WEBVTT

00:00:00.750 --> 00:00:02.610
We have seen multiple times in our life

00:00:02.610 --> 00:00:08.055
that distance can be viewed as rate times
time.

00:00:08.055 --> 00:00:10.990
Now what I wanna do in this video is use
this fairly

00:00:10.990 --> 00:00:15.730
simple formula right over here, this
fairly simple equation, to understand that

00:00:15.730 --> 00:00:20.140
units can really be viewed as algebraic
objects, that you can kind

00:00:20.140 --> 00:00:24.360
of treat them like variables as we work
through a formula or equation.

00:00:24.360 --> 00:00:27.480
Which could be really, really helpful to
make sure that

00:00:27.480 --> 00:00:33.000
our, that we're getting results in units
that actually make sense.

00:00:33.000 --> 00:00:38.030
So for example, if someone were to give
you a rate, if they were to say a

00:00:38.030 --> 00:00:43.360
rate of, let's say, 5 meters

00:00:43.360 --> 00:00:48.960
per second, and they were to give you a
time, a time of 10

00:00:48.960 --> 00:00:53.220
seconds, then we can pretty, in a pretty
straight forward way, apply this formula.

00:00:53.220 --> 00:00:57.770
We say well distance is equal to our rate,
5 meters per

00:00:57.770 --> 00:01:03.440
second, times our time, times our time
which is 10 seconds.

00:01:03.440 --> 00:01:07.050
And what's neat here is we can treat the
units,

00:01:07.050 --> 00:01:11.160
as I just said, like algebraic constructs,
kind of like variables.

00:01:11.160 --> 00:01:13.640
So this would be equal to, well

00:01:13.640 --> 00:01:16.010
multiplication doesn't matter what order
we multiply in.

00:01:16.010 --> 00:01:17.760
So we can change them, the order.

00:01:17.760 --> 00:01:21.340
This is the same thing as 5 times 10.

00:01:21.340 --> 00:01:30.877
5 times 10 times meters per second, times
meters per seconds, times seconds.

00:01:30.877 --> 00:01:32.593
And if we were to treat our units as these

00:01:32.593 --> 00:01:35.141
kinds of algebraic objects, and we could
say, hey look,

00:01:35.141 --> 00:01:37.273
we have seconds divided by seconds, or you
have a

00:01:37.273 --> 00:01:40.930
seconds in the denominator multiplied by
seconds in the numerator.

00:01:40.930 --> 00:01:42.660
Those are going to cancel out.

00:01:42.660 --> 00:01:47.370
And 5 times 10, of course is, 5 times 10,
of course is 50.

00:01:47.370 --> 00:01:50.900
So we would be left with 50, and the

00:01:50.900 --> 00:01:54.540
units that we're left with are the meters,
50 meters.

00:01:54.540 --> 00:01:55.750
So that's pretty neat.

00:01:55.750 --> 00:01:56.690
The units worked out.

00:01:56.690 --> 00:02:00.900
When we treated the units out like
algebraic objects, they worked out so that

00:02:00.900 --> 00:02:06.430
our end units for distance were in meters,
which is a unit of distance.

00:02:06.430 --> 00:02:07.910
Now you're saying, okay, that's, that's
cute and

00:02:07.910 --> 00:02:09.310
everything, but this seems like a little
bit

00:02:09.310 --> 00:02:14.290
of too much overhead to worry about when
I'm just doing a simple formula like this,

00:02:14.290 --> 00:02:18.020
but what I wanna show is that even with a
simple, with a simple formula like

00:02:18.020 --> 00:02:22.490
distance is equal to rate times time, what
I just did could actually be quite useful.

00:02:22.490 --> 00:02:25.570
And this thing that I'm doing is actually
called dimensional analysis.

00:02:25.570 --> 00:02:27.150
And it's useful for something as simple as

00:02:27.150 --> 00:02:29.370
distance equals rate times time, but as
you go

00:02:29.370 --> 00:02:32.340
into physics and chemistry and
engineering, you'll see

00:02:32.340 --> 00:02:35.880
much, much, much more I would say hairy
formulas.

00:02:35.880 --> 00:02:38.090
And when you do the dimensional analysis,
it makes

00:02:38.090 --> 00:02:40.620
sure that your, that the math is working
out right.

00:02:40.620 --> 00:02:42.660
It makes sure that you're getting the
right units.

00:02:42.660 --> 00:02:45.960
But even with this, let's try a slightly
more complicated example.

00:02:45.960 --> 00:02:49.170
Let's say that our rate is, let's say,

00:02:49.170 --> 00:02:52.930
let's keep our rate at 5 meters per
second.

00:02:52.930 --> 00:02:55.510
But let's say that someone gave us the
time.

00:02:55.510 --> 00:02:57.860
Instead of giving it in seconds, they give
it in hours.

00:02:57.860 --> 00:03:01.920
So they say the time is equal to 1 hour.

00:03:01.920 --> 00:03:04.250
So now let's try to apply this formula.

00:03:04.250 --> 00:03:09.596
So we're gonna get distance is equal to 5
meters per second, 5

00:03:09.596 --> 00:03:15.820
meters per second, times time, which is 1
hour, times 1 hour.

00:03:15.820 --> 00:03:17.190
But what's that gonna give us?

00:03:17.190 --> 00:03:20.410
Well the 5 times the 1, so we multiply the
5 times the 1.

00:03:20.410 --> 00:03:22.900
That's just going to give us 5.

00:03:22.900 --> 00:03:25.770
Well then we have to, remember we have to,
the units algebraically.

00:03:25.770 --> 00:03:27.648
Where we're going to do our dimensional
analysis.

00:03:27.648 --> 00:03:35.570
So it's 5, that we have meters per second,
times hours, times hours.

00:03:35.570 --> 00:03:38.670
Or you could say 5 meter hours per second.

00:03:38.670 --> 00:03:41.000
Well this doesn't look like a, this isn't
a, a,

00:03:41.000 --> 00:03:42.970
a set of units that we know, that, that
makes

00:03:42.970 --> 00:03:45.490
sense to us, this doesn't feel like our
traditional units

00:03:45.490 --> 00:03:48.350
of distance, so we wanna cancel this out
in some way.

00:03:48.350 --> 00:03:49.950
And it might jump out of you, well if, if
we can

00:03:49.950 --> 00:03:52.070
get rid of this hours, if we could express
it in terms

00:03:52.070 --> 00:03:55.240
of seconds, then that would cancel here
and we'd be left with

00:03:55.240 --> 00:03:58.902
just the meters, which is a unit of
distance that we're familiar with.

00:03:58.902 --> 00:04:00.410
So how do we do that?

00:04:00.410 --> 00:04:04.710
Well we'd wanna multiply this thing by
something that has hours in

00:04:04.710 --> 00:04:07.150
the denominator, and seconds in the

00:04:07.150 --> 00:04:10.330
numerator, times essentially, seconds per
hour.

00:04:10.330 --> 00:04:13.160
Well how many seconds are there per hour?

00:04:13.160 --> 00:04:19.540
Well, there are 3600, let me do this in a,
I'll do this color, there are 3600 seconds

00:04:19.540 --> 00:04:27.510
per hour, or you could even say that there
are 3600 seconds for every 1 hour.

00:04:27.510 --> 00:04:30.885
So when you, now when you multiply, these
hours will

00:04:30.885 --> 00:04:35.535
cancel with these hours, these seconds
will cancel with those seconds,

00:04:35.535 --> 00:04:38.385
and we are left with, we are left with 5
times

00:04:38.385 --> 00:04:43.232
3600, what is that, that's 5 times 3000
would be 15000.

00:04:43.232 --> 00:04:46.126
5 times 600 is another 3000.

00:04:46.126 --> 00:04:51.956
So that is, it's equal to 18000, and the
only units that

00:04:51.956 --> 00:04:57.786
we're left with, we just have the meters
there 18 oh, it's

00:04:57.786 --> 00:05:03.310
18000, 18000, 18000 meters, right?

00:05:03.310 --> 00:05:04.540
And so, this is, we're done.

00:05:04.540 --> 00:05:08.300
We've now expressed our distance in terms
of units that we recognize.

00:05:08.300 --> 00:05:13.960
If you go 5 meters per second, for 1 hour,
you will go 18000 meters.

00:05:13.960 --> 00:05:17.020
But let's just use our little dimensional
analysis muscles a little bit more.

00:05:17.020 --> 00:05:18.540
What if, what if we didn't want the answer

00:05:18.540 --> 00:05:20.550
in meters, but we wanted the answer in
kilometers?

00:05:20.550 --> 00:05:22.000
What could we do?

00:05:22.000 --> 00:05:27.570
Well we could take that 18000 meters,
18000 meters, and if

00:05:27.570 --> 00:05:30.700
we could multiply it by something that has
meters in the

00:05:30.700 --> 00:05:36.190
denominator, meters in the denominator,
and kilometers in the numerator, then

00:05:36.190 --> 00:05:39.760
these meters would cancel out and we'd be
left with the kilometers.

00:05:39.760 --> 00:05:42.760
So what could we multiply it so we're not
really changing the value?

00:05:42.760 --> 00:05:45.354
Well we want to multiply it by essentially
1, so

00:05:45.354 --> 00:05:49.480
we wanna write equivalent things in the
numerator and the denominator.

00:05:49.480 --> 00:05:55.730
So 1 kilometer is equivalent to 1000
meters.

00:05:55.730 --> 00:05:58.540
So one way to think about it, we're just
multiplying this thing by 1.

00:05:58.540 --> 00:06:01.670
One kilometer over 1000 meters.

00:06:01.670 --> 00:06:03.260
Well, one kilometer is 1000 meters.

00:06:03.260 --> 00:06:05.670
So this thing is equivalent to 1.

00:06:05.670 --> 00:06:07.970
But what's in need is when you multiply,
we

00:06:07.970 --> 00:06:12.446
have meters cancelling with meters, until
you're left with

00:06:12.446 --> 00:06:16.570
18000 divided by 1000 is equal to 18, and

00:06:16.570 --> 00:06:20.010
then the only units we're left with is the
kilometers.

00:06:21.020 --> 00:06:23.980
And we are done, we have re-expressed our

00:06:23.980 --> 00:06:27.560
distance instead of in meters, in terms of
kilometers.