might see in introductional physics class

but more importantly to see they are really just common sense ideas

So let's just start with a simple example

Let's say that and for the sake of this video

keep things that magnitudes and velocities

that's the direction of velocity etc.

let's just assume that if I have a positive number that it means

for example postive velocity that it means I'm going to the right

let's say I have a negative number we won't see in this video

let's assume we are going to the left

In that way I can just write a number down only operating in one dimension

you know that by specifying the magnitude and the direction if I say

velocity is 5m/s that means 5m/s to the right

if I say negative 5m/s that means 5m/s to the left

let's say just for simplicitiy, say that we start with initial velocity

we start with an initial velocity of 5m/s

once again I specify the magnitude and the direction because of this

convention here, we know it is to the right

let's say we have a constant acceleration

we have a constant acceleration 2m/s^2 or 2m per second square

and once again since this is positive it is to the right

and let's say that we do this for a duration

so my change in time, let's say we do this for a duration of 4

I will just use s, second and s different places

so s for this video is seconds

So I want to do is to think about how far do we travel?

and there is two things how fast are we going?

after 4 seconds and how far have we travel over the course of those 4 seconds?

so let's draw ourselves a little diagram here

So this is my velocity axis, and this over here is time axis

we have to draw a straighter line than that

So that is my time axis, time this is velocity

This is my velocity right over there

and I'm starting off with 5m/s, so this is 5m/s right over here

So vi is equal to 5m/s

And every second goes by it goes 2m/s faster

that's 2m/s*s every second that goes by

So after 1 second when it goes 2m/s faster it will be at 7

another way to think about it is the slope of this velocity line is

my constant accleration, my constant slope here

so it might look something like that

So what has happend after 4s?

So 1 2 3 4 this is my delta t

So my final velocity is going to be right over there

I'm writing it here because this get into the way of veloctiy

so this is v this is my final velocity what would it be?

Well I'm starting at 5m/s

So we are doing this both using the variable and concretes

Some starting with some initial velocity

I'm starting with some initial velocity

Subscript i said i for initial and then each second that goes by

I'm getting this much faster so if I gonna see how much faster have I gone

I multiply the number of second, I will just multiply the number second

it goes by times my acceleration, times my acceleration

and once again, this right here, subscript c saying that is a constant

acceleration, so that will tell my how fast I have gone

If I started at this point and multiply the duration time with slope

I will get this high, I will get to my final velocity

just to make it clear with the numbers, this number can really be anything

I'm just taking this to make it concrete in your mind

you have 5m/s plus 4s plus, I wanna do it in yellow

plus 4s times our acceleration with 2m per second square

and what is this going to be equal to?

you have a second that is cancelling out one of the second down here

You have 4 times, so you have 5m/s plus 4 times 2 is 8

this second gone, we just have 8m/s

or this is the same thing as 13m/s which is going to be our final velocity

and I wanna take a pause here, you can pause and think about it yourself

this whole should be intuitive, we are starting by going with 5m/s

every second goes by we are gonna going 2m/s faster

so after 1s it would be 7 m/s, after 2s we will be 9m/s

after 3s we will be 11m/s, and after 4s we will be at 13 m/s

so you multiply how much time pass times acceleration this is how much

faster we are gonna be going, we are already going 5m/s

5 plus how much faster? 13 m/s

so this right up here is 13m/s

So I will take a little pause here hopefully intuitive

and the whole play of that is to show you this formula you will see

in many physics book is not something that randomly pop out of there

it just make complete common sense

Now the next thing I wanna talk about is what is the total distance

that we would have travel?

and we know from the last video that distance is just the area under

this curve right over here, so it's just the area under this curve

you see this is kind of a strange shape here how do I caculate this area?

and we can use a little symbol of geometry to break it down into two

different areas, it's very easy to calculate their areas

two simple shapes, you can break it down to two, blue part is the

rectangle right over here, easy to figure out the area of a rectangle

and we can break it down to this purple part, this triangle right here

easy to figure out the area of a triangle

and that will be the total distance we travel

even this will hopefully make some intuition

because this blue area is how far we would have travel if we

are not accelerated, we just want 5m/s for 4s

so you goes 5m/s 1s 2s 3s 4s

so you are going from 0 to 4 you change in time is 4s

so if you go 5m/s for 4s you are going to go 20 m

this right here is 20m

that is the area of this right here 5 times 4

this purple or magentic area tells you how furthur than this are you going

because you are accelerating because kept going faster and faster and faster

it's pretty easy to calculate this area

the base here is still 5(4) because that's 5(4) second that's gone by

what's the height here?

The height here is my final velocity minus my initial velocity

minus my initial velocity

or it's the change in velocity due to the accleration

13 minus 5 is 8 or this 8 right over here

it is 8m/s

so this height right over here is 8m/s

the base over here is 4s that's the time that past

what's this area of the triangle?

the area of this triangle is one half times the base which is 4s

4s times the height which is 8m/s

times 8m/s second cancel out

one half time 4 is 2 times 8 is equal to 16m

So the total distance we travel is 20 plus 16 is 36m

that is the total I could say the total displacement

and once again is to the right, since it's positive

so that is our displacement

What I wanna do is to do the exact the same calculation

keep it in variable form, that will give another formula many people often memorize

You might understand this is completely intuitive formula

and that just come out of the logical flow of reasoning

that we went through this video

what is the area once again if we just think about the variables?

well the area of this rectangle right here is our initial velocity

times our change in time, times our change in time

So that is the blue rectangle right over here, and plus

what do we have to do? we have the change in time

once again we have the change in time

times this height which is our final velocity

which is our final velocity minus our initial velocity

these are all vectors, they are just positive if going to the right

we just multiply the base with the height

that will just be the area of the entire rectangle

I will take it by half because triangle is just half of that rectangle

so times one half, so times one half

so this is the area, this is the purple area right over here

this is the area of this, this is the area of that

and let's simplify this a little bit

let's factor out the delta t, so you factor out the delta t

you get delta t times a bunch of stuff

v sub i your initial velocity we factor this out

plus this stuff, plus this thing right over here

and we can distribute the one half

we factor the one half, we factor the delta t out, taking it out

and let's multiply one half by each of these things

so it's gonna be plus one half times vf, times our final velocity

that's not the right color, I will use the right color so you would understand

what I am doing, so this is the one half

so plus one half times our final velocity

final velocity minus one half, minus one half times our initial velocity

I'm gonna do that in blue, sorry I have trouble in changing color today

minus one half times our initial velocity, times our initial velocity

and what is this simplify do? we have something plus one half times something else

minus one half of the original something

so what is vi minus one half vi?

so anything minus its half is just a positive half left

so these two terms, this term and this term will simplify to

one half vi one half initial velocity plus one half times the final velocity

plus one half times the final velocity

and all of that is being multiplied with the change in time

the time that has gone by

and this tells us the distance, the distance that we travel

another way to think about it, let's factor out this one half

you get distance that is equal to change in time times factoring out the one half

vi plus vf, vi no that's not the right color vi plus vf

so this is interesting, the distance we travel is equal to one half of

the initial velocity plus the final velocity

so this is really if you just took this quantity right over here

it's just the arithmetic, I have trouble saying that word

it's the arithmetic mean of these two numbers, so I'm gonna define,

this is something new, I'm gonna call this the average velocity

we have to be very careful with this

this right here is the average velocity

but the only reason why I can just take the starting velocity and ending velocity

and adding them together and divide them by two since you took an average

of two thing it's some place over here

and I take that as average velocity

it's because my acceleration is constant

which is usually an assumption in introductory physics class

but it's not always the assumption

but if you do have a constant acceleration like this

you can assume that the average velocity is gonna be the average of

the initial velocity and the final velocity

if this is a curve and the acceleration is changing you could not do that

but what is useful about this is if you wanna figure out the distance

that was travelled, you just need to know the initial velocity

and the final velocity, average their two

and multiply the times it goes by

so in this situation our final velocity is 13m/s

our initial velocity is 5m/s

so you have 13 plus 5 is equal to 18

you divided that by 2, you average velocity is 9m/s

if you take the average of 13 and 5

and 9m/s times 4s gives you 36m

so hopefully it doesn't confuse you I just wannt show you

some of these things you will see in your physics class

but you shouldn't memorize they can all be deduced