Video Transcript
In this video, we’re going to learn
about position, distance, and displacement. What each term means, how they
compare to one another, and how to work with them practically. As we start out, imagine for a
moment that you are a world-class swimmer and are just finishing up a long career
involving world championships and Olympic medals. Over the course of your career, you
swam tens, if not hundreds of thousands, of laps in a pool. Your talent and ability has led you
to the top of the winners podium on more than one occasion. And tonight, at the end of your
long career, you’re being celebrated and inducted into the international swimmers
hall of fame.
During the evening’s events, the
master of ceremonies unveils a special banner honoring your lifetime
achievement. The MC explains that a careful
accounting has been performed of all of your swimming both training and competition
over the years. On this special evening, your total
career swimming displacement is being recognized. And it is zero meters! Wait, that can’t be right. The audience is surprised. Not least of all, you are. The master of ceremonies is
flustered. But it turns out, with a better
understanding of distance displacement and position, we can find that this result
does make sense. To get to the bottom of this, let’s
consider what the definitions of position, distance, and displacement are.
Say that we have a
three-dimensional coordinate system, 𝑥, 𝑦, and 𝑧. And imagine further that we have an
insect, such as a fly, moving around in this coordinate space. At any given moment, we can draw a
vector starting at the origin and going towards the fly’s position. That vector, which we can call 𝑝,
is known as the position of the object, in this case our fly, relative to the origin
of a coordinate axis we’ve defined. So that’s what the position vector
of an object is. It’s the object’s location relative
to some defined origin. And notice that position is a
vector quantity. Going back to our fly, we know that
flies often move around within a space. And this one is no different. It follows a path seemingly
randomly throughout the open space. If we take a snapshot of the fly’s
position at a second moment in time, we can draw in its position vector calling it
𝑝 sub two. And with these two vectors defined,
we now have a start point and an end point for a journey that the fly makes.
If we drew a vector from the fly’s
initial position to the fly’s final position, we can call that vector 𝑑. And it would represent the fly’s
displacement over this journey. That’s the second term whose
meaning we wanted to come to understand, displacement. And we see that displacement also
is a vector quantity. What about distance? What is the distance that the fly
travelled? To find that out, we need to follow
the fly’s path through every twist and turn that it takes. It’s that total path length which
is the fly’s distance travelled. We can call that capital 𝐷. Unlike position and displacement,
distance is a scalar quantity. It doesn’t have a direction
associated with it. And we can go further in showing
the difference between this scalar quantity distance and the vector quantity’s
displacement and position.
Notice that when it comes to
distance, the location of our origin, or our coordinate system in general, doesn’t
matter. The distance the fly travels is
independent of our coordinate system. On the other hand, the position of
the fly and its displacement are very much dependent on the location of our origin
and our coordinate system in general. One last thing about the difference
between distance and displacement. Imagine that our fly continued on
its way lazily buzzing about the room and finally ending up at the same location at
which we started. And we ask the question, what is
the fly’s distance travelled over this journey? To find that out, we would
calculate the total length of all of its travels.
But now, what if we ask what is its
displacement over the loop? When we recall that displacement is
equal to the vector difference between initial and final position, if initial and
final position of an object are the same, then that displacement vector is zero. So over this path, even though the
fly may have travelled a great distance, its displacement is zero. This is the answer to our question
of how a hall of fame swimmer could have a lifetime swimming displacement of zero
meters. Every time a swimmer swim a lap,
they start and end in essentially the same position. So their displacement in total is
zero. Now that we have an idea of what
these terms mean, let’s get some practice using them with a few example
questions.
On a journey, a cyclist rides
3.0 kilometers west and then turns around and rides 2.0 kilometers east. Assume that east corresponds to
positive displacement. What is the cyclist’s
displacement from the starting point of their journey to its end point? What distance does the cyclist
travel during their journey? What is the magnitude of the
cyclist’s displacement from the starting point of their journey to its end
point?
In this exercise, we want to
solve first for the cyclist’s displacement. Then we want to solve for the
distance that they’ve travelled on their journey. And finally, we wanna solve for
the displacement magnitude of the cyclist’s trip. Let’s start by drawing a sketch
of their journey. If the cyclist begins on their
journey at a location we call the origin, we’re told that they ride to the west
3.0 kilometers and then turn around and return to the east a distance of 2.0
kilometers. We want to solve for their
displacement. We can call it lower case
𝑑. The distance they travel on
their journey, we can call it capital 𝐷. And finally, we want to know
the magnitude of their displacement.
In order to solve for the
displacement the cyclist experiences over their journey, we mark out their
starting and ending locations along our axis. And we draw a vector starting
at the start point and ending at their ending location. That vector, joining their
starting and ending point, is their displacement. And looking at our distance
scale, we see that that displacement is negative 1.0 kilometers. So we record that value for
lowercase 𝑑. Notice that this answer shows
the vector nature of displacement. We’re told that movement
towards the east is movement in the positive direction. So our minus sign shows that
indeed displacement is a vector.
Next, we want to solve for the
distance the cyclist travelled on their journey. To figure this out, we move
alongside the cyclist all throughout their journey, recording each bit of
distance we cover. We see that on the first leg of
their journey, the cyclist moves 3.0 kilometers. Then on the second leg, they
move 2.0 kilometers. So the distance travelled is
the sum of these parts. So the cyclist’s total distance
travelled is 5.0 kilometers. And finally, we want to solve
for the magnitude of the displacement of the cyclist. We know that the magnitude of
any vector — in this case, the vector 𝑑 for displacement — is a positive
value. The magnitude of the cyclist’s
displacement is equal to the absolute value of negative 1.0 kilometers, or 1.0
kilometers. This is a measure of the
distance between the cyclist’s start point and end point.
Let’s now look at one more example
helping us understand the difference between distance, displacement, and
position.
Which of the following
statements comparing position, distance, and displacement is correct? An object may record a nonzero
distance while recording a displacement of zero. An object may record a distance
of zero while recording a nonzero displacement. An object may record a nonzero
distance while maintaining a position of zero. An object may record a nonzero
displacement while maintaining a position of zero.
In this exercise, we want to
evaluate each of these four statements and find which one of the four is
correct. Let’s begin with the first
statement. An object may record a nonzero
distance while recording a displacement of zero. That’s accurate. We can imagine an object moving
in such a way that it starts and ends at the same location. And yet, it’s travelled a total
distance in order to complete that journey. In that case, its distance
travelled would indeed be nonzero. But its displacement, that is
the separation from its start and end point, would be zero. We highlight this first
statement as correct. Even though we know this is the
right one, let’s continue on and look at the remaining statements. The second statement says that
an object may record a distance of zero while recording a nonzero
displacement. We know this statement can’t be
accurate because distance is always equal to or greater than displacement.
The third statement says that
an object may record nonzero distance while maintaining a position of zero. If an object maintains a
position of zero, that means that it doesn’t move at all. In which case, it couldn’t
record a nonzero distance. So we know this third statement
is not accurate either. And finally, the fourth
statement says an object may record a nonzero displacement while maintaining a
position of zero. Once again, if an object
maintains a position of zero, then its displacement as well as its distance
cannot be anything other than zero. This means it’s unable to
record a nonzero displacement. So this fourth statement is not
correct either.
Let’s summarize what we’ve learned
about position, distance, and displacement. Position, distance, and
displacement are all terms that describe object location as well as the past motion
of that object. Also, position and displacement are
vectors, while distance is a scalar quantity. This means that we can define
distance without reference to a coordinate frame or an origin. But position and displacement
require that framework. And finally, distance and
displacement do not equal one another, if the moving object’s direction changes. When an object moves while changing
direction, then the distance it travels will be greater than the magnitude of the
displacement it experiences.
