We have seen multiple times in our life
that distance can be viewed as rate times time.
Now what I wanna do in this video is use this fairly
simple formula right over here, this fairly simple equation, to understand that
units can really be viewed as algebraic objects, that you can kind
of treat them like variables as we work through a formula or equation.
Which could be really, really helpful to make sure that
our, that we're getting results in units that actually make sense.
So for example, if someone were to give you a rate, if they were to say a
rate of, let's say, 5 meters
per second, and they were to give you a time, a time of 10
seconds, then we can pretty, in a pretty straight forward way, apply this formula.
We say well distance is equal to our rate, 5 meters per
second, times our time, times our time which is 10 seconds.
And what's neat here is we can treat the units,
as I just said, like algebraic constructs, kind of like variables.
So this would be equal to, well
multiplication doesn't matter what order we multiply in.
So we can change them, the order.
This is the same thing as 5 times 10.
5 times 10 times meters per second, times meters per seconds, times seconds.
And if we were to treat our units as these
kinds of algebraic objects, and we could say, hey look,
we have seconds divided by seconds, or you have a
seconds in the denominator multiplied by seconds in the numerator.
Those are going to cancel out.
And 5 times 10, of course is, 5 times 10, of course is 50.
So we would be left with 50, and the
units that we're left with are the meters, 50 meters.
So that's pretty neat.
The units worked out.
When we treated the units out like algebraic objects, they worked out so that
our end units for distance were in meters, which is a unit of distance.
Now you're saying, okay, that's, that's cute and
everything, but this seems like a little bit
of too much overhead to worry about when I'm just doing a simple formula like this,
but what I wanna show is that even with a simple, with a simple formula like
distance is equal to rate times time, what I just did could actually be quite useful.
And this thing that I'm doing is actually called dimensional analysis.
And it's useful for something as simple as
distance equals rate times time, but as you go
into physics and chemistry and engineering, you'll see
much, much, much more I would say hairy formulas.
And when you do the dimensional analysis, it makes
sure that your, that the math is working out right.
It makes sure that you're getting the right units.
But even with this, let's try a slightly more complicated example.
Let's say that our rate is, let's say,
let's keep our rate at 5 meters per second.
But let's say that someone gave us the time.
Instead of giving it in seconds, they give it in hours.
So they say the time is equal to 1 hour.
So now let's try to apply this formula.
So we're gonna get distance is equal to 5 meters per second, 5
meters per second, times time, which is 1 hour, times 1 hour.
But what's that gonna give us?
Well the 5 times the 1, so we multiply the 5 times the 1.
That's just going to give us 5.
Well then we have to, remember we have to, the units algebraically.
Where we're going to do our dimensional analysis.
So it's 5, that we have meters per second, times hours, times hours.
Or you could say 5 meter hours per second.
Well this doesn't look like a, this isn't a, a,
a set of units that we know, that, that makes
sense to us, this doesn't feel like our traditional units
of distance, so we wanna cancel this out in some way.
And it might jump out of you, well if, if we can
get rid of this hours, if we could express it in terms
of seconds, then that would cancel here and we'd be left with
just the meters, which is a unit of distance that we're familiar with.
So how do we do that?
Well we'd wanna multiply this thing by something that has hours in
the denominator, and seconds in the
numerator, times essentially, seconds per hour.
Well how many seconds are there per hour?
Well, there are 3600, let me do this in a, I'll do this color, there are 3600 seconds
per hour, or you could even say that there are 3600 seconds for every 1 hour.
So when you, now when you multiply, these hours will
cancel with these hours, these seconds will cancel with those seconds,
and we are left with, we are left with 5 times
3600, what is that, that's 5 times 3000 would be 15000.
5 times 600 is another 3000.
So that is, it's equal to 18000, and the only units that
we're left with, we just have the meters there 18 oh, it's
18000, 18000, 18000 meters, right?
And so, this is, we're done.
We've now expressed our distance in terms of units that we recognize.
If you go 5 meters per second, for 1 hour, you will go 18000 meters.
But let's just use our little dimensional analysis muscles a little bit more.
What if, what if we didn't want the answer
in meters, but we wanted the answer in kilometers?
What could we do?
Well we could take that 18000 meters, 18000 meters, and if
we could multiply it by something that has meters in the
denominator, meters in the denominator, and kilometers in the numerator, then
these meters would cancel out and we'd be left with the kilometers.
So what could we multiply it so we're not really changing the value?
Well we want to multiply it by essentially 1, so
we wanna write equivalent things in the numerator and the denominator.
So 1 kilometer is equivalent to 1000 meters.
So one way to think about it, we're just multiplying this thing by 1.
One kilometer over 1000 meters.
Well, one kilometer is 1000 meters.
So this thing is equivalent to 1.
But what's in need is when you multiply, we
have meters cancelling with meters, until you're left with
18000 divided by 1000 is equal to 18, and
then the only units we're left with is the kilometers.
And we are done, we have re-expressed our
distance instead of in meters, in terms of kilometers.