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This video has to do with graphs of motion.
To understand that we need to understand how the graph is laid out.
For all these sorts of graphs
time is over here.
Time is the independent variable.
Time marches on with or without you.
Where you are
depends upon what time it is
but what time it is does not depend upon where you are.
So time will be the horizontal axis for all of these.
The vertical axis could be
the position,
could be the velocity,
could be the acceleration.
There is no way to properly interpret the graph if you do not know what that
axis is
so you need to look and be sure you know what this axis is all about.
We also need to understand the meanings of these things.
So x is the position of an object,
v is the velocity, a is the acceleration
and we need to know something about mathematical definitions for those
things.
X, simply the position on a number line.
Velocity
has to do with
how that position changes
with respect to time.
How quickly
that position is changing
if we look at acceleration
acceleration is
how
that velocity is changing with time
how quickly it changes as time goes on
so we need to know those things
where the time axis is
what might be on the
vertical axis and indeed what is in any particular graph on vertical axis
and we need to know these sorts of definitions over here
the rest of it is tied into the concept of slope
and slope
is equal to
a rise
over a run whatever that means well let's find out what that means
if we have the graph of something
and there's a line on it
we start at the left
most part on that line or we pick a place towards the left of the line
such as that
we then pick a place to the right
we start at the left most line and
think about going to the right
and as i go to the right what happens to my line? Oh,
my line went up
that's a positive slope
this is my rise
this is my run
take one divided by the other we get the slope
if on the other hand
it had looked something like this
i start at the left
have a point to the right
start at the leftmost point
move straight across to the right what happened to my line? Oh, it went
down. That is a negative slope
i have a negative rise in that case and my run is still over here
on the other hand
if we did something like this
we had a line like so we would start at the left go to the right
i would
move horizontally over until i got to my
my point here
and i see that it goes neither up nor down
this has zero slope. this would have zero rise.
Now one thing to keep in mind with slopes is that the slope is independent
of where are the line is.
in other words
it here's a graph and
i haven't written any labels on these axes because it doesn't really matter
what these are graphs of
this
over here is a positive slope
and it doesn't matter where that line is
here's a positive slope
here's a positive slope here's a positive slope here's a positive slope
here's a positive slope here's a positive slope
those are all
positive slopes does not matter where
those lines are on the graph
when you're looking at how sloped
they are
same sort of thing here if i have negative slopes
there's a negative slope there's negative slope
there, there, there. It doesn't matter where the line is
if it has a negative slope
that negative slope is independent of whether we were in a positive region of
the first quadrant second quadrant and so on
all that matters is the steepness
and
finally if we have a line with zero slope it doesn't matter where that line
is on the graph
all of those lines
have zero slope. So now
we can start looking more specifically
at some of these graphs
let's take a look at a graph of
position
and time
there's position
and there's time
let's say that that graph
looked like this
What is it saying?
it's saying that the object started in the positive region of the number line
and progressively went to
higher and higher positive numbers
in other words
if we drew our little number line off to the side over here
and these are positive numbers and
negative numbers here
here's my zero this started in positive numbers
and moved to bigger and bigger positive numbers it moved in a positive
direction
it has a
positive velocity
if we look at the slope of this line if i look at my rise over here
my rise is actually
a change in this vertical axis and this vertical axis is my x-axis
so this is a change in x
if i look at my run over here
my run is a change in the
time axis
my run is delta t
so if i go and look at my slope formula
i have slope
is equal to rise over run
well rise over run in this case is my delta x
over my delta t
but that's just the definition of a velocity so the slope
on this kind of a graph
is the velocity
and we can talk about instantaneous
and average in a bit
likewise if i make a graph
velocity
versus time
and let's say it looked like that
and i have my
rise
my rise in this case is a change in the velocity of the object
my run in this case
is a change
in the time just like it was before
so the slope
which is
again the rise of the run
the rise in this case is delta v change in the velocity
the run in this case is delta t
the change in the time
so the slope on this kind of a graph velocity and time
is actually
giving us the definition of acceleration
so the slope
on graph of position versus time
gives us the velocity
the slope on a graph of velocity versus time
gives us the acceleration
now we're ready to start interpreting some of those things