In the last video, I said that we started off with the change
in distance, so we said that we know
the change in distance.
These are the things that we are given.
We're given the acceleration, we're given the initial
velocity, and I asked you how do we figure out what the
final velocity is?
In the last video-- if you don't remember it, go watch
that last video again-- we derived the formula that vf
squared, the final velocity squared, is equal to the
initial velocity squared plus 2 times
the change in distance.
You'll sometimes just see it written as 2 times distance,
because we assume that the initial distance is at point
0, so the change in distance would just
be the final distance.
We could write it either way, and hopefully, at this point,
you see why I keep switching between change in
distance and distance.
It's just so you're comfortable when you see it
either way.
This is for the situation when we didn't
know what the vf was.
Let's say we want to solve for time instead.
Once we solve for the final velocity, we could actually
solve for time, and I'll show you how to do that, but let's
say we didn't want to go through this step-- how can we
solve for time directly, given the change in distance, the
acceleration, and the initial velocity?
Let's go back once again to the most basic distance
formula-- not the distance formula, but how distance
relates to velocity.
We know that-- I'll write it slightly different this time--
the change in distance over the change in time is equal to
the average velocity.
We could have rewritten this as the change in distance is
equal to the average velocity times the change in time.
This is change in time and change in distance.
Sometimes we'll just see this written as d equals-- let me
write this in a different color, so we have some
variety-- velocity times time, or d equals rate times time.
The reason why I have change in distance here, or change in
time, is that I'm not assuming necessarily that we're
starting off at the point 0 or at time 0.
If we do, then it just turns out to the final distance is
equal to the average velocity times the final time, but
let's stick to this.
We want to figure out time given this set of inputs.
Let's go from this equation.
If we want to solve for time, or the change in time, we
could just could divide both sides by the average
velocity-- actually, no, let's not do that.
Let's just stay in terms of change in distance.
I've wasted space too fast, so let me clear
this and start again.
We're given change in distance, initial velocity,
and acceleration, and we want to figure out what the time
is-- it's really the change in time, but let's just assume
that we start time 0, so it's kind of the final time.
Let's just start with the simple formula: distance, or
change in distance-- I'll use them interchangeably, with a
lower case d this time-- is equal to the average velocity
times time.
What's the average velocity?
The average velocity is just the initial velocity plus the
final velocity over 2.
The only reason why we can just average the initial and
the final is because we're assuming constant
acceleration, and that's very important, but in most
projectile problems, we do have constant acceleration--
downwards-- and that's gravity.
We can assume, and we can do this-- we can say that the
average of the initial and the final velocity is the average
velocity, and then we multiply that times time.
Can we use this equation directly?
No. we know acceleration, but don't know final velocity.
If we can write this final velocity in terms of the other
things in this equation, then maybe we can solve for time.
Let's try to do that: distance is equal to-- let me take a
little side here.
What do we know about final velocity?
We know that the change in velocity is equal to
acceleration times time, assuming that time
starts a t equals 0.
The change in velocity is the same thing is vf minus vi is
equal to acceleration times time.
We know that the final velocity is equal to the
initial velocity plus acceleration times time.
Let's substitute that back into what I was
writing right here.
We have distance is equal to the initial velocity plus the
final velocity, so let's substitute this expression
right here.
The initial velocity, plus, now the final velocity is now
the initial velocity, plus acceleration times time, and
then we divide all of that by 2 times time.
We get d is equal to-- we have 2 in the numerator, we have 2
initial velocity, 2vi's plus at over 2, and all
of that times t.
Then we can simplify this.
This equals d is equal to-- this 2 cancels out this 2, and
then we distribute this t across both terms-- so d is
equal to vit plus-- this term is at over 2, but then you
multiply the t times here, too-- so it's at squared over
2 plus at squared over 2.
We could use this formula if we know the change in
distance, or the distance-- this actually should be the
change in distance, and the change in time-- is equal to
the initial velocity times time plus acceleration times
squared divided by 2.
Let me summarize all of the equations we have, because we
really now have in our arsenal every equation that you really
need to solve one dimensional projectile problems-- things
going either just left, right, east, west, or north, south,
but not both.
I will do that in the next video.
Let's summarize everything we know.
We know the change in distance divided by the change in time
is equal to velocity-- average velocity, and it would equal
velocity if velocity's not changing, but average when
velocity does change-- and we have constant acceleration,
which is an important assumption.
We know that the change in velocity divided by the change
in time is equal to acceleration.
We know the average velocity is equal to the final velocity
plus the initial velocity over 2, and this assumes
acceleration is constant.
If we know the initial velocity, acceleration, and
the distance, and we want to figure out the final velocity,
we could use this formula: vf squared equals vi squared plus
2a times-- really the change in distance, so I'm going to
write the change in distance, because that sometimes matters
when we're dealing with direction-- change in
distance, but so you'll sometimes just
write this as distance.
Then we just did the equation-- I think I did this
in the third video, as well, early on-- but we also learned
that distance is equal to the initial velocity times time
plus at squared over 2.
In that example that I did a couple of videos ago, where we
had a cliff-- actually, I only have a minute
left in this video.
I will do that in the next presentation.
I'll see you soon.