### Video Transcript

In this video, we will learn how to
determine the speeds of objects relative to other objects.

To understand relative speed, let’s
begin with speed of a different type. Say that here we have a running
track with a runner moving on it. If the runner moves ahead at a
speed of three meters per second, that tells us that for every second of time that
passes, the runner moves down the track three additional meters. This means that the runner speed of
three meters per second is three meters per second relative to this track. The track, of course, isn’t
moving. And this means we can call the
runner’s speed of three meters per second the runner’s absolute speed.

The absolute speed of any object is
the speed of that object as it moves relative to something that doesn’t move. As we see here, the thing that
doesn’t move is the track. When we talk simply about the speed
of an object, often we mean its absolute speed. Compared to that moving object’s
motion, say, the motion of this runner, there’s something which is not moving
relative to that object. And this lets us know the moving
object’s absolute speed.

Now let’s say that our scenario
changes. We have the second runner moving
towards the first at an absolute speed of four meters per second. That is for every one second of
time this runner passes over four meters of the track. We know the speed of the first
runner compared to the track is three meters per second and the speed of the second
runner compared to the track is four meters per second. But what is the speed of this first
runner compared to the second runner? Finding this out means solving for
the relative speed of these runners.

The speed of one object relative to
another can be defined mathematically like this. It equals the change in distance
between those two objects divided by a change in time. Let’s apply this relationship to
our runners. We’re talking about a change in
distance over time. Let’s say that that amount of time
is one second. We’ve seen that in one second of
time this runner will move three meters along the track. Then over that same second of time
— from what we’ve called zero seconds up to one second — the second runner moves a
distance of four meters along the track. We know this because the second
runner’s speed relative to the track is four meters every second. For our runners then, over a time
interval of one second, their total change in distance is three meters plus four
meters.

Now the reason we use one second as
our change in time is because this second of time during which the first runner
covered three meters is the same second of time for which the second runner covered
four. It’s not two separate seconds, so
we don’t use, say, two seconds. Rather, it all happens in one
second of time and so that is our change in time. Three plus four is seven. So, the relative speed of our
runners is seven meters per second.

Now here’s something
interesting. This relative speed is the speed of
the first runner relative to the second runner, and it’s the speed of the second
runner relative to the first. Comparing either moving object to
the other, the relative speed between them is seven meters per second. Later on, we’ll get some more
practice using this equation.

For now, notice that we’ve called
this an equation for relative speed, but that we can still use this equation, even
if the two objects we’re considering aren’t both in motion. For example, if we wanted to solve
for the relative speed of the second runner compared to the track, then in that
case, over a time interval of one second, the change in distance between those
objects would be four meters. The speed of the second runner
compared to the track and the speed of the track compared to the second runner is
four meters per second.

As we said, because one of these
objects isn’t in motion — that is, the track isn’t moving — we call this the
absolute speed of the runner. An object’s relative speed can also
be its absolute speed if the thing its speed is measured relative to is not in
motion. All that to say we can use this
equation to solve for the speed between any two objects even if the objects aren’t
both moving. Let’s now get some practice working
with relative speed through some examples.

A blue object moves across a grid
of lines spaced one meter apart. The object moves for two
seconds. The arrows show the distance that
the object moves in each second. What is the speed of the object
relative to the grid lines that it crosses?

Looking at these grid spaces, we’re
told that the side of each square has a length of one meter. We’re also told that the arrows —
there’s one here and one here — show the distance that the object moves over each
second of time. In the first second, the object
moves one grid space. That’s one meter of distance. In the second second, it also moves
one meter. We want to find the speed of this
object relative to the grid lines. To do this, let’s recall that the
relative speed between two objects equals the change in distance between them
divided by the change in time.

In this scenario, our blue object
is definitely in motion, while our grid lines are not; they don’t move at all. So, the only change in distance
between our objects will be caused by the motion of our blue object. The amount of time over which the
blue object is in motion is one second plus one second. And we saw that in each of those
seconds the blue object moves one grid space or one meter. Therefore, the blue object has
moved two meters in two seconds. Its speed then is one meter per
second. This is the speed of the blue
object relative to the grid lines. And notice since the grid lines
aren’t moving at all, it’s also the absolute speed of the blue object.

Whenever we have two objects where
one object is moving and one is not, the relative speed between them is the same as
the moving object’s absolute speed.

Let’s look now at another
example.

A blue object and an orange object
move across a grid of lines spaced one meter apart. The arrows show the distances that
the objects move in each second. What is the speed of either object
relative to the other?

We’re told here that these grid
spaces each have a side length of one meter and that the arrows show the distances
moved by the orange and blue objects in each second of time. Knowing this, we want to solve for
the speed of either of these objects relative to the other. To get started, let’s recall the
mathematical expression for relative speed. The speed of one object relative to
another is the change in distance between them divided by the change in time.

For our blue and orange objects,
we’ll say that they had their original positions at a time of zero seconds. Then each object moved for one
second to reach its final position. The total change in time, then, is
just one second. That’s the time period over which
both objects were moving. Over that time, we see the orange
object moved one, two, three grid spaces; that’s three meters. Meanwhile, the blue object moved
one, two grid spaces or two meters. The change in distance between
these two moving objects is three meters plus two meters. We get a result of five meters per
second.

Note that because these objects are
approaching one another, their relative speed is how many meters closer one object
gets to the other object per second. If instead they have been moving
away from one another, their relative speed is how many meters away one object gets
from the other in each second of time. So, five meters per second is the
speed of the orange object relative to the blue object. And it’s also the speed of the blue
object relative to the orange object. That is, it’s the speed of either
object relative to the other.

Let’s look at another example.

A blue object and an orange object
move across a grid of lines spaced one meter apart. The arrows show the distances that
the objects move in each second. What is the speed of either object
relative to the other?

To find the relative speed between
these objects, let’s start by recalling the mathematical equation for relative
speed. For two objects, the relative speed
between them equals the change in distance between them divided by the change in
time. So, for our orange and blue
objects, we’ll first figure out how the distance between them changes. And then, we’ll see over what
interval of time that change occurs.

We see that the initial position of
the orange object is here and the initial position of the blue object is here. That is, at first, these objects
are separated by one, two, three, four grid spaces. Each grid space is one meter of
distance, so four grid spaces is four meters. Both of these objects then move and
so that they reach final positions at these locations. At the end of this time interval,
the orange and blue objects are one, two, three grid spaces apart. The change in distance between them
then is the larger distance minus the smaller one. All this change occurred over some
time interval.

We’re told that in our diagram the
arrows show the distances that the objects move in each second of time. Since there’s one arrow for each
object, we know that each object moved for one second. The total change in time then is
just the one second over which both the orange and blue objects were in motion. Four meters minus three meters is
one meter. So, the relative speed between
these objects is one meter per second.

Note that to calculate this
relative speed, we needed to subtract one distance from another. This happens whenever our two
objects are moving in the same direction. If they had been moving apart, we
could have approached the problem differently. All that said, the speed of the
orange object relative to the blue and the blue relative to the orange is one meter
per second.

Let’s look now at one last
example.

A yacht passes a small island upon
which a person is standing. A person on the yacht walks through
a doorway and then along the deck in the opposite direction to the motion of the
yacht. The person on the island sees the
person on the yacht move in the direction of the motion of the yacht. Which is greater, the speed of the
yacht or the speed of the person walking?

Looking at this picture, we see a
yacht moving along to the left, while a person on the yacht walks to the right. All this is observed by a person
standing still on this island. We want to know which speed is
greater, the speed of the person walking on the yacht’s deck or the speed of the
yacht itself. To answer this question, let’s
imagine watching the yacht through the eyes of the person standing still. At one instant in time, the yacht
is here and the person on the yacht is standing in the doorway. Then later, the yacht has moved
over here and the person has moved to the back of the yacht. The question is, which object moves
faster, the yacht in moving from here over to here or the person in moving from the
doorway to the back of the yacht?

We’ve seen that while the yacht is
moving to the left, the person on the yacht is walking to the right. Here’s what that means. Between these two instants in time,
if the person on the yacht is walking faster than the yacht is moving through the
water, then overall in the eyes of the person standing on the island, the person on
the yacht will move to the right. Again, that’s what would happen if
the person on the yacht was walking faster than the yacht traveled itself. On the other hand, if the yacht
moves faster than the person walks, then we would expect the person overall to move
to the left as the yacht moves.

The question is, which of these two
outcomes do we see? At the first instant in time, the
person on the yacht is here, while at a later instant that person is here. That means the person on the
yacht’s overall motion is to the left, even though they were walking to the right on
the yacht. We could say then that the speed of
the yacht has overcome the speed of the person; the speed of the yacht must be
greater. Our answer is that the speed of the
yacht is greater than the speed of the person walking.

Let’s now finish up our lesson by
reviewing a few key points. In this video, we learn that the
absolute speed of an object is the speed of that object relative to another object
that does not move. On the other hand, relative speed
is the speed between two objects that may both be moving. The equation for finding relative
speed is the change in distance between two objects divided by the change in
time. We can use this equation whether
both objects are moving, one is moving and the other is not, or even if both objects
are still. This is a summary of relative
speed.