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
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