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# Same rate with different units

Video transcript

So I have a car here, and
let's say that in 3 hours, this car is able to
travel 150 kilometers. So what I want to think
about in this video is, what are some reasonable
ways to express the rate at which this car
traveled for those 3 hours? And I encourage you
to pause this video and think about it for yourself. One, you could
calculate the rate. But also think about
different units that you could use
to express that rate, and which ones would
be useful-- which ones what would be
reasonable, and which ones would be unreasonable. So let's just remind
ourselves what rate even is. So you could think
about distance as being equal to
rate times time. Or you could imagine if you
divide both sides by time, you could imagine that
distance divided by time is equal to rate. So they've given us a distance,
they've given us a time. So we could just divide the
distance divided by the time to figure out the rates. And I'm going to
keep the units here. Because it's really important
to recognize that the units, to some degree, they can also
be manipulated algebraically. Now they aren't variables, but
they follow the same rules, I guess I should say,
as a variable would. So for example, if
I said, look rate is distance divided by time. So I could say that my
rate in this situation is going to be 150 kilometers
divided by 3 hours. So if we just look at
the numeric part of this, what's 150 divided by 3? Well that's going to be 50. So this is going
to be equal to 50. But we can keep
the units the way that they are, right over here. This is 50 kilometers per hour. This is what I meant
by saying look, we're dividing this quantity,
expressed in kilometers, by this quantity,
expressed in hours. We can divide the numeric
part, 150 divided by 3. But then we could leave at
the units in that relationship that they were before. So you can kind of algebraically
keep them that way. And you'll see in a
second we're going to manipulate them
a lot more, using what's often called the
dimensional analysis. But anyway, this is a reasonable
way to express a rate. 50 kilometers per hour. I can imagine this. I can imagine that
in 1 hour, you're going to go 50 kilometers. Let's think about other ways
that we could represent that. So 50 kilometers
per hour-- and this is where we're really going to
do some dimensional analysis with our units. So 50 kilometers per hour. Let's say we want to
express it in terms of kilometers per second. So how could we write
50 kilometers per hour, in terms of
kilometers per second? Well it's always good, actually,
as a first approximation, to just think about it. If you went this far in an hour,
then the number of kilometers you go in a second, is that
going to be less, or more? Well a second's a much,
much shorter period of time. There's 3,600
seconds in an hour. So you're going to go
1/3,600 of this distance. But let's think about
how we would actually work out with the units. Well, we want to get rid of
this hours in the denominator. And the plural,
obviously the grammar doesn't hold up
with the algebra, but this could be hour or hours. So we could think
about well, 1 hour-- I'll write an hour in
the numerator that's going to cancel with this
hour in the denominator. But we want it in
terms of seconds. So 1 hour is equal
to how many seconds? Well 1 hour is equal
to 3,600 seconds. This is what I meant
by saying that using dimensional analysis, which
is what I'm doing right now, we can essentially
manipulate these units, as we would traditionally
do with a variable. So we have hours
divided by hours. And so when we do
the multiplication, we can multiply
the numeric parts. So we have 50 times
1, divided by 3,600. Let me write that. 50 times 1 over 3,600. And then our units left
are kilometers per second. Or I could say seconds. So we can play around with
the plural and singular parts of it, but I'll just write
it as kilometers per second. And so this is 50/3,600. And this fits our intuition. In a second, you're going to go
1/3,600 as far as you would go in an hour. But let's actually think
about what this is equal to. 50/3,600-- so this is
going to be the same thing, as-- Let me just
simplify it over here. So 50/3,600 is the
same thing as 5/360, which is the same thing as-- let
me write it this way-- 10/720. And I did that way because
that makes it clear that that's the
same thing as 1/72. So you could write
this as, you're going, this is equal to 1/72 of
a kilometer per second. Now I would claim
that this is not so reasonable of units for
this example right over here. 1/72 of a kilometer
every second? That doesn't help me too much. I guess I'll know
that in 72 seconds I will have gone a kilometer. But this is something that's
kind of strange for me to conceptualize. If I wanted to get
my calculator out, 1/72, 1 divided by
72-- if someone said, hey, I'm going 0.0139
kilometers in a second, that doesn't seem to make a
lot of conceptual sense to me. So I would say that
this, right over here, is a very reasonable
way to express our rate. This one seems like more
of an unreasonable way. But we could salvage this. Because we're going 1/72
of a kilometer per second. Now this is a small
number, but how could we make it much larger? Well, what if we
thought in terms of, not kilometers per
second, but if we thought in terms of
meters per second. A kilometer's 1,000 meters. So if we think about this in
terms of meters per second, we're going to get a
larger number here, in fact larger by
a factor of 1,000. So let's think about that. Let's try to convert this
kilometers to meters. So how would we do that? Well, once again, if
we have kilometers in the denominator
this kilometer will cancel with that kilometer. And we want a meter
in the numerator. So we want to think about
how many meters are there per kilometer. Well there's 1,000
meters per kilometer. Kilometers cancel out, and we
are left with 1,000 times 1-- I'll just write that
as 1,000-- over 72. And now we're left
with in the numerator, we're left with the
unit meters per second. And I know I keep writing
second in different ways. Oftentimes actually you'll see
people write second like that. So actually let me
just go with that. So s is second, is seconds,
is sec, just like that. So is this fairly reasonable? Well actually, this
feels pretty good. Let's get our calculator out
and figure out what that is. So 1,000 divided by 72
gives us 13, if I round, that's about 13.9. So this is approximately equal
to 13.9 meters per second. Which I can visualize. I can imagine how
far 13.9 meters is, and of doing that
distance in one second. So this actually also
seems like a reasonable way to express the rate. So I could say hey, this thing's
going 50 kilometers per hour. I can imagine it going roughly
13.9 meters per second. So this is reasonable as well. But to say it's going 1/72
of a kilometer per second doesn't really
seem to make sense. And also, if I try to think
about it in terms of meters per hour, that also
would be strange. Actually, I encourage
you to calculate it. Try to convert this right
over here to meters per hour. Then we would say well, we
could use the same thing here. That's going to be 1,000
meters for every 1 kilometer. Kilometers cancel out. And I'm going to be
left with 50 times 1,000 is 50,000 meters per hour. So I have trouble
imagining-- well, obviously if I convert
to kilometers in my head I could imagine it-- but this
is kind of a crazy large number. 50,000 meters per hour. So at least in my eyes,
using kilometers per hour to describe this
rate seems useful. Describing this rate as meters
per second seems useful. But describing it as kilometers
per second, or meters per hour, seem a little bit unusual.