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# Finding average speed or rate

Using the formula for finding distance we can determine Usian Bolt's average speed, or rate, when he broke the world record in 2009 in the 100m. Watch. Created by Sal Khan.

Video transcript

SALMAN KHAN: I have
some footage here of one of the most exciting
moments in sports history. And to make it
even more exciting, the commentator is
speaking in German. And I'm assuming that
this is OK under fair use, because I'm really using
it for a math problem. But I want you to
watch this video, and then I'll ask you
a question about it. [CHEERING] COMMENTATOR: [SPEAKING GERMAN] SALMAN KHAN: So you see,
it's exciting in any language that you might watch it. But my question to you is,
how fast was Usain Bolt going? What was his average
speed when he ran that 100 meters right there? And I encourage you to watch
the video as many times as you need to do it. And now I'll give you a little
bit of time to think about it, and then we will solve it. So we needed to figure out
how fast was Usain Bolt going over the 100 meters. So we're really thinking
about, in the case of this problem, average
speed or average rate. And you might already be
familiar with the notion that distance is equal
to rate or speed-- I'll just write
rate-- times time. And I could write
times like that, but once we start doing algebra,
the traditional multiplication symbol can seem very confusing
because it looks just like the variable x. So instead, I will
write times like this. So distance is equal
to rate times time. And hopefully, this makes
some intuitive sense for you. If your rate or your
speed were 10 meters per second-- just as an example. That's not necessarily
how fast he went. But if you went 10
meters per second, and if you were to do
that for two seconds, then it should hopefully
make intuitive sense that you went 20 meters. You went 10 meters per
second for two seconds. And it also works
out mathematically. 10 times 2 is equal to 20. And then you have seconds in
the denominator and seconds up here in the numerator. I just wrote seconds
here with an s. I wrote it out there. But they also cancel
out, and you're just left with the units of meters. So you're just left
with 20 meters. So hopefully this
makes intuitive sense. With that out of the
way, let's actually think about the problem at hand. What information do
we actually have? So do we have the distance? So what is the distance
in the video we just did? And I'll give you a second
or two to think about it. Well, this race
was the 100 meters. So the distance was 100 meters. Now, what else do we know? Do we know-- well, we're
trying to figure out the rate. That's what we're
going to figure out. What else do we know out of
this equation right over here? Well, do we know the time? Do we know the time? What was the time that it
took Usain Bolt to run the 100 meters? And I'll give you another few
seconds to think about that. Well luckily, they were
timing the whole thing. And they also showed
that it's a world record. But this right over
here is in seconds. It's how long it took Usain
Bolt to run the 100 meters. It was 9.58 seconds. And I'll just write
s for seconds. So given this information here,
what you need to attempt to do is now give us our rate in
terms of meters per second. I want you to think if
you could figure out the rate in terms of
meters per second. We know the distance,
and we know the time. Well, let's substitute
these values into this equation
right over here. We know the distance
is 100 meters. And it's equal to-- we
don't know the rate, so I'll just write
rate right over here. And let me write it
in that same color. It's equal to rate times--
and what's the time? We do know the time. It's 9.58 seconds. And we care about rate. We care about solving for rate. So how can we do that? Well, if you look at this right
hand side of the equation, I have 9.58 seconds times rate. If I were able to
divide this right hand side by 9.58 seconds,
I'll just have rate on the right hand side. And that's what I
want to solve for. So you say, well, why don't I
just divide the right hand side by 9.58 seconds? Because if I did that,
the units cancel out, if we're doing
dimensional analysis. Don't worry too much if that
word doesn't make sense to you. But the units cancel out,
and the 9.58 cancels out. But I can't just divide one
side of an equation by a number. When we started off, this
is equal to this up here. If I divide the
right side by 9.58, in order for the equality
to still be true, I need to divide the left
side by the same thing. So I can't just
divide the right side. I have to divide the
left side in order for the equality
to still be true. If I said one thing is
equal to another thing, and I divide the other thing
by something, in order for them to still be equal, I have
to divide the first thing by that same amount. So I divide by 9.58 seconds. So on our right
hand side-- and this was the whole point--
these two cancel out. And then on the
left hand side, I'm left with 100 divided by 9.58. And my units are
meters per second, which are the exact units that
I want for rate, or for speed. And so let's get the calculator
out to divide 100 by 9.58. So I've got 100 meters
divided by 9.58 seconds gives me 10 point--
this says we've got about three
significant digits here-- so let's say 10.4. So this gives us 10.4. And I'll write it
in the rate color. 10.4-- and the units are
meters per second-- meters per second is equal to my rate. Now, the next question. So we got this in
meters per second. But unfortunately,
meters per second, they're not the--
when we drive a car, we don't see the speedometer
in meters per second. We see either kilometers
per hour or miles per hour. So the next task I have for
you is to express this speed, or this rate-- and this
is his average speed, or his average rate,
over the 100 meters. But to think about this in
terms of kilometers per hour. So try to figure out
if you can rewrite this in kilometers per hour. Well, let's just take
this step by step. So I'm going to
write-- so let me just go down here, start over. So I started off with 10.4. And I'll write meters in
blue, and seconds in magenta. Now, we want to get to
kilometers per hour. Right now we're
meters per second. So let's take baby steps. Let's first think about it in
terms of kilometers per second. And I'll give you
a second to think about what we would
do this to turn this into kilometers per second. Well, the intuition here, if I'm
going 10.4 meters per second, how many kilometers
is 10.4 meters? Well, kilometers is a much
larger unit of measurement. It's 1,000 times larger. So 10.4 meters will be a much
smaller number of kilometers. And in particular, I'm
going to divide by 1,000. Another way to think
about it, if you want to focus on
the units, we want to get rid of this in meters,
and we want a kilometers. So we want a
kilometers, and we want to get rid of these meters. So if we had meters
in the numerator, we could divide by meters here. They would cancel out. But the intuitive way to
think about it is we're going from a smaller
unit, meters, to a larger unit, kilometers. So 10.4 meters are going
to be a much smaller number of kilometers. But if we look at it
this way, how many meters are in 1 kilometer? 1 kilometer is equal
to 1,000 meters. This right over here, 1
kilometer over 1,000 meters, this is 1 over 1. We're not changing
the fundamental value. We're essentially just
multiplying it by one. But when we do this,
what do we get? Well, the meters cancel out. We're left with
kilometers and seconds. And the numbers, you get
10.4 divided by 1,000. 10.4 divided by 1,000
is going to give you-- so if you divide by 10,
you're going to get 1.04. You divide by 100,
you get 0.104. You divide by 1,000,
you get 0.0104. So that's just 10.4
divided by 1,000. And then our units are
kilometers per second. So that's the
kilometers, and then I have my seconds right over here. So let me write the equal sign. Now, let's try to convert
this to kilometers per hour. And I'll give you a
little bit of time to think about that one. Well, hours, there's
3,600 seconds in an hour. So however many kilometers
I do in a second, I'm going to do 3,600
times that in an hour. And the units will
also work out. If I do this many
in a second, so it's going to be times 3,600, there
are 3,600 seconds in an hour. And another way to
think about it is we want hours in the denominator. We had seconds. So if we multiply
by seconds per hour, there are 3,600
seconds per hour, the seconds are
going to cancel out, and we're going to be left
with hours in the denominator. So seconds cancel out, and we're
left with kilometers per hour. But now we have to multiply
this number times 3,600. I'll get the calculator
out for that. So we have 0.0104 times 3,600
gives us, I'll just say 37.4. So this is equal to 37.4
kilometers per hour. So that's his average speed
in kilometers per hour. And now the last thing I want to
do, for those of us in America, we'll convert into
imperial units, or sometimes called English
units, which are ironically not necessarily used in the UK. They tend to be used in America. So let's convert this
into miles per hour. And the one thing I will
tell you, just in case you don't know, is that 1.61
kilometers is equal to 1 mile. So I'll give you a
little bit of time to convert this
into miles per hour. Well, as you see
from this, a mile is a slightly larger
or reasonably larger unit than a kilometer. So if you're going 37.4
kilometers in a certain amount of time, you're going to
go slightly smaller amount of miles in a certain
amount of time. Or in particular, you're
going to divide by 1.61. So let me rewrite it. If I have 37.4
kilometers per hour, we're going to a larger unit. We're going to miles. So we're going to divide by
something larger than one. So we have one-- let
me write it in blue-- 1 mile is equal to
1.61 kilometers. Or you could say there's one
1.61th mile per kilometer. It also, once again,
works out with units. We want to get rid of the
kilometers in the numerator. So we would want it
in the denominator. We want a mile in the numerator. So that's why we have a
mile in the numerator here. So let's once again
multiply, or I guess in this case
we're dividing by 1.61. And we get-- let's just divide
our previous value by 1.61. And we get 23 point--
I'll just round up-- 23.3. This is equal to 23.3. And then we have miles per hour. Which is obviously very fast. He's the fastest human. But it's not maybe as fast
as you might have imagined. In a car, 23.3 miles per
hour doesn't seem so fast. And especially relative
to the animal world, it's not particularly
noteworthy. This is actually slightly
slower than a charging elephant. Charging elephants have been
clocked at 25 miles per hour.