as a function of time given constant acceleration and initial velocity

I want to plot displacement velocity, final velocity, and acceleration

all those function of time,

just so we really understand what's happening

as the ball is going up and then down

So we know this is our displacement function of time,

we know what our final veloctiy is going to be,

as a function of time we talked about in our last video

the final velocity is going to be the initial velocity

plus our acceleration times change in time

right if we start we some initial velocity,

and you multiply the acceleration with time

this part tell you how much faster or slower you are going to go

between initial velocity or that will be your I guess current velocity

the final velocity at that point in time

and of course the acceleration we know

our acceleration is pretty straigth forward

the acceleration due to gravity is just gonna be negative 9.8 m/s*s

once again the negative being the convention that

it is in the downward direction

our initial velocity is going to be upper direction 19.6m/s

so let's plot this out a little bit, let's plot these out

so the first graph I wanna do will be my displacement verses time

so this axis right over here is going to be time

and let me just call it change in time, actually I'll just call it time

let's just call it time and then this axis right over here

I will call it displacement and we leave some marker here

so let's say that this is 5m, 10m, 15m and 20m

and the time, this is zero, this is 1 this is 2 this is 3 & this is 4

seconds, so this is in second here

this is meters 5,10,15,20 and then so this is displacement

displacement graph, and I wanna do at the same time a velocity graph

let me draw my velocity graph like this, a little bit different

so this is the velocity will be going up and down, so we have positive

and negative values here, time will only be positive

so once again I care about 1s 2s 3s and 4s

and velocity I'm gonna call this, this is going to be 10m/s

10m/s this is 20m/s, this will be negative 10m/s

and this will be negative 20m/s, all of these are in m/s

this right here is velocity, this axis right here is time

so this is my velocity graph, and why don't we draw an acceleration graph

over here, to some degree it's easiest of them all

so the acceleration graph, so I will just do the fraction here and assume

that the acceleration is a constant, so this is 1s 2s 3s and 4s

and let's call this negative 10 and all of these are meters per s square

so we know our acceleration is 9.8 m/s*s so the acceleration

the entire time over the four seconds

the acceleration over the four seconds is going to be negative 9.8

it's gonna be a constant acceleration the entire time

let's figure out the displacment and velocity,

let me draw a table here, in one column I will do change in time

and you can sometime do that as time

let's figure out what our final velocity is

I should say the current velocity of velocity at that time

and in this column I will figure out what displacement is

and I will do it in time 0 1 2 3 4 so in 0 zero second gone by

when 1 second has gone by, when 2s 3s 4s has gone by

Actually let me call it change in time axis

this is essentially how many seconds it has gone by

So this is my change in time, let me make it clear that this graph here

is acceleration graph acceleration I will put it on the screen

Alright, so let's fill these things out

so times zero, what is our velocity?

well if we use this expression right here time zero or delta t equal zero

this expression right here is gonna be zero, and it's just going to be

initial velocity, in the last video we gave our initial velocity

is going to be 19.6m/s, so it is going to be 19.6m/s

I will plot that over here, time zero, it's going to be 19.6 m/s

what is our initial displacement at time zero?

Our change in time zero, so you look at this up here

delta t is zero so this expression here is going to be zero

so we haven't done any displacement yet when no time has gone by

So we have done no displacement, we are right over there

now what happen after 1s? 1s has gone by

what is now our velocity? well our initial velocity right here is 19.6m/s

that was a given, and our acceleration is negative 9.8m/s*s

so it's negative right over there and you multiply that by delat t

in every situation, so in this situation we are gonna multiply it by 1

delta t is 1, so you have 19.6 minus 9.8 that gives exactly 9.8m/s

and the unit we got cause we multiply here is second

so its give us meters per second

19.6m/s minus 9.8m/s one of these seconds go away multiply by second

give you 9.8m/s so after 1s our velocity is now half of what it was before

so we are now going 9.8m/s, let me draw a line here

9.8m/s, now what is our displacement? so you look up here

Let me rewrite this displacement formular here with all the infomation

So we know that displacement is going to be equal to initial veloctiy

which is 19.6m/s

now I won't write the units here just for the sake of space

times change in time, times our, use the same color to see what is what

times our change in time, plus one half let me be clear one half

times negative 9.8m/s*s so one half times a is going to be

I rewrite this right over here cause this is gonna be negative 9.8m/s*s

times one half, so this is going to be negative 4.9

All I did is one half times negative 9.8 over here

It is important that is why the vector quantity is start to matter

because if you put a positive here you wouldn't have the obeject slowing

down as you went up, because you will have gravity somehow accelerating

as it went up, but it's actually slowing it down

it's pulling it, it's accelerating it in downward direction

so that's why you have a negative right over there,

that was out convention at the beginning

of last video, up is positive, down is negative, so let's focus

So this part right over here, negative 4.9 m/s*s times delat t square

times delta t, times delta t square, this will make it a little bit easier

Although it still, let me get the calculator out

So when one second has past, I let my trusty TI-85 out now

when 1s past it's 19.6 times 1,

well that's just 19.6 minus 4.9 times 1 square

So that's just minus 4.9, mius 4.9, gives us 14.7 meters

So 14.7 meters So after 1s, the ball has travel 14.7 meters in the air

So that's roughly over there, now what happen after 2s?

I'll do the same agenda, so after 2s,

our velocity is 19.6 minus 9.8 times 2, times 2 this is 2s has gone by

well 9.8 times 2, 9.8 times 2 square seconds gives us 19.6 m/s

so these just cancel out, so we get out velocity now is zero

So after two seconds our velocity now is zero

let me make it so this thing should more look like a line

I don't get a sense, so this is let me just draw the line like this

so our velocity now is zero after 2 seconds

what is our displacement?

So literally we at the point with the ball has no velocity

exactly two seconds, it kind of gone up,

and right for that exact moment of time it's stationary

and then what do we have going on in our displacement?

We have 19.6 let me get the calculator out for this

We can do it by hand but for the sake of quickness

19.6 times 2 minus 4.9 times 2s squared, this is 2s squared

so that's times four, so that gives us 19.6 meters

So we have 19, we are at 19.6 meters after 2 seconds

we are 19.6 meters in the air

now let's go to 3s, so after 3s, our velocity now

it's 19.6 m/s minus 9.8 times 3, we could do that in our head but just

verified it for us, let me get the calculator out

it's 19.6-9.8*3, that gives us 9.8 m/s, negative 9.8m/s

So after 3s, our velocity is now negative 9.8 m/s, what is that mean?

it's now going in downward direction, 9.8 m/s,

so this is our velocity graph, and what is our displacement?

So once again, let's get the calculator out

you are getting the hang of this

at anytime I encourage you to pause it and try it for yourself

So now what is this is, Okay I'm looking at the displacement

Our displacment was delta t with 3s 19.6*3 - 4.9 times

so this is delat t so this is 3s

we are talking about delat t, our change in time is 3s so that's square

So times 9 and that gives us 14.7 meters, so 14.7 meters

So after 3s we are 14.7m again, the same position with 1s

but the difference is now we are moving downard

over here we were moving upwards

and finally what happen after 4s? Or what's our velocity?

Let me just get the calculator out

or you might be figure this out in your head

Our velocity is going to be 19.6 - 9.8 times 4s

just minus 19.6 m/s

So we are going a magnitude of velocity which is the same initially

threw the ball except now it's going at opposite direction

it's now going downward, so it's now going downward

and what is our displacement? get the calculator out

So we have out displacement is 19.6 times 4, 4s has gone by

minus 4.9 times 4 square which is 16 times, which is equal to zero!

the displacement here is zero!

We are back on the ground

So if you plot the displacement, you will actually got a parabola

a downward opening parabola that looks something like this

I best draw it relatively neatly

So my check to do it better than that

Dotted line, dotted line is always easier to adjust than its streamed

so if you plot displacement verses time it look something like this

it's velocity just downward sloping line, and the acceleration is constant

and the whole reason why I wanted to this is I wanted to show you

that velocity the whole time is decreasing at constant rate

and that make sense because that's the rate which the velocity increases

and decreases as the acceleration,

and the acceleration base on our convention is downward

so that why it's decreasing, we have a negative slope here

we have a negative slope of negative 9.8 m/s*s

and just to think about what's happening for this ball

I know this video is getting long as it goes

I'm gonna draw the vectors for velocity

So I'm gonna do that in orange, or maybe I will do that in blue

So velocity in blue, so right when we start,

it has a positive velocity of 19.6m/s,

so I will draw a big vector like this

19.6m/s that's velocity, but after 1s is 9.8m/s so half of that

so then its maybe would look something like this

9.8m/s, then at the speed right over here has the velocity of zero

then as you go to 3s, the magnitude of the velocity is 9.8m/s

but it is now downwards, so it looks like this

and then finally right when it hits the ground,

right before it hits the ground it has a negative velocity of 19.6m/s

So it would look like this, roughly like this

If I use the same scale over here,

but what was the acceleration the entire time?

well the entire time the acceleration is negative

It's 9.8m/s*s I will do that in orange

So the acceleration over here

Negative, now I wanna do that in orange

The acceleration is negative 9.8m/s*s

The acceleration is constant the entire time

So last one is negative 9.8m/s*s

it does not change depending on where you are

in the curve when you are near the surface of the earth

So hopefully that clarify the things a little bit and gives you

what happen when you throw up a projectile into the air