What we're gonna do when we are analyzing any problems
using this variable that I've just introduced, the positions and the velocities
is to look at an object in two snapshots, so it did something initially,
and then we will let it evolve on its own using the laws of physics
and then we take another snapshot we think about what is doing finally.
For the initial snapshot, what I do is I defined its position,
its velocity and its acceleration in that initial time.
What we do to denote that I'm talking about the initial time
is to put a little zero as you can see with these variables
below the letter, so we see an x-sub-zero and the v-sub-zero.
These are just telling you that this is our initial snapshot, our first look at the object.
The way we speak about this, because I don't wanna keep saying x-sub-zero or v-sub-zero
is to say x nought or v nought, and this just a way of speaking
so that we can quickly talk about these variables,
so you'll hear me say very often to the next few lectures x nought or v nought.
And so, this is the initial position, velocity, and acceleration of our object.
We can look at the difference snapshot of our object a little bit of time later,
if this was where it was initially and I let time evolve where it starts at 1 meter
and maybe, I'd have to take example values of quarter,
0.250 of a meter per second of motion, and say we give an acceleration
as we just saw of 0.5 meters per second squared and then let it time evolve,
say we let it evolve for three seconds.
Then we look at where the apple is, after we've let it evolve in time for 3 seconds
and then we'll look at the variables again there,
so the apple at this time has a new position, a new velocity, and a new acceleration.
And you can see what these are by looking at the apple,
at least the position because you can see where it is relative to our coordinates it's at position 4 meters.
If that's the case, then we can find the velocity and the acceleration using equations
that we'll come to later, but the important thing is that you look at these two boxes,
the initial box, and the final box and know that for any problem,
we can always take two snapshots and you can choose where those snapshots are going to be.
So, for example, I could have chosen two different times supposing I needed for a given problem,
to look at my apple non-initially in that three seconds, but initially, end only at two seconds later.
That is also something I can do, and we can look at the final or the new positions,
velocity and acceleration of our apple after only two seconds have evolved.
Looking at our graph on the bottom, you can see that the new position of our apple
is at 2.5 meters and has a new velocity, which we'll be able to solve using the equations
we'll get to soon, turns out to be 1.250 meters per second,
and in this case, has the same acceleration,
because we've just kept the acceleration consistent through this entire time interval.
For the next example, what I'd like to do is return to this basic picture
where we have initial and final separated by 3 seconds
and introduced some ways of talking about the change from initial to final and what that looks like.
So just taking this example like I said of 3 seconds,
there are 3 quantities we can talk about when we're discussing the change
from an initial position or initial set up to a final set up.
First, is how is the position changing, how did the position change from where it is now,
to where it was before. We call this the displacement,
and in fact anytime we're talking about a different or a distance
between two timeframe or two snapshots of an object,
you can always find the distance or the difference between initial and final,
by taking what we call a delta x, which is just a notation meaning that the change in x,
so when you see delta it's just telling you to change in x.
By taking the final position, minus the initial position
that might sound like a strange definition for change
which you might not be familiar with but you can see it works very well,
because if I ask you how far do this apple go,
when starting at 1 meter and going all the way to 4 meters, you would say it moved 3 meters.
But the way you find this mathematically, in case you were met with a much more complicated situation
than an apple moving by 3 meters, as you can take where it is finally,
and subtract where it was initially. And you can see as we hope
we get 4 minus 1 or 3 meters of displacement as this apple move off to the side,
but we don't need to just talk about the displacement in space,
we could also ask about what about the velocity of our object,
if this object is changing its velocity and it has an initial velocity
and it has a final velocity, what is the average velocity over this time interval.
The reason we say the word average here is simply because,
instead of asking its velocity which is changing moment, to moment, to moment.
I just want the average velocity over the entire time interval,
and you get this average velocity over this entire 3 seconds,
what I would do is take the displacement, the total change in position
and divide it by the total change in time, which in our case is 3 seconds.
Doing this we see that from our displacement of 3 meters
we have 3 meters divided by 3 seconds, so the total average velocity of our object is 3 meters
every 3 seconds, so 1 meter per second of motion.
Finally, as you might expect we can ask about the average acceleration of our object,
in this object as we saw we kept the acceleration the same the entire time,
we held it at 0.5 meters per second squared.
But we can still ask ourselves about the equation for the average acceleration
and it would be the exact same, it'll follow the same principles as the first two,
which is that you take your change in velocity and divide that by your change in time
and see how the velocity is changing per time.
The total changed in velocity we just found was 1 meter per second,
but the final velocity minus the initial velocity would be a simple way to do this
as well if those were the variables that you knew.
The final velocity we saw from our problem is 1.5 meters per second,
while the initial velocity was only, 0.250 meters per second
in the same time change of 3 seconds and so we can do this very simply,
subtract 1.750 minus 0.250 and see that we have as we hope and expected
an average velocity which is equal to the, our average acceleration, sorry,
which is equal to the acceleration we held our object at
through the entire interval, which is a half a meter per second squared.