### Video Transcript

In this video, we’re talking about
graphing velocity. Graphing velocity means plotting
out the velocity of some object against time. And we can do this for an object
that is or is not in motion. In order to understand how this all
works, let’s consider a 𝑣-versus-𝑡, a velocity-versus-time, graph and let’s
consider an object whose motion can be displayed this way as a horizontal line. Even though we haven’t made any
markings on the axes of this graph to indicate different velocities and times, we
can still say that whatever the velocity of this object, that velocity is staying
the same. It has a constant value all
throughout this time period.

In contrast to this, we could also
have an object whose motion can be shown this way on a velocity-versus-time
graph. We still have a straight line, but
now the slope or the gradient of this line is positive. This means that as time goes on,
the velocity of our object is increasing. So, its velocity at this point in
time, say, is less than its velocity at this point in time, which is less than its
velocity at this point, and so on. So then, in this instance, we have
an object whose velocity is staying constant, while, in this case, the object’s
velocity is increasing. And it turns out that there’s
another way that we can represent this same object motion. We can do this using what’s called
a displacement-versus-time graph. This is because, as we’ll see,
there’s a connection between an object’s velocity and its displacement.

And at this point, let’s recall the
definitions of those terms. Velocity indicates an object’s
speed and its direction. Therefore, velocity is a
vector. While displacement is the
straight-line change in an object’s position, displacement is also a vector. We can see that the way an object
moves is related to the straight-line change in that object’s position. This is how velocity and
displacement are connected. So, if we look back at this
velocity-versus-time graph and the fact that velocity is constant here, we can
translate this object motion to a displacement-versus-time graph.

First, we’ll make sure the start
point of our two lines matches up, and then we can say that if our object first
starts moving at this point in time, at that instant, we can choose its displacement
to be zero. An object’s initial displacement
could in general be positive or negative or zero, but, in this case, we’ll choose it
to be zero for simplicity. Therefore, we’ll put a dot on the
horizontal axis of our displacement-versus-time graph.

Now, we’ve noted that we don’t know
what this exact velocity value is. But we do see that it’s above the
horizontal axis, and, therefore, we can assume that this velocity is positive. And this means that, as time goes
on, this positive velocity will translate into a positive displacement on our
𝐷-versus-𝑡 graph. So, we could say that after some
amount of time has passed, our object’s displacement has increased to this positive
value.

And then, after more time has
passed, its displacement has gotten to here and so on until we get to the end of a
line that’s graphed on our velocity-versus-time plot. When we connect these points, we
see what the displacement-versus-time curve would look like for this particular
velocity-versus-time graph. Because the velocity here is
constant as well as positive, our displacement versus time is positive as well, and
it’s always increasing.

But then, what about a
velocity-versus-time curve like this, which already has a nonzero slope or gradient
to it? This also can be represented on a
displacement-versus-time curve. Once again, this object’s
displacement starts at zero. But then, instead of increasing at
a constant rate like the displacement did over here, in this case, our displacement
does increase. But it does so at an increasing
rate. Here’s how we can think of
this.

At the beginning of our
velocity-versus-time curve right here, we have a positive value to our object’s
velocity. Like we saw over here with our
constant velocity, this would indicate an object’s displacement that would steadily
increase. But rather than staying the same,
this object’s velocity increases itself at a steady rate. And because the object’s velocity
is always going up, the object’s displacement not only increases, but it does so at
a faster and faster rate.

The proof for that is that the
slope or the gradient of our displacement-versus-time curve is increasing over
time. That is, this graph is curving
upward. So then, if we have an object that
has a constant positive velocity, that indicates an object’s displacement which is
positive and always increasing at the same rate. On the other hand, if the velocity
of our object itself is increasing, also at a steady rate, then that object’s
displacement will go up. But it will do so at a higher and
higher rate over time.

Now, taking a look again at our
velocity-versus-time curves, we know that these don’t represent all the possible
graphs we could have. For example, what if an object’s
velocity versus time looked like this? That is, the object’s velocity is
zero all throughout this interval. If we think about an object whose
velocity is zero, that means it’s not in motion. It’s stationary. And that means that its
displacement doesn’t change in time. If the displacement starts out at
zero, then it will remain at zero. So, in this case, we have a
velocity-versus-time and a displacement-versus-time curve, which are the same. This only happens when velocity is
zero.

Now, if we look at this
velocity-versus-time curve, we see we have a positive constant velocity. And if we look at this one that we
just drew, we see we have a zero but constant velocity. But what if we had a
velocity-versus-time graph that showed a constant negative velocity? To show what that looks like, let’s
clear away our middle graphs, and then we’ll shift the velocity-versus-time curves
on the far-right graphs into the ones in the middle. And then, we’ll redraw these two
graphs on the right so they more easily admit negative values. Okay, so like we said, this graph
here is of a positive constant velocity, while this one is of a constant zero
velocity.

Let’s say that on this graph to the
far right, we have an object moving with a constant negative velocity. This is possible because we are
talking about velocities which are vectors and therefore can have positive as well
as negative values. So we have, in this case, a
constant and negative velocity of our object. Here’s how that translates to our
corresponding displacement-versus-time graph.

First, like before, we can say that
our object begins with a displacement of zero. It hasn’t moved at all yet, and so
its displacement from its start point is still zero. But then, our velocity-versus-time
curve tells us our object does start to move, but in a negative direction. This means our object will also
experience displacement in a negative direction. As time passes, this constant
negative velocity displaces the object farther and farther in the negative
direction, until, finally, at the end of its motion, our object ends up here. And if we connect these dots, we
know how the object’s displacement versus time curve.

Now that we have these three pairs
of graphs for a positive, a zero, and a negative constant velocity, respectively, we
can make an observation. Notice that for the
displacement-versus-time curve here, the slope or gradient of this curve is
positive. And this corresponds to a positive
velocity. Likewise, note that the slope of
this displacement-versus-time curve is zero. And that corresponds to a velocity
of zero. And lastly, that the slope of this
displacement-versus-time curve is negative. And that corresponds to a negative
constant velocity. We now see that it’s possible to
start with a velocity-versus-time curve and generate a displacement-versus-time
curve from it, like we’ve done here. But this process can also be
reversed. We could start with a
displacement-versus-time graph, calculate the slope of that line, and that will tell
us the corresponding velocity of our object. This is another way that velocity
and displacement relate to one another.

Now, we’ve mentioned that there’re
different kinds of velocity-versus-time curves, and, so far, we haven’t yet looked
at one of these curves that crosses over the horizontal axis. In other words, we haven’t yet
looked at a graph that looks like this. The velocity starts out positive,
then goes to zero, and then becomes negative. Now, earlier, we defined this term
velocity this way. We said that the velocity of an
object indicates the speed of that object, that is, the rate at which it’s
traveling, and that it also indicates the direction the object is moving in. Now, this is in contrast to the
speed of an object. An object’s speed simply indicates
its rate of motion, in other words, what we’ve called its speed under our velocity
heading.

So, what if we were to do this? What if we were to plot this
velocity-versus-time curve on this speed-versus-time graph? How would that curve look
transposed, we could say, from velocity to speed? Well, the important thing to see is
that while velocity indicates an object’s direction, speed does not. So, while the velocity of the
object we’re considering did reach negative values here, where it’s below the
horizontal axis, that’s not possible on our speed-versus-time graph. Speeds are always positive or zero,
never negative. And that goes back to the fact that
speed doesn’t tell us about an object’s direction. So, we say that however an object
moves or doesn’t move, its speed will always be greater than or equal to zero.

On our corresponding
speed-versus-time curve, we would start out at the same value as our
velocity-versus-time curve. And until we got to zero, these two
curves will look the same. But then, as we said, because
speeds are never negative, we won’t continue on following the 𝑣-versus-𝑡 line. Instead, what will happen, and this
is quite interesting, is that that line will be reflected about the horizontal
axis. What we mean by that is that, at
every time value, the distance our dashed line is from the horizontal axis is equal
to the distance our real speed-versus-time curve is from that same axis. It’s just that the
speed-versus-time curve has a positive value, whereas the dash line goes into
negative territory.

This difference between speed and
velocity comes down to the fact that velocity is a vector quantity while speed is a
scalar. Speed tells us how fast an object
is moving, but not its direction. And therefore, speed is always
positive or zero, while an object’s velocity does include direction information and
therefore can be less than zero. Now, velocity and speed aren’t the
only vector–scalar pair that we know of. We also know about displacement and
distance. Now, the way these two quantities
relate is like this. Say that we had an object that
traveled this path, starting here and ending here.

To calculate the distance traveled
by that object, we would measure out all the ground covered by the object on its
journey. That total amount would equal the
distance traveled. On the other hand, the object’s
displacement is equal to the straight-line distance that it travels from start to
finish. And since displacement is a vector,
the direction the object moves is recorded as well. So then, what if we switched out
our velocity-versus-time and speed-versus-time curves for displacement versus time
and then distance versus time? And now, it’s our object’s
displacement that goes from positive to zero to negative.

What, we wonder, is the
corresponding distance-versus-time curve? One way to help us, as we find out,
is to imagine a scenario that would generate this displacement-versus-time
curve. Let’s say that we have an origin
point for displacement, and that origin point is right here as an 𝑥. And let’s say further that we
decide that motion in this direction to the right from that origin point is what
we’ll consider positive motion. That, therefore, means that motion
the opposite way is considered negative. Now, if we have an object that
starts out here relative to our origin, then the initial displacement of that object
is positive, as we see on our displacement-versus-time curve.

But then, say that our object
starts to move toward this origin and eventually reaches it. On our graph, that corresponds to
moving to this point where the line crosses over the horizontal axis. If our object then keeps moving
past this origin, it then goes into negative values of displacement. And that’s represented by this
portion of our graph. Now, because the slope of our
displacement-versus-time curve is constant, that means the rate at which our object
moves, in this case, from right to left, is also constant. But anyway, now that we understand
how an object could travel to generate a curve like this, we want to translate this
curve to our distance-versus-time graph. We can start right here where the
object begins its motion.

Now, recalling that the distance an
object travels begins at its start point and then counts up as the object begins to
move, we can recognize that as our object just begins to move here, the distance it
has traveled at this initial instant is zero. So, our distance-versus-time curve
starts out on the horizontal axis, with our distance traveled being zero. But then, as our object moves
along, the distance that it has traveled steadily increases. The fact that we’re getting closer
to what we’ve called the origin doesn’t impact distance because distance is simply
an indication of what we could call the total ground covered. So, our distance-versus-time curve,
as our object moves towards the origin, would look like this.

Because distance is a scalar
quantity, it can’t be negative. And so, we see that these values
are all either positive or zero. And then, as our object journeys on
past the origin point into what we’ve called the negative direction, while our
displacement goes into negative territory, our distance traveled does not, but
continues on in a positive direction. At the end of our object’s journey
then, it has a negative displacement from what we’ve called the origin point over
here. While on the other hand, the total
distance it has traveled is positive. This is how
displacement-versus-time and distance-versus-time graphs can compare to one
another. And note that, as we made this
comparison, it was helpful to sketch out a scenario that could lead to the graph
that we were given, in this case, displacement versus time.

Let’s summarize now what we’ve
learned about graphing velocity. Starting off in this lesson, we saw
that, in general, the velocity-versus-time curve representing the motion of an
object could either indicate a constant velocity like here, or a velocity that
changes in time. We then saw that if we were to
translate such velocity-versus-time curves into displacement-versus-time curves,
then a velocity with a constant positive value leads to a displacement-versus-time
graph with a constant positive slope. While a velocity with a steadily
increasing value over time leads to a displacement-versus-time graph with a steadily
increasing slope.

Along with this, we contrasted
velocity-versus-time graphs with speed-versus-time graphs. And we saw that while velocity is a
vector and can therefore achieve negative values, speed, which is a scalar quantity,
is always positive or zero, never negative. And lastly, in a similar way, we
contrasted displacement, which is a vector, with the distance, which is a
scalar. And just like with velocity and
speed, we saw that while an object’s displacement can be negative, the distance it
has traveled is never negative, but is always either zero or positive. This is a summary of graphing
velocity.