Video Transcript
In this video, we will learn how to
distinguish between the speed and the velocity at which an object moves between
points. To get started, imagine we have an
object and that this object moves along a curved path. The object ends up here, and we can
say the distance the object traveled is the length of this curved arc. On the other hand, if we drew in
the object’s displacement, that would look like this.
Notice that displacement is a
vector. The direction of the arrow shows us
the overall direction of object motion. We also see that displacement is
measured as a straight-line distance between a starting and ending position. Displacement and distance are quite
different from one another. That difference is important when
we talk about the speed of this object compared to its velocity. Object speed is defined as a
distance traveled divided by the time taken to travel that distance. Just as distance is a scalar
quantity, so is speed. Velocity is different. It’s equal to displacement divided
by time. And since displacement is a vector,
velocity is too.
We can now compare the speed of
this object to its velocity. When we figure out its speed, the
amount of time we would use would be equal to the amount of time we would use to
figure out its velocity. Therefore, the difference between
this object’s velocity and speed comes down to the differences between its
displacement and distance traveled. Because our object is following a
curved path, the distance that it travels is greater than the magnitude of
displacement. In this particular case then, we
can say that our object’s speed is greater than the magnitude of its velocity.
And then, as we said, another
difference between speed and velocity is that velocity is a vector. Speed, which depends on distance
and time, is a scalar quantity. Whenever a moving object follows a
curved path, like our object does here, this statement will be true. The object’s speed will always be
greater than the magnitude of its velocity. On the other hand, if we have an
object that moves in a straight line, then the distance traveled by the object will
equal the magnitude of its displacement. For a straight-line path where an
object never changes direction, speed is equal to the magnitude of velocity. This is so because, as before, the
time we used to calculate these two terms is the same. Knowing all this about speed and
velocity, let’s look at some examples.
Two aircraft fly along the path
shown, each flying for the same amount of time. Which color arrow shows the path of
the aircraft with greater speed? (A) The blue arrow, (B) the green
arrow, (C) the speeds are the same.
In our diagram, we see these two
moving aircraft. The one following the blue arrows
moves like this, and the aircraft following the green arrow like this. To determine which aircraft has the
greater speed, let’s recall that, in general, the speed of an object equals the
distance that object travels divided by the time taken to cover that distance. In our question, we’re told that
each of these aircraft flies for the same amount of time. This means the aircraft with the
greater speed will be the one that travels the greater overall distance.
Our diagram shows us that each
aircraft moves up and to the right the same amount. This means we can take this green
arrow and slide it on top of the blue arrows, and it will look like this. It now becomes clear which arrow
involves the greater distance between the initial and final positions. The distance covered by the green
arrow is less than the distance covered by the two blue arrows. Since the blue arrow distance is
greater, so is the speed of the plane following this arrow. We choose answer (A). The blue arrow shows the path of
the aircraft with greater speed.
Let’s look at another example.
If an object moves in a straight
line at a uniform speed, which of the following is correct? (A) The speed becomes a vector
quantity. (B) The speed is the magnitude of
the velocity of the object.
Let’s say that here is our object
and that this is its path of motion. Traveling at a uniform or constant
speed, the object ends up here. So this is the distance the object
travels, while this arrow shows the object’s displacement. Because our object has moved in a
straight line, the distance and the magnitude of the displacement, the length of
this arrow, are the same. Mathematically, we can write that
this way. Distance equals the magnitude of
displacement.
Now, our second answer option,
option (B), talks about the speed of the object as well as its velocity. We can recall general equations for
speed and for velocity. The speed of an object equals the
distance it travels divided by the time taken to cover that distance, while
velocity, a vector quantity, equals displacement divided by time. In this equation for velocity, if
we took the magnitude of displacement, then on the left-hand side we would have the
magnitude of velocity. And now let’s recall that in our
particular scenario, with our object moving in a straight line at a constant speed,
the distance the object travels is indeed equal to the magnitude of its
displacement.
Looking at our equations, that
means we can say the magnitude of displacement here equals this distance here. Going further, for our object
moving in a straight line, the time used to calculate speed is equal to the time
used to calculate the magnitude of velocity. We can therefore write that our
object’s speed is equal to the magnitude of its velocity. This corresponds to answer option
(B). Note that option (A) can’t be
correct because speed is always a scalar, not a vector, quantity. For an object moving in a straight
line at a uniform speed, the speed of the object equals the magnitude of the
object’s velocity.
Let’s look now at another
example.
Two aircraft fly along the paths
shown, each flying for the same amount of time. Which color arrow shows the path of
the aircraft that flies between its initial and final positions at the greater
velocity? (A) The blue arrow, (B) the green
arrow, (C) the velocities are the same.
Comparing these two aircraft, we
see they each travel between an initial and a final position that are separated by
the same amount of vertical and horizontal distance. We want to know which one travels
between the initial and final positions at the greater velocity. Let’s remember a general equation
for velocity. It’s equal to an object’s
displacement divided by the time taken to be displaced some amount. The fact that we’re considering
displacement is key.
Displacement, we recall, only takes
into consideration the start and endpoints of an object’s motion. So the displacement of the plane
following the green arrow is given by this arrow, and the displacement of the plane
following the blue arrows is given by this arrow. These displacements are the
same. And since the planes both flew for
the same amount of time between their initial and final positions, we can say that
since each plane had the same displacement and the same amount of time elapsed, the
velocities of both aircraft are the same. As a side note, the speeds of these
aircraft are not the same. But since velocity depends on
displacement and both planes have the same displacement, their velocities are indeed
equal.
Let’s look now at one last
example.
An aircraft follows the curved line
shown. Which has the greater magnitude,
the aircraft’s speed or its velocity? (A) Speed, (B) velocity.
Let’s begin by thinking about the
general difference between speed and velocity. Speed equals the distance an object
travels divided by the time taken to cover that distance. Our aircraft covers a distance
equal to the length of the curved line. If we divide this distance by the
time it takes for the aircraft to move from its initial to its final position, we
get the aircraft’s speed. Velocity, on the other hand, is
equal to an object’s displacement divided by time.
Displacement can be different from
distance in two ways. First, displacement is a vector,
while distance is a scalar. Second, the magnitude of an
object’s displacement is equal to the straight-line distance between that moving
object’s initial and final positions. In other words, this pink arrow
represents our aircraft’s displacement. In this question, we’re to compare
the magnitude of our aircraft’s speed with the magnitude of its velocity. The magnitude of an object’s
velocity in general equals the magnitude of its displacement over time. For our aircraft, the magnitude of
its displacement is equal simply to the length of this pink arrow.
Now, here’s the question. Which is greater, the magnitude of
this displacement or this distance along the curved path? We can tell that the distance is
greater. And that means that our aircraft’s
speed is greater than the magnitude of its velocity. By the way, this is always true
when an object moves along a curved path. For our answer, we choose option
(A).
Let’s now finish our lesson by
summarizing a few key points. In this video, we saw that speed is
a scalar quantity. Written as an equation, speed
equals distance divided by time. In contrast, velocity is a vector
quantity. Written as an equation, velocity
equals displacement divided by time. This is a summary of velocity.
