Hey it's professor Dave, let's do some
graphical analysis.
One of the best ways to do
quantitative analysis is to make graphs.
When we do this, we are plotting data
points that correspond to two variables
and in physics one of these will often
be time, so we can label the x-axis with
a t, for time, and we could make tick marks
for individual seconds or multiples of
10 seconds or whatever we want. On the
y-axis we will put some other variable
and right now we want to learn what
happens when this variable is position,
velocity, or acceleration. Let's go
through an example and generate all
three of these graphs. Say you are
walking on a path in the woods on a fine
autumn day. You walk the one-kilometer
path at a rate of 100 meters per minute
all the way to the end of the path, then
you sit and rest on a bench for another
10 minutes thinking about what to have
for lunch. Once you've got an idea you
get very excited so you begin to run
back the way you came at a rate of two
hundred meters per minute but after you
get halfway back you become tired and
you slow down to 50 meters per minute.
Let's represent all of this on a graph
of displacement versus time. If the point
where you started is the origin and this
point on the y-axis is 1000 meters where
the bench is, we can draw lines to
represent each section of your walk.
First you went the whole kilometer at a
speed of 100 meters per minute so over
10 minutes you went a thousand meters.
This line represents that activity
because it goes from the origin, the
beginning of the walk, to a point with
coordinates of 10 minutes and 1000
meters. Be sure to understand that this
line does not represent a path in two
dimensions. All the motion is along one
line, but you are also moving forward in
time and that's what the x-axis
represents. Now you are at the bench and
you sit and look at the trees for 10
minutes while thinking about snacks.
This means you will move forward in time
by 10 minutes but your position will not
change, so this is represented by this
horizontal line segment ending at a
point with coordinates of 20 minutes and
1000 meters. Then, very excited about your
pizza, you jog halfway back or 500 meters
at a rate of two hundred meters per
minute, so this should take 2.5 minutes.
This line segment will have a steeper
slope because you are covering more
distance in less time, and we can see
that we are now at coordinates of 22.5
minutes and 500 meters. Suddenly you get
a cramp and you have to finish the
journey at 50 meters per minute for the
last 500 meters which will take 10
minutes, leaving you at a point with
coordinates of 32.5 minutes and 0 meters,
back to the place where you started
but later in time. One thing we will want
to be able to do is to take this data
and draw a graph for the same activity
but with velocity on the y-axis. To do
this we just translate each line segment.
For the first bit we see that we went
1,000 meters in 10 minutes so that's a
velocity of 100 meters per minute, so we
can draw a horizontal line segment at
100 meters per minute extending until
the 10-minute mark, because you moved
with constant velocity during that time.
Then you were at rest for 10 minutes so
your velocity was zero. We can draw
another horizontal line segment for that
time period right on the axis. Then you
began moving in the negative direction
so you had a negative velocity. That
sounds strange but remember that
velocity is different from speed in that
we are also reporting direction so here
if we went 500 meters in the negative
direction over two-and-a-half minutes
you had a velocity of negative 200
meters per minute, which is down here
below the axis. Then for the last 10
minutes you had a velocity of negative
50 meters per minute. And there is our
graph of velocity versus time. Let's
point out a couple things right now.
First we notice that we only have
velocity data when the object is moving.
When at rest, the velocity is 0 because
there is no displacement over that time
period. Also this is quite over
simplified because it is impossible to
go from a standstill to some velocity in
an instant.
In reality there must be an acceleration
or deceleration when starting or
stopping so that means if we were trying
to be as accurate as possible we would
quickly ramp up to this velocity and
then ramp down to rest even if it's
over just a second or two.
This is important because we will now
want to graph acceleration over time, and
just the way that we can only have
velocity data when the object is moving
we can only get acceleration data when
velocity is changing so there will be a
positive acceleration as we just start
moving but it will then come to zero as
we walk with constant velocity, then
acceleration will be negative as we
slow to a stop. It will be negative
again as we begin to walk back and then
back to zero but then slightly
positive when we slow down our
movement in the negative direction, back
to 0 and positive again when we stop,
as we require positive acceleration to
stop moving in the negative direction. We
will only have a non-zero value for
acceleration when velocity is changing.
In this way every point on the
acceleration graph is telling us about
the slope on the velocity graph for the
same point in time, and every point on
the velocity graph is telling us about
the slope on the position graph for the
same point in time. If you are well
versed in calculus you know this means
that velocity is the derivative of
position with respect to time and
acceleration is the derivative of
velocity with respect to time, which
explains why we find instantaneous
velocity on a position curve by looking
at the tangent line. The slope of this
line gives us the velocity at that
instant, and the slope of the tangent
line on a velocity curve will give us
the instantaneous acceleration. We will
cover all kinds of calculus in the
upcoming mathematics course but for now
we are going to stick to physics without
complex math, since we are focusing on
basic concepts. All we want to come away
with now is the ability to draw and
analyze these graphs for position,
velocity, or acceleration, so let's check comprehension.
Thanks for watching, guys. Subscribe to my
channel for more tutorials, support me on
patreon so I can keep making content, and
as always, feel free to email me: