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In the last video, I said that
we started off with the change

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in distance, so we
said that we know

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the change in distance.

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These are the things
that we are given.

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We're given the acceleration,
we're given the initial

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velocity, and I asked you how
do we figure out what the

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final velocity is?

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In the last video-- if you don't
remember it, go watch

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that last video again-- we
derived the formula that vf

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squared, the final velocity
squared, is equal to the

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initial velocity squared
plus 2 times

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the change in distance.

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You'll sometimes just see it
written as 2 times distance,

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because we assume that the
initial distance is at point

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0, so the change in distance
would just

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be the final distance.

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We could write it either way,
and hopefully, at this point,

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you see why I keep switching
between change in

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distance and distance.

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It's just so you're comfortable
when you see it

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either way.

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This is for the situation
when we didn't

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know what the vf was.

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Let's say we want to solve
for time instead.

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Once we solve for the final
velocity, we could actually

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solve for time, and I'll show
you how to do that, but let's

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say we didn't want to go through
this step-- how can we

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solve for time directly, given
the change in distance, the

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acceleration, and the
initial velocity?

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Let's go back once again to
the most basic distance

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formula-- not the distance
formula, but how distance

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relates to velocity.

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We know that-- I'll write it
slightly different this time--

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the change in distance over the
change in time is equal to

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the average velocity.

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We could have rewritten this as
the change in distance is

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equal to the average velocity
times the change in time.

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This is change in time and
change in distance.

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Sometimes we'll just see this
written as d equals-- let me

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write this in a different
color, so we have some

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variety-- velocity times time,
or d equals rate times time.

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The reason why I have change in
distance here, or change in

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time, is that I'm not assuming
necessarily that we're

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starting off at the point
0 or at time 0.

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If we do, then it just turns out
to the final distance is

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equal to the average velocity
times the final time, but

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let's stick to this.

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We want to figure out time
given this set of inputs.

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Let's go from this equation.

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If we want to solve for time,
or the change in time, we

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could just could divide both
sides by the average

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velocity-- actually, no,
let's not do that.

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Let's just stay in terms
of change in distance.

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I've wasted space too fast,
so let me clear

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this and start again.

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We're given change in distance,
initial velocity,

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and acceleration, and we want
to figure out what the time

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is-- it's really the change in
time, but let's just assume

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that we start time 0, so it's
kind of the final time.

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Let's just start with the simple
formula: distance, or

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change in distance-- I'll use
them interchangeably, with a

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lower case d this time-- is
equal to the average velocity

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times time.

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What's the average velocity?

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The average velocity is just the
initial velocity plus the

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final velocity over 2.

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The only reason why we can just
average the initial and

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the final is because we're
assuming constant

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acceleration, and that's very
important, but in most

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projectile problems, we do have
constant acceleration--

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downwards-- and that's
gravity.

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We can assume, and we can do
this-- we can say that the

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average of the initial and the
final velocity is the average

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velocity, and then we multiply
that times time.

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Can we use this equation
directly?

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No. we know acceleration, but
don't know final velocity.

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If we can write this final
velocity in terms of the other

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things in this equation, then
maybe we can solve for time.

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Let's try to do that: distance
is equal to-- let me take a

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little side here.

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What do we know about
final velocity?

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We know that the change in
velocity is equal to

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acceleration times time,
assuming that time

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starts a t equals 0.

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The change in velocity is the
same thing is vf minus vi is

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equal to acceleration
times time.

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We know that the final velocity
is equal to the

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initial velocity plus
acceleration times time.

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Let's substitute that
back into what I was

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writing right here.

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We have distance is equal to the
initial velocity plus the

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final velocity, so let's
substitute this expression

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right here.

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The initial velocity, plus, now
the final velocity is now

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the initial velocity, plus
acceleration times time, and

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then we divide all of that
by 2 times time.

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We get d is equal to-- we have
2 in the numerator, we have 2

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initial velocity, 2vi's plus
at over 2, and all

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of that times t.

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Then we can simplify this.

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This equals d is equal to-- this
2 cancels out this 2, and

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then we distribute this t across
both terms-- so d is

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equal to vit plus-- this term
is at over 2, but then you

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multiply the t times here, too--
so it's at squared over

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2 plus at squared over 2.

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We could use this formula
if we know the change in

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distance, or the distance-- this
actually should be the

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change in distance, and the
change in time-- is equal to

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the initial velocity times time
plus acceleration times

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squared divided by 2.

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Let me summarize all of the
equations we have, because we

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really now have in our arsenal
every equation that you really

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need to solve one dimensional
projectile problems-- things

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going either just left, right,
east, west, or north, south,

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but not both.

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I will do that in
the next video.

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Let's summarize everything
we know.

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We know the change in distance
divided by the change in time

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is equal to velocity-- average
velocity, and it would equal

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velocity if velocity's not
changing, but average when

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velocity does change-- and we
have constant acceleration,

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which is an important
assumption.

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We know that the change in
velocity divided by the change

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in time is equal to
acceleration.

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We know the average velocity is
equal to the final velocity

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plus the initial velocity
over 2, and this assumes

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acceleration is constant.

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If we know the initial velocity,
acceleration, and

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the distance, and we want to
figure out the final velocity,

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we could use this formula: vf
squared equals vi squared plus

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2a times-- really the change in
distance, so I'm going to

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write the change in distance,
because that sometimes matters

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when we're dealing with
direction-- change in

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distance, but so you'll
sometimes just

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write this as distance.

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Then we just did the equation--
I think I did this

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in the third video, as well,
early on-- but we also learned

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that distance is equal to the
initial velocity times time

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plus at squared over 2.

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In that example that I did a
couple of videos ago, where we

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had a cliff-- actually,
I only have a minute

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left in this video.

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I will do that in the
next presentation.

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I'll see you soon.

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