you do this I'm looking for displacement and I

want to find velocity in the X Direction I'm not directly given velocity in the X Direction but I can find velocity in the X Direction by resolving this into components uh recall that the S of our

angle gives us the vertical Vector so our velocity in the y direction initially equals 10

m/s and our velocity in the X Direction initially equals uh 10 rad 3 m/s let me do pull out my calculator so that gives us

17.32 m/s now I have velocity in the X Direction velocity in the Y um next thing I need is time T I have that's not given to me

so what I need to do I need to find a way to find time T I'm going to consult a different equation that I have this equation tells me that my final velocity in the X

Direction equals my initial velocity in the X Direction plus my acceleration time time T So if I were to try and use that in

the X Direction uh my final velocity in the X Direction and my initial velocity in the X direction will be the same because my

acceleration in the X direction is zero so that's not helpful to me however we can try it in the y direction uh let's write that out my

final velocity in the y direction equals my initial velocity in the y direction plus the acceleration in the y direction time time T I have my initial velocity

in the y direction I just derived it out right here I got that my acceleration in the y direction meaning y direction being top down guys what is the

acceleration that we all experience pulling us down it's going to be a acceleration due to gravity G and because the acceleration is pulling us in the negative y direction the acceleration is going to be

-10 our velocity uh in the y direction initially is -10 we want to find T do we have the final velocity in the y

direction well uh let's look at the path that this ball would follow if we launched it okay if the ball's path is

going to look like this it's going to look like an inverse Parabola if we were to go from point A to point B in sometime T it would take t/ 2 to get to

the middle right here and the reasons that's true is because we have a constant acceleration okay so if I know my final

velocity in the y direction is zero and it's zero because at the very top of the arc you're not traveling upwards anymore and at that

exact instant you're not really traveling downwards anymore so at the very top of the arc your velocity in the y direction is zero and we can use that to solve for

the half of the time so we can rewrite that equation as 0 = 10 - 10 T /

2 0 = 10 - 5 t t = 2

seconds okay now we've just solved for our time let me go back to this equation I'm looking for the displacement I have

my initial ual velocity in the X Direction I now have my time T I know that the acceleration in the X direction is zero because there's nothing pushing me along and I know my time T so now I

can write and solve this equation I think I just colored in my suit that's depressing anyway let's write it out our displacement Delta x equals v

x * t + 12 a sub X t^2 Delta X that's what we're solving for equals VX

17.32 m/s * time 2 seconds + 12 0 * T

^2 the 0 makes this whole term go to zero therefore Delta x equals our initial velocity in the X Direction 17.32 * 2 we're going to well I don't

need my calculator it's going to be

before it lands thank you guys for watching