Video Transcript
A car travels six kilometers north
and then eight kilometers south. What is the car’s net displacement
to the north from its starting position?
Okay, so this is a question about
displacement. We have a car that we are told
travels six kilometers north and then eight kilometers south. We’re then asked to find what the
net displacement of the car is to the north relative to its starting position. Let’s begin by drawing out the
car’s journey. The car begins at some point which
we’ll mark with an 𝑥. It then travels north for six
kilometers. After this, it travels south for
eight kilometers, and so it ends its journey at this point here. We’re asked to find the
displacement of the car to the north.
We should recall that the
displacement of an object is the straight-line distance between the start position
and the end position of that object and that it includes the direction that this
distance is in. Since displacement has both a
magnitude and a direction associated with it, this means that displacement is a
vector quantity. Now that we have our definition of
displacement, we can draw an arrow onto our diagram to represent the displacement of
the car. We simply need to begin at the
starting position and draw our arrow pointing in a straight line to the ending
position. So the net displacement of the car
at the end of its journey is represented by this blue arrow that we’ve added to the
diagram.
Now it looks like this arrow is
pointing to the south, but we’re asked to find the displacement to the north. Luckily, this doesn’t present a
problem. Since north and south are exactly
opposite directions, we can think of south as being negative north. In fact, if we draw out the compass
directions, we see that it looks like a set of axes. And if this was a set of axes, we’d
be talking about positive 𝑦 and negative 𝑦. So thinking of south as negative
north does make sense.
Getting back to the question, to
actually calculate the car’s displacement like we’re asked, we see from our
definition of the displacement that we need to find the straight-line distance
between the start and the end points of the car’s journey. In our diagram, this distance is
represented by the blue arrow. We can see from our diagram that
the length of this blue arrow is given by the length of this arrow here which
represents the car’s distance traveled south minus this arrow here representing the
distance the car travels north. And since our blue arrow is
pointing in the south direction, this is the displacement to the south.
And so we have that the car’s
displacement to the south is given by the length of the southward arrow, which is
eight kilometers, minus the length of the northward arrow, which is six
kilometers. Doing the subtraction, we get a
value of two kilometers. But what we have found is the
displacement to the south, and what the question was asking us for is the
displacement to the north. However, recall that we already
said that south is equivalent to negative north, and so the displacement to the
north is given by the negative of the displacement to the south. In our case, we have that the car
is displaced two kilometers to the south, and so the displacement to the north is
given by negative two kilometers.
Since displacement is a vector
quantity, we need both the magnitude and the direction in order to define it. But the direction is already
implicitly defined by the fact that the question asks for the displacement to the
north, so we don’t actually need to quote a direction along with our answer. We simply need to take the care to
include the negative sign to indicate that the displacement is actually to the
south, in other words, a negative amount in the north direction.
And so our answer to the question
is that the net displacement of the car to the north from its starting position is
negative two kilometers.
