I just want to follow up on the last video

Where we threw balls up in the air and saw how long they stayed up in the air

and we used that to figure out how fast we inicialy threw the ball

and how high they went in the air

and in the last video we did it with specific numbers

and in this video I just want to see if we can derive

some interesting formulas so we can do the computations

really fast in our brains while we're playing this game

on some type of a field and we don't necessarily have, have any,

any paper around

so lets say that the ball is in the air for delta T

delta T is equal to time in the air

time in the air

then we know that the time up is going to be half that

which is the same thing as the time down

the time up is going to be equal to delta T

we'll do that the same color

is going to be equal to the time in the air divided by two

so how, What was our initial velocity?

Well all we have to do is remind ourselves that the change in velocity

the change in velocity which is the same thing as the final velocity minus

the initial velocity

so the final velocity, remember, we're just talking about

half of the path of this ball so the time that get's released

and it goes and its, and it's going as kind of it's

maximum upward velocity it goes slower and slower and slower

all the way until it's stationary for just a moment then starts going down

again, remember the acceleration is costed downwards

this entire time so what is the final velocity if we just

consider half of this time well the time is zero so

it's going to be zero minus our

inicial velocity, minus our inicial velocity

when it was taking off

thats our change in velocity

this is our change in velocity, this is our change in velocity

is going to be equal to the acceleration of gravity

the acceleration of gravity

now negative nine point eight meters per second square

or the acceleration due to gravity

when an object is in freefall

to be technicaly correct times,

times, times the time we are going up

so times delta T up witch is the same thing

I won't even write it delta T up

is the same thing as our total time in the air

our total time in the air, divided by two and so we get we get, negative

the initial velocity, is equal to this thing divided be two is going to be

four point nine meters per second square

we still have our negative out in front

times our delta T, times our delta T, and remember this is our total time

in the air, not just the time up, this is our total time in the air

and then we multiply both sides times the negative and we get

our initial velocity, is just going to be equal to four point nine, four point nine

meters per second square times the total time, the total time

that we are in the air, or you could say, the or you could say

it's the, it's going to be nine point eight meters per second square times half

of the time that we're in the air, either of those will get you the same

calculation, so lets figure out our total distance,

or the distance that we travel in that first in the time of, so that will give us the peek distance, remember that distance

or should i say displacement in this situation, displacement is equal to

average velocity, average velocity times the change in time, the change in time that we care

about is the time of, so that is our delta T over two, our total time

our total time divided by two, this is our, this is our time of

time of, and whats our average velocity? well the average velocity

if we assume constant acceleration is your initial velocity plus your final velocity

over two, its really just the mean of the two things. well we know what our initial

velocity is our initial velocity is this thing over here, so this thing is

this thing over here, our final velocity, remember we're just talking about

the first half of the time the ball's in the air so it's final velocity

is zero we're talking when it gets to this peak point over here

its from two videos ago, that peak point right over there, so our average

velocity is just going to be, our average velocity is just going to be, this stuff divided by two

so it's going to be four point nine meters per second square, times delta T

times delta T over, over two, so this right here this is our average velocity

velocity average so lets stick that back over here

so our maximum displacement

our maximum displacement is going to be our average velocity, so that is going to be

four point nine meters per second square times delta T, times delta T, all of that over two

and then we multiply that again times the total times the time of,

so times delta T over two again, this is the same thing, and these are the same thing

and then we can simplify it, our maximum displacement is equal to

four point nine meters per second squared times delta T squared, times delta T squared

all of that over, all of that over four and then we can just divide four point nine

divided by four, four point nine divided by four is, what is it, it's one point

one point two, one point two, two, five I believe, let me just get my calculator out

I don't want to do that in my head, get this far and make a careless mistake

four point nine divided by four is one point two two five

so this is, so our maximum displacement is going to be one point two two five times our total

time in the air, total time in the air squared, witch is a pretty, witch is a pretty

straight forward, witch is a pretty straight forward calculation

so this is, this is our max displacement, kind of how far do we, how high are we getting

displaced, right when, right when that ball is stationary, or is, is , is no net velocity

just for a moment and starts decelerating downwards, so we can use that is a ball is in the air

for five seconds we can verify our computation from the last video

it would, our max velocity is one point two two five times five squared

which is twenty five, will give us thirty point six two five, that's what we got in the last video

if the balls in the air for, i don't know two point three seconds, so its one point two, two, five

times two point three squared, then that means that it went

six point four, eight meters in the air, so anyway I just wanted to give you a, a simply expression

that gives you, that gives you the maximum displacement from the ground,

assuming air resistance is negligible as a function of the total time in the air

Nice formula to calculate the displacement of the object in the air!