Usain Bolt's Average Speed

Figuring out Usain Bolt's average speed over 100m when he broke the world record in 2009

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 okay under
fair use

because I'm really using it for a, a, a
math problem.

But I want you to watch this video and
then I'll ask you a question about it.

>> [FOREIGN]

>> So you see it's exciting in any
language that you might watch it.

But my question to you is how fast was
Ussain 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 will give you a little bit of

time to think about it, and then we will
solve it.

[BLANK_AUDIO]

So we needed to figure out how fast was
Ussain 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, distance is equal to rate

or speed, I'll just write rate, times
time.

I could write times like that, but once we
start doing algebra, the traditional

multiplication symbol can seem very
confusing cuz

it looks just like the variable x.

So instead, I will write times like this.

So distance is equal to rate The times is
equal to rate times time.

And hopefully this makes some intuitive
sense for you.

If your rate or your speed were ten meters
per second just as an

example, that's not necessarily how fast
he went, but if you went ten meters

per second and if you were to do that for
For two seconds, for two seconds,

seconds, then it should hopefully make
intuitive sense that you went 20 meters.

You went ten meters per seconds for two
seconds.

And it also works out mathematically, ten
times two 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, 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,

let's actually think about the problem,
the problem at hand.

What information do we actually have?

So we, do we have the distance?

So, what is, what is the distance in the
video we just did?

And I'll give you a second or two to think
about it.

[BLANK_AUDIO]

Well, this race was 100 meters.

So, the distance was 100.

100 meters.

Now, what else do we know?

Well, we're trying to figure out the rate,
that's what we're gonna figure out.

What else do we know out of, out of this,
out of this equation over here?

Well do we know the time?

Do we know the time?

What was the time that it took Ussain 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've not,

and they've also showed that it's a, it's
a world record.

But this right over here is in seconds, is
how

long it took Usain Bolt to run the 100
meters.

It was 9.58, 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 can 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.

The distance is 100 meters.

And its equal to, we don't know the rate,
so I'll just write rate right over here.

Its, and I'm going to write in that same
color.

Its equal to rate, rate times, and what's
the time?

We do know the time.

It's 9.58 seconds.

9.58, 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 wait why don't I just

divide the right-hand side by 9.58
seconds, 9.58 seconds?

Cuz if I did that the 9, 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
cancel out.

But I can't just divide one side of the
equation by a number.

When we started off, this is equal to this
up here.

If I divide it, if I divide the right side
by 9.58, in order for the

equality To still be true, I needed to
divide to 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 quality 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 the 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 a hundred 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, let's
see.

We've got about three significant digits
here.

So let's say, 10.4.

So this gives us 10.4 and I'll right in
the rate color.

10, 10.4 and the units are meters per
second, meters,

meters per second is equal to, is equal to
my rate.

Now, the next question, so we've got this
in meters per second

but unfortunately meters per second is,
they're not the, you know, when

we drive a car, we don't see the, the
speedometer in meters

per second, we see either kilometers per
hour or miles per hour.

So, the next task I, I have for you, is to
express this speed, or this rate.

And this is his average speed, or his
average rate, over the hundred meters.

But to think about this in terms of
kilometers per, kilometers per hour.

So try to figure out if you can, if you
can rewrite this in kilometers per hour.

Well let's just take this step by step.

So, I'm gonna write, so let me just go
down here, start over.

So, I'm just, I started off with 10.4, and
I'll write meters in blue, meters in

blue, and seconds, seconds in magenta.

Now, we wanna get the kilometers per hour,
I know our 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 second to think about
what we

would do to this, to turn this into
kilometers per second.

Well, the intuition here, if I'm going
10.4 meters

per second, how many, how many kilometers
is 10.4 meters?

Well, kilometers is a much larger unit of
measurement, it's 1000 times larger.

So, 10.4 meters will be a much smaller
number of kilometers.

And in particular, I'm gonna divide by
1000.

Another way to think about it, if you
wanna focus on the

units, we wanna get rid of this meters and
we want a kilometers.

So, we wanted kilometers, and we wanna get
rid of these meters.

So, if, if we had meters in the numerator,
we

could divide by meters here, and they
would cancel out.

But, the intuitive way to think about it
is, we're

going from a smaller unit, meter, to a
larger unit, kilometers.

So, 10.4 meters is going to be a much
smaller number of kilometers.

But if we look at it this way, how many,
how many meters are in one kilometer?

One kilometer is equal to 1,000 meters.

This right over here, 1 kilometer over
1,000 meters.

This is over 1 over 1, we're sent, we're
not changing the fundamental value.

We're essentially just multiplying it by
one.

But when we do this, when we do this what
do we get?

Well, the meters cancel out, we're left
with kilometers and

seconds, and then the numbers you get 10.4
divided by 1,000.

10.4 divided by 1,000 is going to you, so
if you divide by 10,

you're gonna get 1.04, you divide by 100,
you get 0.104.

You divide by 1000, you get 0.0104, so
that's just 10.4 divided by 1000.

And then our units are kilometers,
kilometers per second.

So that's the kilometers And then I have
my seconds right over here.

Now, 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, there's 3,600 seconds
in an hour, so however many kilometers

I do in a second, I'm gonna do 3,600 times
that in an hour.

And the units will also work out.

If I'm going, I'm, right now.

Well, if I do this many in a second, so
it's gonna be times 3600.

There are 3600 seconds in an hour.

3600 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 3600 seconds per hour.

The seconds are going to cancel out and we

are 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 3600.

I'll get the calculator out for that.

I'll get the calculator out for that.

So we have, we have 0.0104, 0.0104 times
3600, 3600 gives us 37.,

I'll just stay 37.4 so this is equal to

37.4, 37.4

kilometers.

Kilometers, kilometers per hour.

Kilometers per hour.

So that's his average speed in kilometers
per hour.

And then, the last thing I wanna do, for
those of you, for those of us in America.

We'll convert into imperial units or
sometimes called English units.

Which are ironically.

Not necessarily used in UK, they tend to
be used in America.

So lets convert this into miles per hour
and the one thing I will tell you just in

case you don't know is that their 1.61
kilometers

is equal to 1 is equal to 1 mile.

So I'll give you little bit of time to
convert this into miles per hour.

[BLANK_AUDIO]

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

gonna go slightly smaller amount of miles
in a certain amount of time.

Or in particular, you're gonna divide by
1.61.

So let me rewrite it.

If I have 37.4 kilometers per hour,
kilometers

per hour, we're going to a larger unit.

We're going to miles.

So we're gonna divide by something larger
than one.

So we have one, we have one, let me right
it in blue.

One mile, one mile is equal to, is equal
to 1.61 kilometers.

Or you could say there's 1, 1.61 mile per
kilometer.

And also once again works out with units.

We wanna get rid of the kilometers in the
numerator, so we would

want it in the denominator and 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, so
we get.

Let's just divide our previous value by
1.61, 1.61,

and we get 23, I'll just round up, 23.3.

This is equal to 23.3, 23.

23.3 and then we have miles, miles, miles
per hour.

20 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.

You know, in the 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.