Understanding Linear Motion: Position, Velocity, and Acceleration Explained

hey everyone welcome to the first video on linear motion these videos will introduce us to

kinematics by starting with Motion in a straight line or one-dimensional motion then in later videos we'll cover emotion

in two dimensions so in this video we'll talk about position and displacement

velocity and acceleration these are the key elements that we'll need for learning

kinematics so before we begin what is kinematics well it's a part of physics that studies

the motion of objects without considering the forces that cause the motion

so basically kinematics focuses on how objects move or the way they move and the path that they take over time

we'll learn how to describe an object's motion using things like position velocity and acceleration and in

kinematics we can answer questions like where will an object be at a certain point in time what direction is it going

and how fast is it moving on the other hand Dynamics focuses on why objects move or what causes them to

move this includes things like forces energy mass and momentum when learning physics we usually start

with kinematics and learn how to describe the motion of objects first then in later sections we'll learn

Dynamics and what causes that motion so let's get into kinematics first we'll start with position and

displacement so position is pretty simple we're going to describe position as

where an object is at a single point in time for example we might say the position of

this calculator is on the desk but the calculator could be any place on this desk if we want to be more specific

we could say something like the calculator is on the left side of the desk

this is sort of how we're used to describing position in everyday life as another example here's a flag

attached to a flagpole how would we describe the position of the flag well here we might say the flag is up

or here we might say the flag is down again this is how we would normally describe position

but in physics we need to be extremely specific about where things are the laws of physics deal with exact

positions and exact times we can't plug things like up or down or the left side of the desk into our physics equations

so how do we be more specific well we use numbers if we were to hold up a tape measure so

the end is at the bottom of the flagpole we can measure exactly where the flag is by using numbers

and since all objects have some length or width we'll need to pick a point on the object to measure

most of the time we'll pick the middle so let's measure from the middle of this flag

now instead of saying up or down we can more precisely say the flag is four feet from the bottom

or the flag is two feet from the bottom now we're describing the exact position of the flag on this flagpole

also notice how we're measuring the position of the flag relative to the bottom of the flagpole where the

position would be zero we always have to measure position relative to some point

and we don't have to pick the bottom what if we measure the position of the flag from the top

now the top of the flagpole has a position of zero and if the flag was in the same position

as it is on the left now we would say the flag is four feet from the top

in both ways of measuring are fine all we need to do is pick our zero point that we're measuring from

now since the flag can only move up or down this flagpole in a straight line we're measuring position along one

dimension we can describe the position of this flag by using only a single number like four feet

one dimensional motion or linear emotion is what we'll be warning first so as we start to talk about the

position velocity and acceleration of objects I think it'll help if we stick with one real world example that we can

relate everything back to so let's use a car driving along a flat straight Road

just like we held the tape measure up to our flagpole to measure the position of the flag we're going to use this number

line to show the position of the car along the road in physics we represent the horizontal

position of an object using the variable X position could have any unit of length

or distance such as inches feet miles or kilometers but we'll be using the SI unit of

position which is meters or a lowercase M for short as a reference one meter is equal to

about 3.3 feet or about one yard so let's say the car starts at a position of 5 meters but then it drives

to a position of 15 meters this is what we call displacement displacement is the change in position

of an object here's our equation for displacement in this little triangle here is called

Delta it's a letter of the Greek alphabet and it means the change in something

so in this case it means the change in position the way we actually calculate

displacement is the final position minus the initial position let's take a closer look at this

so you'll notice in this equation we have the variable X which means position but we also have these tiny letters f

and I when we have a variable with a small letter or number at the bottom right the

larger wider is our variable and the small letter is called a subscript the variable X means position and the

subscript f means final so together as a whole they mean final position

and x sub I or just x i means initial position so what was the displacement of the car

the initial position or x i is 5 meters and the final position or XF is 15 meters

from our equation we see that the displacement is the final position minus the initial position so in our case that

would be 15 meters minus 5 meters which gives us 10 meters so when the car drove forward its

displacement was 10 meters so now that we have an idea of what position is let's take a look at how we

could graph the position of this car over time as it moves to do that we'll start by listing the

position of the car at different points in time as if someone was standing on the side of the road with a stopwatch

and recording the position of the car at different times so let's say the car is driving forward

like this at the very beginning the position of the car is zero meters

and after one second the car's position is 5 meters after two seconds the car's position is

10 meters and after three seconds the position is 15 meters

now we have a table showing the car's position at different points in time so what would a graph of the car's

position over time look like we'll start with an empty graph where the x-axis or the horizontal axis

represents time and the y-axis or the vertical axis represents the position of the car on

the road now it might seem weird that we're putting the car's horizontal position on

the vertical axis of the graph but the reason is we always put time on the horizontal axis

that will make more sense throughout the course but for now just know that we always put time on the horizontal axis

of a graph so how do we graph the position of this car over time

we start by plotting the points that we have in the table on the left when time equals zero seconds the position of the

car is zero meters so we'll plot that point on the graph where time is zero and position to zero

next when time equals one second the position of the car is 5 meters so we'll plot the point where time equals one and

position equals five we also have points for when time equals two seconds and the position is 10

meters and when time equals three seconds and the position is 15 meters

now that we have some data points on our graph we can connect the points with lines like this

and there we go for this car's motion we've graphed the position versus time and now we have another way to visualize

that motion next let's talk about velocity velocity is similar to speed but we'll

talk about why it's different from speed later on velocity is defined as the displacement

of an object divided by the amount of time it takes that object to travel which is the same thing as the change in

position over time there are some common units we could use to describe velocity or speed such as

miles per hour or kilometers per hour but the unit we use in physics the SI unit of velocity is meters per second

abbreviated as M over s for reference one meter per second is roughly 2.2 miles per hour or 3.6

kilometers per hour so we represent velocity using the variable V and we represent time using

the variable t so here's our equation for the average velocity of an object

we could say the average velocity equals the displacement over time or we could write out each of the

variables separately let's take a closer look at this equation

so V with the subscript AVG stands for average velocity XF is the final position x i is the

initial position TF is the final time and TI is the initial time

altogether this equation says that the average velocity of an object equals the final position minus the initial

position divided by the final time minus the initial time

another way to say it is the average velocity equals the change in position over the change in time

and here are the SI units for the variables in this equation time is in seconds

position is in meters and velocity is in meters per second an important thing when we use physics

equations like this one the math only works out if we plug in numbers using SI units we can only get velocity in meters

per second if we plug in position in meters and time in seconds of course you could use any units you

want like inches and hours but then your velocity will be in inches per hour and things could get a little crazy

especially if you're using multiple equations in a problem so just remember to stick with the SI

units when we're using equations and one more thing I want to note here is that this equation that we just

learned is the same thing as this equation you might see this equation in class or

on an equation sheet these are the exact same equation but the variables have just been rearranged

all right so let's do an example problem to try out this equation a car is driving along a straight highway

it travels 800 meters in 35 seconds what is the car's average velocity first let's draw a picture to see what

this looks like it's pretty simple we just have a car on a straight road and it travels 800

meters next we're going to write down what we know

for this problem we know that the change in the car's position or the displacement of the car is 800 meters

we also know that the change in time is 35 seconds next we'll list out the equations that

we might be able to use since this problem wants us to find average velocity we'll use the equation

that we just learned since we know the displacement and the change in time we can just use the first

part of our equation average velocity equals displacement divided by change in time

we would plug in 800 meters for the displacement in 35 seconds for the change in time

and don't forget that we always need to keep track of our units using our calculator we'll do 800 divided by 35

and we'll find that the average velocity equals 22.86 meters per second and just a side note sometimes we might

write our units using a fraction like this and sometimes we'll write it out all in one line by using a slash

so you might have asked why didn't we use the other part of our equation with the initial and final variables

well we can try that too here we're using the equation average velocity equals the final position minus the

initial position divided by the final time minus the initial time the question is where do we plug in our

800 meters in 35 seconds well since the problem doesn't say what the initial position of the car is or

what the initial time is then we're always going to assume that the initial values are zero if they're not given

in this case our picture might look a little bit different now we can see that the car has values

for its initial position and for its final position at the initial position we assume that

time equals zero seconds and the position is 0 meters at the final position time is 35 seconds

and the position is 800 meters now we can plug in 800 meters for the final position 0 meters for the initial

position 35 seconds for the final time and zero seconds for the initial time and don't forget your units

800 minus 0 is 835 minus 0 is 35. now what we're left with is the same equation as last time and we find that

the average velocity of the car is 22.86 meters per second so we could have done this either way

the thing to remember is that if we're not given an initial value we have to assume the initial value is zero

but what if we are given both the initial and the final values let's use the example from when we made

a table of a car's position at different points in time so here's the new problem

a car is driving along a straight highway while someone records the car's position at different points in time

that's where we get our table from what is the car's average velocity from when time equals 2 seconds to time

equals three seconds first here's a picture of this scenario next we're going to write down what we

know in this case we're given the table of the car's position at different times next we're going to write down the

equations that we could use to solve the problem but this time we're specifically given

an initial point and a final point so we'll have to use the second part of our equation

our initial point is when time equals two seconds and the car's position is 10 meters

and the final point is when time is three seconds and the position is 15 meters

now in our equation we can plug in 15 meters for the final position 10 meters for the initial position 3 seconds for

the final time and two seconds for the initial time 15 minus 10 gives us 5 meters and three

minus 2 gives us 1 second 5 divided by 1 gives us 5 so the average velocity of the car between these two points is 5

meters per second so those are ways we could use the equation for average velocity

so just like we did with position let's see if we can graph a car's average velocity over time

we'll use the same motion of the car moving from when we graph the position and then we can use the same data points

that we had for the times and positions of the car what we're going to do is calculate the

average velocity of the car as it travels between each of these positions using our equation for average velocity

between the first and second positions the average velocity equals 5 meters minus zero meters divided by one second

minus zero seconds which gives us 5 meters per second between the second and third positions

the average velocity equals 10 meters minus 5 meters divided by two seconds minus one second

and that gives us 5 meters per second again and last between the third and fourth

positions the average velocity is 15 meters minus 10 meters divided by three seconds minus two seconds

which again gives us 5 meters per second all right great so now let's graph that again the horizontal axis will represent

time but this time the vertical axis is going to represent the velocity of the car

now notice how each of these average velocities that we found do not correspond to a specific time point

we didn't calculate that the car was traveling 5 meters per second at exactly one second or two seconds

we only found the average velocity of the car as it traveled between two points

so instead of plotting points on our graph let's just draw lines for each of these time intervals

this isn't how you would normally graph things but we'll clear that up in a second

so as the car traveled from zero seconds to one second the average velocity was 5 meters per second

and the same thing is true from one to two seconds and two to three seconds

so here's a nice graph that lets us visualize the average velocities that we found

but what did we graph exactly we were given the car's position at different points in time

then we found the car's average velocity as it drove between each of these points then we plotted that average velocity

across each time interval and it appears that the car is traveling at a constant 5 meters per second the

entire time but do we know that for sure when you're driving your speed might

vary a little bit right is it possible that the car's velocity went up and down like this

but the average for each time interval was 5 meters per second that actually is possible but we don't

know because we didn't graph the car's true velocity at every single instant in time only the averages

so how could we do that instead well every car has a speedometer which tells the driver how fast the car is

moving usually it's in units of miles per hour or kilometers per hour but for this

example let's say the speedometer reads in meters per second what if one of the passengers in the car

watched the speedometer and wrote down the car's velocity at different points in time

at the start of this journey they noticed the speedometer reads 5 meters per second

at exactly one second they notice that it reads 5 meters per second again and at two seconds it reads 5 meters per

second and at three seconds it still reads five meters per second

now what we have is a table showing the car's true velocity at specific points in time

and this is what we call the instantaneous velocity that means the car's velocity at any

specific instant this is different than when we found the car's average velocity between time

points however since the car's velocity actually didn't change over time it turns out that the instantaneous

velocities are the same as the average velocities so now we can graph the instantaneous

velocity and each point in our table corresponds with one point on the graph at zero seconds the velocity is 5 meters

per second at one second it's still 5 meters per second and at 2 seconds and 3 seconds the velocity is still 5 meters

per second now we can connect our data points with a line and we have a graph of the car's

instantaneous velocity over time and if the person in the car had recorded velocity every half second we

could add more points to our graph and so on until we have a graph of the car's true instantaneous velocity at every

single point in time so here's the question what would the car have to do so that

the velocity graph increases over time like this well for that we would need acceleration

acceleration is defined as the change in velocity of an object over time acceleration might seem a little bit

less relatable than velocity but it's still something that most of us are used to

if you've ever heard someone say something like my car can go 0 to 60 in four seconds

well they're talking about acceleration what they're really saying is my car can go from zero miles per hour to 60 miles

per hour in four seconds that means the acceleration of the car is its change in velocity which is 60

miles per hour over the period of time which is 4 seconds

60 divided by 4 is 15. so the acceleration of their car is 15 miles per hour per second

that's the same as saying the car can go 15 miles per hour faster every second when the car is accelerating after every

second that passes the speedometer moves up another 15 miles per hour and this unit might look strange but

that's what acceleration is velocity miles per hour divided by time seconds

written another way the unit of acceleration is a unit of position per unit of time

and that whole thing per unit of time so what would the SI unit of acceleration be

well our SI unit of velocity is meters per second and our SI unit of time is seconds

so this unit would be meters per second per second mathematically that's actually the same

as meters divided by seconds times seconds which is the same as meters divided by a

second squared and that's our SI unit for acceleration meters per second squared abbreviated M

over s squared and we'll be representing acceleration with the variable a

so a car is one example of an object that can accelerate what's another example

well it turns out that if you drop an object like a ball it'll actually accelerate as it's falling

this is what we call the acceleration due to gravity we don't need to understand gravity yet

or why this happens but it's a good time to mention that all objects that are falling with gravity have the exact same

acceleration this acceleration due to Earth's gravity is 9.8 meters per second squared towards the Earth

and we represent this special value using the variable lowercase G we'll leave it at that for now and we'll

come back to it later so how do we calculate acceleration here's our equation acceleration equals

the change in velocity over the change in time you might notice it looks similar to our

equation for average velocity let's take a closer look so a stands for acceleration VF is the

final velocity v i is the initial velocity TF is the final time and TI is the

initial time so this equation says that the acceleration of an object is the final

velocity minus the initial velocity divided by the final time minus the initial time

another way to say it is the acceleration is the change in velocity divided by the change in time

and here are the SI units for the variables in this equation time is in seconds velocity is in meters

per second and acceleration is in meters per second squared also although it's not super important I

do want to mention that just like our equation for average velocity this here is really the equation for average

acceleration but we won't be dealing with changing accelerations in this course so the

acceleration at any time will always be the same as the average acceleration also it seems that most classes and

equation sheets just call this acceleration so that's what we're going to do in this course

and like before you might see this equation written a different way like this

these two are the exact same equation but with the variables rearranged in different ways

so let's try an example problem a car starts from rest when the driver hits the gas and after 4.5 seconds the

car is going 27 meters per second what is the car's acceleration so first we'll draw a quick picture just

to see what it looks like next we need to write down what we know the problem says that the car starts

from rest in physics when we say an object is at rest that means that it's not moving and

therefore its velocity is zero so we'll say that at the initial point when no time has passed that the initial

time is zero seconds and since the car started from rest the initial velocity is zero meters per

second we also know that at the final point the final time is 4.5 seconds and the final

velocity is 27 meters per second after that we write the equations that we might use for this problem and here

we're looking at our equation for acceleration since we have an initial and a final

velocity let's use the right part of this equation so acceleration equals the final

velocity minus the initial velocity divided by the final time minus the initial time

next we'll plug in the variables that we know the final velocity is 27 meters per

second and the initial velocity is 0 meters per second the final time is 4.5 seconds and the

initial time is zero seconds and don't forget to keep track of your units

I usually find it easiest to write it out like this with each number and its unit in parentheses so that I can keep

them together if this looks a little confusing here's just the math that we're doing

27 minus 0 divided by 4.5 minus 0. if we plug that into our calculator we find that the acceleration of the car is

6 meters per second squared we know the units for our answer because we know the SI unit for acceleration is

meters per second squared and we have followed the rule of only plugging the correct s i units into this equation

so you might have asked why did we have to use the right part of the equation for acceleration could we have used the

other part acceleration equals change in velocity over change in time yeah we definitely could have done that

too instead of using initial and final values for velocity and time we could have used change in velocity

and change in time we might have read the problem and thought well it starts at rest and

speeds up to 27 meters per second so the change in velocity must be 27 meters per second

and it takes 4.5 seconds to speed up so the change in time is 4.5 seconds that way works too and for this problem

it seems simpler but sometimes the car isn't starting at rest and the initial velocity isn't zero

and later on we'll learn that if the car is going the other way the velocity is actually negative so we would have to

write out initial and final velocities anyway just to keep track of everything so long story short writing out initial

and final values is a good habit to make sure we don't mess up but for simple problems like this one we don't need to

alright so just like we did with position and velocity let's try to graph the acceleration of a car over time

this is the new motion of the car like before we can imagine there's someone in the car looking at the

speedometer and writing down velocity of the car at different points in time and this is what we get at zero seconds

the velocity is zero and the car isn't moving at one second the velocity is five

meters per second at two seconds the velocity is 10 meters per second

and at three seconds the velocity is 15 meters per second now what we can do is calculate the

average acceleration of the car between each pair of time points using our equation for acceleration

and I'll do the math for us what we'll find is that the acceleration between seconds 0 and 1 is 5 meters per

second squared between seconds one and two it's five meters per second squared

and between seconds two and three it's still 5 meters per second squared so we can see that the velocity is

increasing over time but the acceleration of the car stays the same and that makes sense the car has some

acceleration and acceleration is the change in velocity over time so the car's velocity changes over time

so what would our graph of acceleration look like as always we have time along our

horizontal axis and now we have acceleration along the vertical axis for the period of zero to one seconds we

know the acceleration is 5 meters per second squared from one to two seconds it's five again and from two to three

seconds it's still five and there we go we have a graph of the car's acceleration over time

however like we covered with velocity this is just a graph of the car's average acceleration plotted across each

time period but could we do what we did with velocity instead of just calculating the

average could we measure the car's acceleration at different points in time actually we could there are things

called accelerometers which measure acceleration and interestingly enough you have one in your phone

so like we said this car has an acceleration causing the car to speed up and the car's velocity to increase

but the actual value of acceleration doesn't change over time in this course and in most courses this

is as far as we're going to go we won't deal with accelerations that change over time

but could we have an object with an acceleration that increases over time like this

Yeah we actually could just like velocity is the change in position over time and acceleration is

the change in velocity over time jerk or jolt is the change in acceleration over time

and it turns out that we could keep going I just learned from Googling this that the next steps would be a jounce or

snap a flounce or a crackle and a pounce or pop

this is not a joke jounce flounce and pounce and snap crackle and pop are all real physics terms look it up on

Wikipedia but yeah for this course we're only going to work with position velocity and

acceleration alright lots of stuff in this video let's do a recap

first we learned about position which has the SI unit of meters and displacement which is the change in

position we also learned how we can graph an object's position over time

next we learn about velocity which is the displacement over time or the change in position over time

the SI unit for velocity is meters per second and we were in the equation for average velocity

we also learned that we could measure and graph an object's velocity over time and last we learned about acceleration

which is the change in velocity over time the SI unit for acceleration is meters

per second squared and we learned the equation for calculating acceleration

and finally we learned how we can graph the acceleration of an object over time so thanks for watching I'll see you in

the next video and we'll learn more about linear motion