Video Transcript
In this video, we will learn how to
distinguish between constant speed and average speed. Let’s start by thinking about what
it means for an object to move with constant speed. Say we have an object here and that
in front of this object we have a series of distances marked out. At zero seconds of time, our object
is right here. But then say that at one second of
time our object is here. At two seconds, it’s here. Here it is at three seconds and
then four seconds and five seconds. So, for every second of time that
passed, our object moved an additional distance of one meter. Whenever a moving object covers
equal intervals of distance over equal intervals of time, we say that it’s moving at
a constant speed; that is, it’s a speed that doesn’t change.
In real life, though, we know that
many moving objects don’t move at a constant speed. Say we have a second object
here. And if we let this object move for
one second of time, we find its new position is at one meter of distance. But then after two seconds have
passed, the object is up here at four meters. And then finally, we’ll say that at
three seconds of time, the object is at five meters. In this case, equal intervals of
time, one second, two seconds, three seconds, do not correspond to equal intervals
of distance. We know then that this second
object is not moving at a constant speed. But we can still understand
something about the rate at which it moves.
We can do this by calculating
what’s called the average speed of this object. We do this by combining all the
distances this object moved over each one of the one-second time intervals for which
it was in motion. We’ll start with this first
interval from zero seconds to one second. The interval of time is one second
long, and over this time our object moved a distance of one meter. We then move on to the next time
interval. When we calculate average speed, we
keep what we could call a running total of all of the times and all of the
corresponding distances traveled. This second time interval, like the
first, is one second long. But over the second interval, the
object moves from one meter up to four meters. That’s a total distance of three
meters.
We then move on to our last
interval. And here also we’ll add that last
interval of time and the last amount of distance. Over this last one-second interval,
our object moves a distance of one meter, from four meters to five meters. If we add up all the values in the
numerator and all those in the denominator of this fraction, we get one plus three
plus one, or five meters, and one plus one plus one, or three seconds. And notice that this result has
units of meters per second. Those are the units of speed. The average speed of this second
object is five-thirds meters per second.
We can use this example to write a
general equation for calculating the average speed of an object. If we call the average speed of an
object 𝑣, then we saw that this is equal to the total change in the object’s
distance traveled divided by the total change in time passed. Whenever we see an equation that
involves a quantity that changes, that change is often represented using this
symbol. The name for this symbol is Δ. It’s a letter in the Greek
alphabet. In the case of average speed, since
we’re talking about a change in distance being divided by a change in time, we can
write this using our Δ symbol. It equals Δ𝑑 over Δ𝑡. The average speed of some object
equals the total change in distance traveled by the object over the change in time
during which that object was moving.
Since this is an equation for
average speed, we can use it even when our object is not moving at a constant speed
like it was here. In fact, it’s instances like those
where average speed is especially useful to know. This equation can be used to
calculate nonconstant as well as constant speeds. Knowing all this, let’s get some
practice working with average speeds through a few examples.
A blue object and an orange object
move across a grid of equally spaced lines. Both objects move for five
seconds. The arrows show distances moved
each second. Which color object has the greater
average speed?
We see here these blue and orange
objects moving across a grid of squares. We’re not told how long each side
length of a square grid is, but we do know that this distance is the same as the
square is wide. We can assume then that all of the
sides of these grid squares have the same length. For these two objects, each arrow
shows how far that given object moved every second. So, for example, the orange object
over the first second moved this distance; then over the second second, this
distance; over the third second, this distance; and so on. Over the entire five-second
interval, we see the orange object covers one, two, three, four, five grid
squares. It does this by moving one grid
square for every second of time.
For the blue object on the other
hand, over the first second, it moves one grid square and then the same thing over
the second second. But over the third time interval,
we see it covers two grid squares. And then over the fourth and fifth
time intervals, it covers one-half of a grid square. Like the orange object, over five
seconds, the blue object covers one, two, three, four, five grid squares. This fact is important for
comparing the average speed of these two objects. In general, the average speed of an
object 𝑣 equals the change in distance that that object experiences divided by the
change in time over which the object was moving.
To compare the average speeds of
our two objects then, we’ll look at the total amount of distance each one moved
divided by the total amount of time each one was moving. For our orange object, we saw that
it moved five spaces in five seconds of time. The blue object too covered a total
distance of five spaces in five seconds. Even though the orange object did
this by moving at a constant speed while the blue object did it moving at a
nonconstant speed, the average speed for each one over the entire journey is the
same. For our answer then, we’ll say that
both objects have the same average speed. This is because, on average, each
one moves the same amount of distance over the same amount of time.
Let’s look now at another
example.
The toy car shown was traveling at
a uniform speed before we started to measure its speed by recording its position
each second. What was the average speed of the
car during the time that the speed was measured?
Looking at the diagram, we know
that up until this moment when the measuring began, the car was moving along at a
uniform, or constant, speed. That means it was traveling equal
intervals of distance over equal intervals of time. But let’s look to see what happens
after we start measuring the car’s motion. After one second of time, the car
has moved one meter of distance. But then look at this. After two seconds of time, the
car’s position is here, which is greater than one meter of distance from its
position after one second. And then over the last second of
time measured, the car moves some distance so that its total distance from when the
measuring started is one meter plus one meter plus one meter, or three meters.
Over the three seconds of time for
which the car’s motion was measured, we want to know its average speed. We can begin solving for this by
recalling that the average speed of an object 𝑣 is equal to its change in distance
divided by its change in time. It’s worth pointing out that these
changes we’re talking about are total changes. That is, in the case of our car,
Δ𝑑 is the total change in distance the car experiences. That’s three meters as we’ve
seen. And then Δ𝑡 is the total
corresponding change in time, and we see that there are three seconds of total time
elapsed.
Δ𝑑 is three meters and Δ𝑡 is
three seconds. When we calculate this fraction, we
solve for the average speed of the car over the measured time interval. Three divided by three is one so
that average speed is one meter per second. Note that because this is an
average speed, we never actually had to figure out how far this distance here is,
that is, how much farther than one meter the car traveled between one second of time
and two seconds. To find average speed, we only
needed to know total change in distance and total change in time. As we’ve seen, that average speed
is one meter per second.
Let’s look now at one more
example.
A blue object and an orange object
move across a grid of equally spaced lines. Both objects move for four
seconds. The arrows show the distances moved
each second. Which color object has greater
average speed?
When we take a look at the motion
of these two objects on the grid, we know, first of all, that all the grid spaces
are of the same size. So, for example, this distance here
is the same as this distance here, and it’s also the same as this distance here, and
so on. For each object, the black arrows
show us how far that object moves every second. In the first second of time, the
orange object moves across one grid space, same with the blue object. Then, over the second time
interval, both objects once again move one grid space. However, over the third second, the
orange object once again moves one grid space, but the blue object moves one, two,
three grid squares. Over the last one-second interval,
both objects once again move across one grid space.
We want to know which object has
greater average speed. The average speed of an object 𝑣
is in general equal to the total change in distance traveled by that object divided
by the total change in time elapsed. For our orange and blue objects,
we’ll calculate their average speed in terms of a number of grid spaces over
time. Because we’re solving for average
speed, we won’t need to look at each individual one-second time interval. Instead, as our equation tells us,
we’ll look for the change in these variables of distance and time overall.
For our orange object then, what is
its total change in distance Δ𝑑? And what is its total change in
time Δ𝑡? This object we see moves four grid
spaces, and it does this in a total time of four seconds. This tells us the orange object has
an average speed of one space per second. When it comes to our blue object,
the total distance traveled here is one, two, three, four, five, six spaces. And this also all happens in a time
of four seconds. Therefore, the blue object has an
average speed of three-halves of one space per second. Since three-halves is greater than
one, our answer is that it’s the blue object that has the greater average speed.
Let’s finish our lesson now by
reviewing a few key points. In this video, we saw that an
object moving at constant speed travels equal distances in equal times. And also we learned that the
average speed of an object equals its change in distance traveled divided by the
change in time passed. Written as an equation, this is 𝑣
equals Δ𝑑 divided by Δ𝑡.
