### Video Transcript

In this video, we’re talking about
units of speed. Across the world in everyday life,
we talk about speed in different ways. Often those differences come down
to the units in which speed is expressed. In this lesson, we’ll see what
those common units are and how to convert from one unit of speed to another.

As we get started, let’s remind
ourselves of the definition of speed. Often symbolized with a lowercase
𝑠, speed is equal to a distance traveled divided by the time it takes to travel
that distance. And when we think about the units
in this expression, we know that the SI base unit for distance is the meter and that
the base unit in this system for time is the second. So this means if we were to write
out the base units of speed, they would be meters per second.

Now, even though meters and seconds
are the base units within the SI system for distance and time, that doesn’t mean
that they’re the only units that could possibly be used. Many of us, for example, have heard
of speeds described not in units of meters per second, but in units called miles per
hour, abbreviated mph. This is another kind of unit in
which speed can be expressed. Or along with this, what about
reporting a speed in the units of kilometers per hour? This is yet another option.

Depending on the particular
context, it’s fairly common to see speeds expressed in all three of these kinds of
units. But what if we hear a speed
reported in a set of units that are not as familiar to us and we want to convert
them to a set that are. Well, in that case, we’ll need to
be able to convert a speed reported in, say, units of kilometers per hour to units
of miles per hour or one reported in miles per hour to units of meters per
second. In general, being able to switch
between different sets of units for expressing a given speed will be helpful to
us.

To do this, we’ll need to know how
these units of distance, miles, kilometers, and meters, relate to one another as
well as how the units of time, hours and seconds, relate. Starting off thinking of the units
of distance, we know that one kilometer is 1000 meters because this prefix kilo-
refers to 1000. Now, when it comes to the
conversion between kilometers and miles, 1.6 kilometers is approximately equal to
one mile. For simplicity, we’ll treat this
relationship as exact, that 1.6 kilometers is exactly one mile.

So far, we’ve talked about how to
convert a kilometer to a meter and vice versa and a kilometer to a mile and back
again. So the only distance conversion we
haven’t yet covered is a mile to a meter or a meter to a mile. But see that, knowing what we know
so far, we can figure it out by steps. Say we have a distance given to us
in units of miles. Well, we know now how to convert
that into units of kilometers. And as a next step, we know how to
convert a distance in kilometers to a distance in meters. Therefore, through a two-step
process, we can convert miles to meters. Or we could go the other way and go
from meters to miles.

Once we’ve learned how to convert
between distances, we’ll want to do the same thing between different units of
time. In particular, we want to convert
between an hour and a second. These are often the time units in a
speed unit. Now, if we think about one hour of
time, we know that that’s equal to 60 minutes or that 60 minutes makes up one
hour. And we also know that everyone of
those minutes has 60 seconds of time in it. So then, to know how many seconds
are in one hour of time, we can multiply 60 minutes by 60 seconds per minute. Now, at first, it may seem like
we’ve done something unallowed here. After all, we’re multiplying one
side of a mathematical equation by a number. But we’re not multiplying it on the
other side by that same number.

But here’s the thing. The number that we’re multiplying
by is actually equal to one. That’s because 60 seconds of time
is the same amount of time as one minute. So in multiplying by 60 seconds
divided by one minute, effectively, we’re multiplying by one. For that reason, mathematically,
it’s permissible to do this to one side of the equation without doing it to the
other. And even though we’re only
technically multiplying by one, this operation still has a helpful impact.

To see what that is, let’s look at
the units on the right-hand side of this expression. What we have is units of minutes
here multiplied by units of seconds here divided by units of minutes. But then, if we multiply and divide
by the same units, in this case minutes, what happens to that unit. Well, it cancels out, which means
on the right-hand side, we’re left with 60 multiplied by 60 seconds. 60 times 60 is 3600, which means we
figured out how many seconds one hour of time is equivalent to.

At this point, we’ve seen how to
convert between and among all these different distance and time units: kilometers,
meters, miles, hours, and seconds. This is by no means an exhaustive
list of all the units we might encounter in units of speed. But they do cover these three
commonly used ways of expressing speed. The best way to become expert at
working with different units of speed is through practice. So let’s look now at a few practice
exercises.

A spacecraft travels at a speed
of 5.5 kilometers per second. What is its speed in meters per
second?

Alright, so we have some
spacecraft. Let’s say that this is it. And we’re told that it’s moving
along at a speed of 5.5 kilometers per second. We’ll call that speed 𝑠. And we see it’s expressed in
units of kilometers, that’s the distance, per second, that’s the time. Just as a side note, this
agrees with our understanding of what speed is. It’s a distance divided by a
time. And here, we have kilometers, a
unit of distance, divided by seconds, a unit of time. So anyway, we have the speed 𝑠
in units of kilometers per second. But we want to express that
speed in a different set of units, in units of meters per second. So the basic question we want
to answer is how many meters per second is 5.5 kilometers per second.

Looking at the units involved
here, we can see that we won’t have to do anything to the units of time. We start out in units of
seconds. And we end up in those same
units. No conversion necessary
there. But we will need to convert the
distance from units of kilometers to units of meters. To do that, we can recall that
one kilometer of distance is equal to 1000 meters. Keeping that in mind, there’s a
mathematical operation we can perform on this original speed, 5.5 kilometers per
second, to convert it to the equivalent speed in units of meters per second.

To do this, we’re going to
multiply our original speed by one. Now, we’re not really going to
multiply by one exactly. But we are going to multiply by
something that is equivalent to one. And we’re going to choose that
something so that when we multiply it by our original speed, out come the units
that we want, meters per second. Here’s how we’ll do this. Recall that we want to convert
this distance unit, kilometers, to a unit of meters. And we’ve already recalled that
one kilometer is 1000 meters. Since that’s true since,
mathematically, the two values on either side of this equality are the same. That means if we write them as
a ratio, like we do here, then that ratio must be equal to one. That’s because the numerator,
1000 meters, is equal to the denominator. So this fraction is one.

But now, let’s think of what
happens when we multiply this fraction by our original speed. If we just focus on the units
for a moment, we have in the numerator units of kilometers times meters. And in the denominator, we have
units of seconds times kilometers. Since that factor of kilometers
appears in both the numerator and denominator once, that means it cancels
out. Once they’re gone, look at what
we have left over, units of meters per second, just the units we wanted to end
up with.

So by multiplying our original
speed by this ratio, even though the ratio is equal to one, it’s effected the
change in units that we wanted to bring about. When we multiply these two
values together, the result has units of meters per second. Knowing that, all we need to
figure out now is the number that goes in front of the units. And that number is equal to 5.5
multiplied by 1000. And that’s equal to 5500. So then, 5.5 kilometers per
second is equal to a speed of 5500 meters per second.

Let’s consider now a second
example.

What value of speed in
kilometers per hour is equal to a value of speed of one meter per second?

Okay, so this example is
telling us that we have a speed of one meter per second. And if we recall the fact that,
in general, speed 𝑠 is equal to a distance 𝑑 divided by the time it takes to
travel that distance 𝑡, then we can see that indeed this number, one meter per
second, is a speed. It’s a distance over a
time. What we want to do is express
this speed in a different set of units, in units of kilometers per hour. The question we want to answer
is how many kilometers per hour is one meter per second. What we’re doing is converting
a speed in one set of units to its equivalent speed in another set. That is, the actual speed of
whatever is in motion isn’t changing. Only the units in which we
express that speed are changing.

To write this one meter per
second speed in units of kilometers per hour, we’ll need to make two
conversions. First, we’ll need to convert
meters to kilometers. And then, we’ll need to convert
seconds to hours. To get a start at doing this,
let’s recall that one kilometer is equal to 1000 meters. In order to reexpress the
distance unit in our speed of one meter per second, in units of kilometers,
we’re going to multiply this speed by a particularly chosen fraction. There are two requirements that
this fraction needs to satisfy. First, the numerator needs to
be equal to the denominator. In other words, the fraction
overall must be equal to one. The second requirement is that,
by multiplying by this fraction, our original speed must then have its speed
expressed in kilometers per second, rather than meters per second.

Using our conversion between
kilometers and meters, we can write out a fraction that fits both these
requirements. Here’s what we’ll do. Keeping in mind that we want
this unit of distance to cancel out and be replaced by a unit of kilometers,
we’ll put the distance of 1000 meters in the denominator of our fraction. In the numerator, we’ll put one
kilometer. After all, one kilometer is
equal to 1000 meters. That means this fraction we’ve
created does equal one, which is one of the conditions we had for it. But then, see how it meets our
second condition too. If we multiply this fraction by
the original speed, then the units of meters cancel out from numerator and
denominator. The remaining units are
kilometers up top and seconds down bottom. At this point, if we multiply
these two values together, we’ll get a result in units of kilometers per
second.

Now we’re headed in the right
direction. That would give us a result
with distance units in kilometers like we want. But recall that we also need to
convert the time units, from seconds to hours. To start doing that, we can
recall that one hour is equal to 3600 seconds. And the reason an hour is 3600
seconds is because one hour is made of 60 minutes. And each minute has 60 seconds
in it. And 60 times 60 is 3600. Now that we’re working on
converting our unit of time, seconds, into units of hours, we’ll once again
multiply our originally given speed by a fraction, which must be equal to one
but will give us the units in time that we want, hours.

Using our conversion between
hours and seconds, the numerator of this fraction is 3600 seconds. And the denominator is one
hour. Since one hour is equal to 3600
seconds, this fraction is equal to one. But looking at the units, we
see that when we multiply this fraction by our original value, the time units of
seconds cancel out with one another. And we’re left with one over
hours. Now that we’ve cleared away all
the cancelled out units, see what’s left. What remains is a speed in
units of kilometers per hour, just like we wanted. To figure out what that value
in kilometers per hour is, we’ll multiply our original speed times one over 1000
times 3600. The result of all that is
3.6.

So then, a speed of one meter
per second is equal to 3.6 kilometers per hour.

Let’s summarize now what we’ve
learned about units of speed. Starting off, we saw that speed is
defined as a distance traveled divided by the time taken to travel that
distance. Written as an equation, it’s often
represented as 𝑠 is equal to 𝑑 divided by 𝑡. Speeds can be expressed in many
different units. Common units for expressing speeds
include meters per second, where meters and seconds are the SI base units of
distance and time, kilometers per hour, and miles per hour, abbreviated mph.

To convert between different speed
units, it’s necessary to convert between units of distance and/or units of time. And lastly, we learned some of
those distance and time conversions. We saw that one kilometer is equal
to 1000 meters. One mile is equal to 1.6
kilometers. And one hour of time is equal to
3600 seconds. Knowing these conversions allowed
us to convert between speeds in meters per second, kilometers per hour, and miles
per hour.