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.
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.