Projectile motion (part 3)

WEBVTT
Kind: captions
Language: en

00:00:01.140 --> 00:00:05.910
In the last video, I said that
we started off with the change

00:00:05.910 --> 00:00:08.119
in distance, so we
said that we know

00:00:08.119 --> 00:00:09.369
the change in distance.

00:00:11.490 --> 00:00:13.350
These are the things
that we are given.

00:00:13.350 --> 00:00:17.610
We're given the acceleration,
we're given the initial

00:00:17.610 --> 00:00:20.910
velocity, and I asked you how
do we figure out what the

00:00:20.910 --> 00:00:22.070
final velocity is?

00:00:22.070 --> 00:00:24.290
In the last video-- if you don't
remember it, go watch

00:00:24.290 --> 00:00:28.960
that last video again-- we
derived the formula that vf

00:00:28.960 --> 00:00:34.640
squared, the final velocity
squared, is equal to the

00:00:34.640 --> 00:00:39.180
initial velocity squared
plus 2 times

00:00:39.180 --> 00:00:40.680
the change in distance.

00:00:40.680 --> 00:00:43.320
You'll sometimes just see it
written as 2 times distance,

00:00:43.320 --> 00:00:47.570
because we assume that the
initial distance is at point

00:00:47.570 --> 00:00:49.470
0, so the change in distance
would just

00:00:49.470 --> 00:00:51.540
be the final distance.

00:00:51.540 --> 00:00:53.620
We could write it either way,
and hopefully, at this point,

00:00:53.620 --> 00:00:55.600
you see why I keep switching
between change in

00:00:55.600 --> 00:00:57.060
distance and distance.

00:00:57.060 --> 00:00:59.740
It's just so you're comfortable
when you see it

00:00:59.740 --> 00:01:01.630
either way.

00:01:01.630 --> 00:01:04.330
This is for the situation
when we didn't

00:01:04.330 --> 00:01:06.230
know what the vf was.

00:01:06.230 --> 00:01:11.330
Let's say we want to solve
for time instead.

00:01:11.330 --> 00:01:13.470
Once we solve for the final
velocity, we could actually

00:01:13.470 --> 00:01:15.620
solve for time, and I'll show
you how to do that, but let's

00:01:15.620 --> 00:01:17.460
say we didn't want to go through
this step-- how can we

00:01:17.460 --> 00:01:20.560
solve for time directly, given
the change in distance, the

00:01:20.560 --> 00:01:23.140
acceleration, and the
initial velocity?

00:01:23.140 --> 00:01:27.620
Let's go back once again to
the most basic distance

00:01:27.620 --> 00:01:30.040
formula-- not the distance
formula, but how distance

00:01:30.040 --> 00:01:32.290
relates to velocity.

00:01:32.290 --> 00:01:36.010
We know that-- I'll write it
slightly different this time--

00:01:36.010 --> 00:01:44.130
the change in distance over the
change in time is equal to

00:01:44.130 --> 00:01:47.520
the average velocity.

00:01:47.520 --> 00:01:56.030
We could have rewritten this as
the change in distance is

00:01:56.030 --> 00:02:00.780
equal to the average velocity
times the change in time.

00:02:00.780 --> 00:02:04.130
This is change in time and
change in distance.

00:02:04.130 --> 00:02:07.510
Sometimes we'll just see this
written as d equals-- let me

00:02:07.510 --> 00:02:09.440
write this in a different
color, so we have some

00:02:09.440 --> 00:02:16.640
variety-- velocity times time,
or d equals rate times time.

00:02:16.640 --> 00:02:19.010
The reason why I have change in
distance here, or change in

00:02:19.010 --> 00:02:22.310
time, is that I'm not assuming
necessarily that we're

00:02:22.310 --> 00:02:24.920
starting off at the point
0 or at time 0.

00:02:24.920 --> 00:02:28.340
If we do, then it just turns out
to the final distance is

00:02:28.340 --> 00:02:31.920
equal to the average velocity
times the final time, but

00:02:31.920 --> 00:02:32.950
let's stick to this.

00:02:32.950 --> 00:02:37.720
We want to figure out time
given this set of inputs.

00:02:45.090 --> 00:02:48.200
Let's go from this equation.

00:02:48.200 --> 00:02:51.670
If we want to solve for time,
or the change in time, we

00:02:51.670 --> 00:02:55.730
could just could divide both
sides by the average

00:02:55.730 --> 00:02:58.600
velocity-- actually, no,
let's not do that.

00:02:58.600 --> 00:03:01.180
Let's just stay in terms
of change in distance.

00:03:05.266 --> 00:03:08.740
I've wasted space too fast,
so let me clear

00:03:08.740 --> 00:03:12.670
this and start again.

00:03:12.670 --> 00:03:23.750
We're given change in distance,
initial velocity,

00:03:23.750 --> 00:03:27.780
and acceleration, and we want
to figure out what the time

00:03:27.780 --> 00:03:31.810
is-- it's really the change in
time, but let's just assume

00:03:31.810 --> 00:03:34.070
that we start time 0, so it's
kind of the final time.

00:03:37.150 --> 00:03:40.230
Let's just start with the simple
formula: distance, or

00:03:40.230 --> 00:03:44.790
change in distance-- I'll use
them interchangeably, with a

00:03:44.790 --> 00:03:48.990
lower case d this time-- is
equal to the average velocity

00:03:48.990 --> 00:03:51.320
times time.

00:03:51.320 --> 00:03:52.570
What's the average velocity?

00:03:56.600 --> 00:04:01.730
The average velocity is just the
initial velocity plus the

00:04:01.730 --> 00:04:04.230
final velocity over 2.

00:04:04.230 --> 00:04:07.565
The only reason why we can just
average the initial and

00:04:07.565 --> 00:04:09.410
the final is because we're
assuming constant

00:04:09.410 --> 00:04:12.240
acceleration, and that's very
important, but in most

00:04:12.240 --> 00:04:14.840
projectile problems, we do have
constant acceleration--

00:04:14.840 --> 00:04:16.890
downwards-- and that's
gravity.

00:04:16.890 --> 00:04:18.899
We can assume, and we can do
this-- we can say that the

00:04:18.899 --> 00:04:22.070
average of the initial and the
final velocity is the average

00:04:22.070 --> 00:04:25.080
velocity, and then we multiply
that times time.

00:04:29.050 --> 00:04:30.700
Can we use this equation
directly?

00:04:30.700 --> 00:04:33.310
No. we know acceleration, but
don't know final velocity.

00:04:33.310 --> 00:04:37.040
If we can write this final
velocity in terms of the other

00:04:37.040 --> 00:04:41.060
things in this equation, then
maybe we can solve for time.

00:04:41.060 --> 00:04:47.960
Let's try to do that: distance
is equal to-- let me take a

00:04:47.960 --> 00:04:49.250
little side here.

00:04:49.250 --> 00:04:51.870
What do we know about
final velocity?

00:04:51.870 --> 00:04:57.850
We know that the change in
velocity is equal to

00:04:57.850 --> 00:05:01.620
acceleration times time,
assuming that time

00:05:01.620 --> 00:05:03.900
starts a t equals 0.

00:05:03.900 --> 00:05:09.040
The change in velocity is the
same thing is vf minus vi is

00:05:09.040 --> 00:05:11.860
equal to acceleration
times time.

00:05:11.860 --> 00:05:14.250
We know that the final velocity
is equal to the

00:05:14.250 --> 00:05:18.860
initial velocity plus
acceleration times time.

00:05:18.860 --> 00:05:21.550
Let's substitute that
back into what I was

00:05:21.550 --> 00:05:22.890
writing right here.

00:05:22.890 --> 00:05:28.170
We have distance is equal to the
initial velocity plus the

00:05:28.170 --> 00:05:31.530
final velocity, so let's
substitute this expression

00:05:31.530 --> 00:05:32.435
right here.

00:05:32.435 --> 00:05:36.835
The initial velocity, plus, now
the final velocity is now

00:05:36.835 --> 00:05:41.910
the initial velocity, plus
acceleration times time, and

00:05:41.910 --> 00:05:48.000
then we divide all of that
by 2 times time.

00:05:48.000 --> 00:05:53.610
We get d is equal to-- we have
2 in the numerator, we have 2

00:05:53.610 --> 00:06:02.510
initial velocity, 2vi's plus
at over 2, and all

00:06:02.510 --> 00:06:05.600
of that times t.

00:06:05.600 --> 00:06:07.730
Then we can simplify this.

00:06:07.730 --> 00:06:12.380
This equals d is equal to-- this
2 cancels out this 2, and

00:06:12.380 --> 00:06:16.260
then we distribute this t across
both terms-- so d is

00:06:16.260 --> 00:06:24.820
equal to vit plus-- this term
is at over 2, but then you

00:06:24.820 --> 00:06:27.300
multiply the t times here, too--
so it's at squared over

00:06:27.300 --> 00:06:34.190
2 plus at squared over 2.

00:06:34.190 --> 00:06:36.480
We could use this formula
if we know the change in

00:06:36.480 --> 00:06:40.490
distance, or the distance-- this
actually should be the

00:06:40.490 --> 00:06:43.190
change in distance, and the
change in time-- is equal to

00:06:43.190 --> 00:06:47.030
the initial velocity times time
plus acceleration times

00:06:47.030 --> 00:06:48.860
squared divided by 2.

00:06:48.860 --> 00:06:51.860
Let me summarize all of the
equations we have, because we

00:06:51.860 --> 00:06:55.630
really now have in our arsenal
every equation that you really

00:06:55.630 --> 00:07:00.400
need to solve one dimensional
projectile problems-- things

00:07:00.400 --> 00:07:03.080
going either just left, right,
east, west, or north, south,

00:07:03.080 --> 00:07:04.000
but not both.

00:07:04.000 --> 00:07:06.340
I will do that in
the next video.

00:07:06.340 --> 00:07:09.060
Let's summarize everything
we know.

00:07:13.820 --> 00:07:19.970
We know the change in distance
divided by the change in time

00:07:19.970 --> 00:07:23.730
is equal to velocity-- average
velocity, and it would equal

00:07:23.730 --> 00:07:26.830
velocity if velocity's not
changing, but average when

00:07:26.830 --> 00:07:29.597
velocity does change-- and we
have constant acceleration,

00:07:29.597 --> 00:07:31.700
which is an important
assumption.

00:07:31.700 --> 00:07:35.870
We know that the change in
velocity divided by the change

00:07:35.870 --> 00:07:40.500
in time is equal to
acceleration.

00:07:40.500 --> 00:07:47.300
We know the average velocity is
equal to the final velocity

00:07:47.300 --> 00:07:51.650
plus the initial velocity
over 2, and this assumes

00:07:51.650 --> 00:07:52.900
acceleration is constant.

00:08:01.870 --> 00:08:04.870
If we know the initial velocity,
acceleration, and

00:08:04.870 --> 00:08:09.540
the distance, and we want to
figure out the final velocity,

00:08:09.540 --> 00:08:20.300
we could use this formula: vf
squared equals vi squared plus

00:08:20.300 --> 00:08:25.603
2a times-- really the change in
distance, so I'm going to

00:08:25.603 --> 00:08:27.500
write the change in distance,
because that sometimes matters

00:08:27.500 --> 00:08:30.890
when we're dealing with
direction-- change in

00:08:30.890 --> 00:08:32.299
distance, but so you'll
sometimes just

00:08:32.299 --> 00:08:34.919
write this as distance.

00:08:34.919 --> 00:08:36.970
Then we just did the equation--
I think I did this

00:08:36.970 --> 00:08:40.440
in the third video, as well,
early on-- but we also learned

00:08:40.440 --> 00:08:47.870
that distance is equal to the
initial velocity times time

00:08:47.870 --> 00:08:53.230
plus at squared over 2.

00:08:53.230 --> 00:08:56.230
In that example that I did a
couple of videos ago, where we

00:08:56.230 --> 00:08:58.160
had a cliff-- actually,
I only have a minute

00:08:58.160 --> 00:08:58.890
left in this video.

00:08:58.890 --> 00:09:01.880
I will do that in the
next presentation.

00:09:01.880 --> 00:09:03.270
I'll see you soon.


Projectile motion (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:05.090
In the last video, I dropped
myself or a penny from the top

00:00:05.090 --> 00:00:06.630
of a cliff.

00:00:06.630 --> 00:00:09.770
We started off at 0 velocity,
obviously because it was

00:00:09.770 --> 00:00:13.290
stationary, and at the bottom,
it was 100 meters per second.

00:00:13.290 --> 00:00:17.660
We used that to figure out how
high the cliff was, and we

00:00:17.660 --> 00:00:20.120
figured out that the cliff
was 500 meters high.

00:00:20.120 --> 00:00:22.310
What I want to do now is let's
do that same problem, but

00:00:22.310 --> 00:00:25.310
let's do in a general form, and
see if we can figure out a

00:00:25.310 --> 00:00:29.270
general formula for a
problem like that.

00:00:29.270 --> 00:00:34.920
Let's say that you have the same
thing, and let's say the

00:00:34.920 --> 00:00:43.210
initial velocity-- you're given
the initial velocity,

00:00:43.210 --> 00:00:46.900
you're given the final velocity,
you're given the

00:00:46.900 --> 00:00:52.410
acceleration, and you want to
figure out the distance.

00:00:52.410 --> 00:00:56.980
This is what you're given,
and you want to

00:00:56.980 --> 00:00:58.230
figure out the distance.

00:01:03.730 --> 00:01:08.620
Doing it the exact same way we
did in that last presentation,

00:01:08.620 --> 00:01:09.650
but now we're now [INAUDIBLE]

00:01:09.650 --> 00:01:15.740
formulas, we know that the
change in distance is equal to

00:01:15.740 --> 00:01:20.950
the average velocity times--
we could actually say the

00:01:20.950 --> 00:01:23.300
change in time, but I'll just
say it with time, because we

00:01:23.300 --> 00:01:24.470
always assume that we
start with time

00:01:24.470 --> 00:01:27.810
equals 0-- times time.

00:01:27.810 --> 00:01:35.880
We know that the average
velocity is the final velocity

00:01:35.880 --> 00:01:43.190
plus the initial velocity
divided by 2, so that's the

00:01:43.190 --> 00:01:47.800
average velocity-- let me
highlight-- this is the same

00:01:47.800 --> 00:01:52.890
thing as this, and then
that times time.

00:01:52.890 --> 00:01:55.510
What's the time?

00:01:55.510 --> 00:01:58.120
You could figure out the time
by saying, we know how fast

00:01:58.120 --> 00:02:00.430
we're accelerating, and we know
the initial and final

00:02:00.430 --> 00:02:05.040
velocity, so we can figure out
how long we had to accelerate

00:02:05.040 --> 00:02:07.580
that fast to get that
change in velocity.

00:02:07.580 --> 00:02:09.280
Another way of saying that, or
probably a simpler way of

00:02:09.280 --> 00:02:15.730
saying that, is change in
velocity, which is the same

00:02:15.730 --> 00:02:19.220
thing as the final velocity
minus the initial velocity is

00:02:19.220 --> 00:02:22.230
equal to acceleration
times time.

00:02:25.560 --> 00:02:29.460
If you want to solve for time,
you could say the time-- if I

00:02:29.460 --> 00:02:34.240
just divide both sides of this
equation by a-- is equal to vf

00:02:34.240 --> 00:02:39.920
minus vi divided by a.

00:02:39.920 --> 00:02:42.210
We could take that and
substitute that into this

00:02:42.210 --> 00:02:45.910
equation, and remember-- this
is all change in distance.

00:02:45.910 --> 00:02:50.190
We say change in distance is
equal to-- let me write this

00:02:50.190 --> 00:02:58.740
term in yellow-- vf
plus vi over 2.

00:02:58.740 --> 00:03:03.650
Let me write this
term in green.

00:03:03.650 --> 00:03:10.520
That's times vf minus
vi over a.

00:03:15.130 --> 00:03:18.800
Then if we do a little
multiplying of expressions on

00:03:18.800 --> 00:03:21.690
the top-- you might have
recognized this-- this would

00:03:21.690 --> 00:03:27.760
be vf squared minus vi squared,
and then we multiply

00:03:27.760 --> 00:03:30.040
the denominators over 2a.

00:03:30.040 --> 00:03:33.570
So the change in distance is
equal to vf squared minus vi

00:03:33.570 --> 00:03:35.630
squared over 2a.

00:03:35.630 --> 00:03:37.710
That's exciting-- let me
write that over again.

00:03:45.000 --> 00:03:53.940
The change in distance is equal
to vf squared minus vi

00:03:53.940 --> 00:03:59.300
squared divided by 2
times acceleration.

00:03:59.300 --> 00:04:01.710
We could play around with this a
little bit, and if we assume

00:04:01.710 --> 00:04:04.290
that we started distance is
equal to 0, we could write d

00:04:04.290 --> 00:04:06.150
here, and that might
simplify things.

00:04:06.150 --> 00:04:10.600
If we multiply both sides by
2a, we get-- and I'm just

00:04:10.600 --> 00:04:12.710
going to switch this to
distance, if we assume that we

00:04:12.710 --> 00:04:15.370
always start at distances
equal to 0.

00:04:15.370 --> 00:04:19.230
di, or initial distance,
is always at point 0.

00:04:19.230 --> 00:04:24.360
We could right 2ad-- I'm just
multiplying both sides by 2a--

00:04:24.360 --> 00:04:31.860
is equal to vf squared minus vi
squared, or you could write

00:04:31.860 --> 00:04:41.030
it as vf squared is equal
to vi squared plus 2ad.

00:04:41.030 --> 00:04:43.330
I don't know what your physics
teacher might show you or

00:04:43.330 --> 00:04:46.400
written in your physics book,
but of these variations will

00:04:46.400 --> 00:04:48.010
show up in your physics book.

00:04:48.010 --> 00:04:49.870
The reason why I wanted to show
you that previous problem

00:04:49.870 --> 00:04:52.470
first is that I wanted to show
you that you could actually

00:04:52.470 --> 00:04:55.900
figure out these problems
without having to always

00:04:55.900 --> 00:04:58.780
memorize formulas and resort
to the formula.

00:04:58.780 --> 00:05:00.930
With that said, it's probably
not bad idea to memorize some

00:05:00.930 --> 00:05:04.600
form of this formula, although
you should understand how it

00:05:04.600 --> 00:05:06.892
was derived, and when
to apply it.

00:05:09.670 --> 00:05:12.230
Now that you have memorized it,
or I showed you that maybe

00:05:12.230 --> 00:05:15.920
you don't have to memorize
it, let's use this.

00:05:15.920 --> 00:05:17.970
Let's say I have the
same cliff, and it

00:05:17.970 --> 00:05:21.490
has now turned purple.

00:05:21.490 --> 00:05:27.070
It was 500 meters high-- it's
a 500 meter high cliff.

00:05:27.070 --> 00:05:29.960
This time, with the penny,
instead of just dropping it

00:05:29.960 --> 00:05:35.470
straight down, I'm going to
throw it straight up at

00:05:35.470 --> 00:05:37.930
positive 30 meters per second.

00:05:37.930 --> 00:05:40.640
The positive matters, because
remember, we said negative is

00:05:40.640 --> 00:05:43.450
down, positive is up-- that's
just the convention we use.

00:05:43.450 --> 00:05:49.370
Let's use this formula, or any
version of this formula, to

00:05:49.370 --> 00:05:56.940
figure out what our final
velocity was when we hit the

00:05:56.940 --> 00:05:59.000
bottom of the ground.

00:05:59.000 --> 00:06:01.040
This is probably the easiest
formula to use, because it

00:06:01.040 --> 00:06:03.240
actually solves for
final velocity.

00:06:03.240 --> 00:06:08.430
We can say the final velocity
vf squared is equal to the

00:06:08.430 --> 00:06:12.780
initial velocity squared-- so
what's our initial velocity?

00:06:12.780 --> 00:06:20.500
It's plus 30 meters per second,
so it's 30 meters per

00:06:20.500 --> 00:06:26.600
second squared plus 2ad.

00:06:26.600 --> 00:06:30.550
So, 2a is the acceleration of
gravity, which is minus 10,

00:06:30.550 --> 00:06:36.840
because it's going down, so it's
2a times minus 10-- I'm

00:06:36.840 --> 00:06:39.060
going to give up the units for
a second, just so I don't run

00:06:39.060 --> 00:06:43.300
out of space-- 2 times minus
10, and what's the height?

00:06:43.300 --> 00:06:45.320
What's the change in distance?

00:06:45.320 --> 00:06:47.950
Actually, I should be correct
about using change in

00:06:47.950 --> 00:06:51.490
distance, because it matters
for this problem.

00:06:51.490 --> 00:06:55.520
In this case, the final distance
is equal to minus

00:06:55.520 --> 00:07:00.550
500, and the initial distance
is equal to 0.

00:07:00.550 --> 00:07:07.170
The change in distance
is minus 500.

00:07:07.170 --> 00:07:08.450
So what does this get us?

00:07:08.450 --> 00:07:15.710
We get vf squared is equal to
900, and the negatives cancel

00:07:15.710 --> 00:07:26.570
out-- 10 times 500 is 5,000, and
5,000 times 2 is 10,000.

00:07:26.570 --> 00:07:37.220
So vf squared is equal
to 10,900.

00:07:37.220 --> 00:07:45.460
So the final velocity is equal
to the square root of 10,900.

00:07:45.460 --> 00:07:46.000
What is that?

00:07:46.000 --> 00:07:53.310
Let me bring over my trusty
Windows-provided default

00:07:53.310 --> 00:07:54.550
calculator.

00:07:54.550 --> 00:08:01.830
It's 10,900, and the
square root.

00:08:01.830 --> 00:08:15.110
It's about 104 meters per
second, so my final velocity

00:08:15.110 --> 00:08:17.530
is approximately-- that
squiggly equals is

00:08:17.530 --> 00:08:22.060
approximately-- 104
meters per second.

00:08:22.060 --> 00:08:23.090
That's interesting.

00:08:23.090 --> 00:08:26.710
If I just dropped something--
if I just drop it straight

00:08:26.710 --> 00:08:28.820
from the top-- we figured out
in the last problem that at

00:08:28.820 --> 00:08:31.810
the end, I'm at 100
meters per second.

00:08:31.810 --> 00:08:35.000
But this time, if I throw it
straight up at 30 meters per

00:08:35.000 --> 00:08:38.559
second, when the penny hits
the ground, it's actually

00:08:38.559 --> 00:08:40.480
going even faster.

00:08:40.480 --> 00:08:43.799
You might want to think about
why that is, and you might

00:08:43.799 --> 00:08:44.380
realize it.

00:08:44.380 --> 00:08:46.960
When I throw it up, the highest
point of the penny--

00:08:46.960 --> 00:08:50.230
if I throw it up at 30 meters
per second, the highest point

00:08:50.230 --> 00:08:53.320
of the penny is going to be
higher than 500 meters-- is

00:08:53.320 --> 00:08:56.250
going to make some positive
distance first, and then it's

00:08:56.250 --> 00:08:59.030
going to come down, so it's
going to have even more time

00:08:59.030 --> 00:09:00.910
to accelerate.

00:09:00.910 --> 00:09:05.660
I think that makes some
intuitive sense to you.

00:09:05.660 --> 00:09:07.890
That's all the time I have
now, and in the next

00:09:07.890 --> 00:09:10.800
presentation, maybe I'll use
this formula to solve a couple

00:09:10.800 --> 00:09:13.420
of other types of problems.


Projectile motion (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:01.740
Welcome back.

00:00:01.740 --> 00:00:05.500
I'm not going to do a bunch of
projectile motion problems,

00:00:05.500 --> 00:00:07.800
and this is because I think
you learn more just seeing

00:00:07.800 --> 00:00:09.610
someone do it, and thinking
out loud,

00:00:09.610 --> 00:00:11.040
than all the formulas.

00:00:11.040 --> 00:00:14.570
I have a strange notion that I
might have done more harm than

00:00:14.570 --> 00:00:16.700
good by confusing you with a lot
of what I did in the last

00:00:16.700 --> 00:00:19.480
couple of videos, so hopefully
I can undo any damage if I

00:00:19.480 --> 00:00:21.430
have done any, or even better--
hopefully, you did

00:00:21.430 --> 00:00:24.530
learn from those, and we'll
just add to the learning.

00:00:24.530 --> 00:00:28.350
Let's start with a
general problem.

00:00:28.350 --> 00:00:32.150
Let's say that I'm at the top
of a cliff, and I jump--

00:00:32.150 --> 00:00:35.750
instead of throwing something,
I just jump off the cliff.

00:00:35.750 --> 00:00:38.770
We won't worry about my motion
from side to side, but just

00:00:38.770 --> 00:00:40.060
assume that I go
straight down.

00:00:40.060 --> 00:00:42.740
We could even think that someone
just dropped me off of

00:00:42.740 --> 00:00:43.750
the top of the cliff.

00:00:43.750 --> 00:00:46.670
I know these are getting kind
of morbid, but let's just

00:00:46.670 --> 00:00:49.120
assume that nothing
bad happens to me.

00:00:49.120 --> 00:00:54.560
Let's say that at the top
of the cliff, my initial

00:00:54.560 --> 00:01:01.160
velocity-- velocity initial-- is
going to be 0, because I'm

00:01:01.160 --> 00:01:06.160
stationary before the person
drops me or before I jump.

00:01:06.160 --> 00:01:16.740
At the bottom of the cliff
my velocity is

00:01:16.740 --> 00:01:23.730
100 meters per second.

00:01:23.730 --> 00:01:26.750
My question is, what is the
height of this cliff?

00:01:29.590 --> 00:01:32.970
I think this is a good time
to actually introduce the

00:01:32.970 --> 00:01:35.440
direction notion of
velocity, to show

00:01:35.440 --> 00:01:37.200
you this scalar quantity.

00:01:37.200 --> 00:01:41.010
Let's assume up is positive,
and down is negative.

00:01:41.010 --> 00:01:44.160
My velocity is actually 100
meters per second down-- I

00:01:44.160 --> 00:01:46.720
could have assumed
the opposite.

00:01:46.720 --> 00:01:49.480
The final velocity is 100 meters
per second down, and

00:01:49.480 --> 00:01:51.860
since we're saying that down
is negative, and gravity is

00:01:51.860 --> 00:01:54.210
always pulling you down, we're
going to say that our

00:01:54.210 --> 00:01:59.030
acceleration is equal to
gravity, which is equal to

00:01:59.030 --> 00:02:02.935
minus 10 meters per
second squared.

00:02:06.000 --> 00:02:08.120
I just wrote that ahead of
times, because when we're

00:02:08.120 --> 00:02:09.780
dealing with anything of
throwing or jumping or

00:02:09.780 --> 00:02:13.340
anything on this planet, we
could just use this constant--

00:02:13.340 --> 00:02:15.950
the actual number is 9.81, but
I want to be able to do this

00:02:15.950 --> 00:02:19.110
without a calculator, so I'll
just stick with minus 10

00:02:19.110 --> 00:02:20.060
meters per second squared.

00:02:20.060 --> 00:02:23.440
It's pulling me down, so that's
why the minus is there.

00:02:23.440 --> 00:02:26.400
My question is: I know my
initial velocity, I know my

00:02:26.400 --> 00:02:30.700
final velocity, right before I
hit the ground or right when I

00:02:30.700 --> 00:02:32.420
hit the ground, what's
the distance?

00:02:35.770 --> 00:02:38.010
In this circumstance, what
does distance represent?

00:02:38.010 --> 00:02:41.060
Distance would be the height of
the cliff, and so how do we

00:02:41.060 --> 00:02:42.310
figure this out?

00:02:45.400 --> 00:02:48.470
What's the only formula that
we know for distance, or

00:02:48.470 --> 00:02:52.090
actually the change in distance,
but in this case,

00:02:52.090 --> 00:02:52.990
it's the same thing.

00:02:52.990 --> 00:02:58.790
Change in distance is equal
to the average velocity.

00:02:58.790 --> 00:03:01.650
When you learned this in middle
school, or probably

00:03:01.650 --> 00:03:03.580
even elementary school, you
didn't say average velocity,

00:03:03.580 --> 00:03:05.720
because you always assumed
velocity was constant-- the

00:03:05.720 --> 00:03:07.990
average and the instantaneous
velocity was

00:03:07.990 --> 00:03:09.560
kind of the same thing.

00:03:09.560 --> 00:03:12.100
Now, since the velocity is
changing, we're going to say

00:03:12.100 --> 00:03:13.790
the average velocity.

00:03:13.790 --> 00:03:16.350
So, the change in distance is
equal to the average velocity

00:03:16.350 --> 00:03:18.210
times time.

00:03:18.210 --> 00:03:20.020
This should be intuitive
to you at this point.

00:03:20.020 --> 00:03:23.530
Velocity really is just distance
divided by time, or

00:03:23.530 --> 00:03:26.290
actually, change in distance
divided by times change in

00:03:26.290 --> 00:03:28.430
time-- or, change in
distance divided by

00:03:28.430 --> 00:03:30.270
change in times is velocity.

00:03:30.270 --> 00:03:32.406
Let me actually change this
to change in time.

00:03:35.030 --> 00:03:38.145
Since we always assume-- or we
normally assume-- that we

00:03:38.145 --> 00:03:42.870
start at distance is equal to 0,
and we assume that start at

00:03:42.870 --> 00:03:46.380
time is equal to 0, we can write
distance is equal to

00:03:46.380 --> 00:03:49.970
velocity average times time.

00:03:49.970 --> 00:03:52.110
Maybe later on we'll do
situations where we're not

00:03:52.110 --> 00:03:55.620
starting at time 0, or distance
0, and in that case,

00:03:55.620 --> 00:03:57.680
we will have to be a little more
formal and say change in

00:03:57.680 --> 00:04:02.915
distance is equal to average
velocity the change in time.

00:04:06.430 --> 00:04:10.250
This is a formula we know,
and let's see what

00:04:10.250 --> 00:04:11.580
we can figure out.

00:04:11.580 --> 00:04:16.959
Can we figure out the
average velocity?

00:04:16.959 --> 00:04:19.459
The average velocity is just
the average of the initial

00:04:19.459 --> 00:04:21.940
velocity and the
final velocity.

00:04:21.940 --> 00:04:26.500
The average velocity is just
equal to the average of these

00:04:26.500 --> 00:04:32.520
two numbers: so, minus 100 plus
0 over 2-- and I'm just

00:04:32.520 --> 00:04:39.230
averaging the numbers-- equals
minus 50 meters per second.

00:04:39.230 --> 00:04:40.790
We were able to figure
that out, so can

00:04:40.790 --> 00:04:42.040
we figure out time?

00:04:44.250 --> 00:04:48.230
We know also that velocity,
or let's say the change in

00:04:48.230 --> 00:04:57.020
velocity, is equal to the
final velocity minus the

00:04:57.020 --> 00:04:58.400
initial velocity.

00:04:58.400 --> 00:05:01.640
This is nothing fancy-- you
don't have to memorize this.

00:05:01.640 --> 00:05:04.390
This hopefully is intuitive to
you, that the change is just

00:05:04.390 --> 00:05:06.755
the final velocity minus the
initial velocity, and that

00:05:06.755 --> 00:05:12.380
that equals acceleration
times time.

00:05:12.380 --> 00:05:15.920
So what's the change in velocity
in this situation?

00:05:15.920 --> 00:05:22.815
Final velocity is minus 100
meters per second, and then

00:05:22.815 --> 00:05:27.260
the initial velocity is 0, so
the change in velocity is

00:05:27.260 --> 00:05:31.290
equal to minus 100 meters
per second.

00:05:31.290 --> 00:05:34.030
I'm kind of jumping in and out
of the units, but I think you

00:05:34.030 --> 00:05:35.780
get what I'm doing.

00:05:35.780 --> 00:05:39.230
That equals acceleration
times time-- what's the

00:05:39.230 --> 00:05:39.560
acceleration?

00:05:39.560 --> 00:05:42.150
It's minus 10 meters per second
squared, because I'm

00:05:42.150 --> 00:05:46.920
going straight down-- minus 10
meters per second squared

00:05:46.920 --> 00:05:48.550
times time.

00:05:48.550 --> 00:05:51.365
This is a pretty straightforward
equation.

00:05:51.365 --> 00:05:54.400
Let's divide both sides by the
acceleration, by the minus 10

00:05:54.400 --> 00:05:59.160
meters per second squared, and
you'll get time is equal to--

00:05:59.160 --> 00:06:03.870
the negatives cancel out, as
they should, because negative

00:06:03.870 --> 00:06:07.290
time is difficult, we're
assuming positive time, and

00:06:07.290 --> 00:06:09.630
it's good we got a positive
time answer-- but the

00:06:09.630 --> 00:06:11.410
negatives cancel out
and we get time

00:06:11.410 --> 00:06:14.620
is equal to 10 seconds.

00:06:14.620 --> 00:06:17.560
There we have it: we figured out
time, we figured out the

00:06:17.560 --> 00:06:20.030
average velocity, and so now we
can figure out the height

00:06:20.030 --> 00:06:21.110
of the cliff.

00:06:21.110 --> 00:06:27.200
The distance is equal to the
average velocity minus 50

00:06:27.200 --> 00:06:32.690
meters per second times
10 seconds.

00:06:32.690 --> 00:06:36.210
The distance-- this is going to
be an interesting notion to

00:06:36.210 --> 00:06:45.240
you-- the distance it's going
to be minus 500 meters.

00:06:45.240 --> 00:06:47.930
This might not make a lot of
sense to you-- what does minus

00:06:47.930 --> 00:06:50.510
500 meters mean?

00:06:50.510 --> 00:06:53.660
This is actually right, because
this formula is

00:06:53.660 --> 00:06:54.910
actually the change
in distance.

00:06:58.780 --> 00:07:00.680
We said if we did it formally,
it would be

00:07:00.680 --> 00:07:01.900
the change in distance.

00:07:01.900 --> 00:07:09.470
So if we have a cliff-- let me
change colors with it-- and if

00:07:09.470 --> 00:07:13.780
we assume that we start at this
point right here, and

00:07:13.780 --> 00:07:21.390
this distance is equal to 0,
then the ground, if this cliff

00:07:21.390 --> 00:07:24.855
is 500 hundred meters high, your
final distance-- this is

00:07:24.855 --> 00:07:29.140
the initial distance-- your
final distance df is actually

00:07:29.140 --> 00:07:33.760
going to be at minus
500 hundred meters.

00:07:33.760 --> 00:07:35.136
We could have done it the other
way around: we could

00:07:35.136 --> 00:07:37.930
have said this is plus 500
meters, and then this is 0,

00:07:37.930 --> 00:07:40.010
but all that matters is really
the change in distance.

00:07:40.010 --> 00:07:42.240
We're saying from the top of the
cliff to the ground, the

00:07:42.240 --> 00:07:45.260
change in distance is
minus 500 meters.

00:07:45.260 --> 00:07:48.250
And minus, based on our
convention, we said minus is

00:07:48.250 --> 00:07:52.990
down, so the change is 500
meters down, and that's height

00:07:52.990 --> 00:07:53.380
of the cliff.

00:07:53.380 --> 00:07:54.370
That's pretty interesting.

00:07:54.370 --> 00:08:04.160
If you go to a 500 meter cliff--
500 is about 1,500

00:08:04.160 --> 00:08:08.960
feet-- so that's roughly the
size of maybe a very tall

00:08:08.960 --> 00:08:10.480
skyscraper, like the
World Trade Center

00:08:10.480 --> 00:08:12.440
or the Sears Tower.

00:08:12.440 --> 00:08:15.860
If you jump off of something
like that, assuming no air

00:08:15.860 --> 00:08:18.930
resistance, which is a big
assumption, or if you were to

00:08:18.930 --> 00:08:21.110
drop a penny-- because a penny
has very little air

00:08:21.110 --> 00:08:24.330
resistance-- if you were to drop
a penny off of the top of

00:08:24.330 --> 00:08:28.800
Sears Tower or a building like
that, at the bottom it will be

00:08:28.800 --> 00:08:31.600
going 100 meters per second.

00:08:31.600 --> 00:08:35.130
That's extremely fast, and
that's why you shouldn't be

00:08:35.130 --> 00:08:38.520
doing it, because that is fast
enough to kill somebody, and I

00:08:38.520 --> 00:08:41.830
don't want to give you any bad
ideas if you're a bad person.

00:08:41.830 --> 00:08:44.710
It's just interesting that
physics allows you to solve

00:08:44.710 --> 00:08:45.950
these types of problems.

00:08:45.950 --> 00:08:47.760
In the next presentation, I'm
just going to keep doing

00:08:47.760 --> 00:08:51.090
problems, and hopefully you'll
realize that everything really

00:08:51.090 --> 00:08:54.880
just boils down to average
velocity-- change in velocity

00:08:54.880 --> 00:08:57.830
is acceleration times time,
and change in distance is

00:08:57.830 --> 00:09:03.380
equal to change in time times
average velocity, which we all

00:09:03.380 --> 00:09:05.180
did just now.

00:09:05.180 --> 00:09:06.940
I'll see you in the
next presentation.


Introduction to motion (part 3)

WEBVTT
Kind: captions
Language: en

00:00:00.950 --> 00:00:02.640
I am back.

00:00:02.640 --> 00:00:03.510
Where were we?

00:00:03.510 --> 00:00:08.750
We were saying that we know that
velocity, or kind of a

00:00:08.750 --> 00:00:11.440
change in velocity, is
acceleration times time.

00:00:11.440 --> 00:00:14.660
I just wrote that a little bit
more formally, really kind of

00:00:14.660 --> 00:00:15.990
incorporating the change
in velocity.

00:00:15.990 --> 00:00:16.580
Right?

00:00:16.580 --> 00:00:19.460
The final velocity is equal to
the initial velocity plus

00:00:19.460 --> 00:00:20.535
acceleration times time.

00:00:20.535 --> 00:00:22.180
I actually could have written
it like this: I could have

00:00:22.180 --> 00:00:27.640
written vf minus vi is equal
to acceleration times time,

00:00:27.640 --> 00:00:30.010
and this is the change
in velocity.

00:00:30.010 --> 00:00:31.190
Actually, that's the way
I should be doing it.

00:00:31.190 --> 00:00:34.280
As you can tell, I kind of do
some of this stuff on the fly,

00:00:34.280 --> 00:00:37.070
but I do that for a reason--
it's because I want you to get

00:00:37.070 --> 00:00:40.460
the same intuition that I
hopefully have, instead of

00:00:40.460 --> 00:00:43.480
just kind of doing it in a very
formal way in a book, and

00:00:43.480 --> 00:00:46.400
sometimes the book doesn't
necessarily make the

00:00:46.400 --> 00:00:49.110
connections in the
most natural way.

00:00:49.110 --> 00:00:52.130
This is going straight from my
brain to this video, and

00:00:52.130 --> 00:00:54.410
hopefully into your brain.

00:00:54.410 --> 00:00:56.560
These are all ways of saying the
same thing, and I actually

00:00:56.560 --> 00:01:01.960
should write this as change in
velocity-- that triangle, or

00:01:01.960 --> 00:01:03.910
delta, just means change.

00:01:03.910 --> 00:01:06.220
The final velocity, my initial
velocity, is equal to

00:01:06.220 --> 00:01:08.590
acceleration times time.

00:01:08.590 --> 00:01:11.200
The average velocity, you could
just figure-- you take

00:01:11.200 --> 00:01:12.830
the final, and you take the
initial, and you average the

00:01:12.830 --> 00:01:16.220
two, and it's equal to this.

00:01:16.220 --> 00:01:21.280
Then I said, we know what the
final velocity is-- this is

00:01:21.280 --> 00:01:24.920
the final velocity, and the
average velocity is this.

00:01:24.920 --> 00:01:27.470
We substitute it for the final
velocity, and then we came to

00:01:27.470 --> 00:01:31.860
this equation for average
velocity.

00:01:31.860 --> 00:01:35.650
Then before I almost ran out of
time, I said I'm going to

00:01:35.650 --> 00:01:38.720
take this formula for the
average velocity-- and I

00:01:38.720 --> 00:01:40.820
really encourage you to just
play around with these

00:01:40.820 --> 00:01:43.100
formulas yourself and derive
it yourself, because it's

00:01:43.100 --> 00:01:45.930
going to pay huge rewards later
on when you forget the

00:01:45.930 --> 00:01:50.650
formulas on your exam, but you
can work it out anyway.

00:01:50.650 --> 00:01:52.430
We have this formula for average
velocity, and let's

00:01:52.430 --> 00:01:57.600
substitute it back into this, so
we can say that distance is

00:01:57.600 --> 00:02:07.290
equal to the average velocity,
and that's this: vi plus at

00:02:07.290 --> 00:02:12.680
over 2 times time.

00:02:12.680 --> 00:02:16.160
If we just distributed that t,
we have the initial velocity

00:02:16.160 --> 00:02:26.440
times time, plus acceleration
times time squared over 2.

00:02:26.440 --> 00:02:29.670
So, distance is equal to the
initial velocity-- let me draw

00:02:29.670 --> 00:02:32.200
a line here, so we don't confuse
things-- distance is

00:02:32.200 --> 00:02:36.400
equal to the initial velocity
times time plus acceleration

00:02:36.400 --> 00:02:40.510
times time squared
divided by 2.

00:02:40.510 --> 00:02:43.250
Sometimes the physics teacher
might just teach at squared

00:02:43.250 --> 00:02:45.100
2-- that's sometimes what people
memorize, and that's

00:02:45.100 --> 00:02:48.490
because in a lot of these
projectile motion problems,

00:02:48.490 --> 00:02:51.460
your initial velocity is 0,
especially when you're

00:02:51.460 --> 00:02:52.440
dropping a rock.

00:02:52.440 --> 00:02:56.610
If your initial velocity is 0,
this term would cancel out.

00:02:56.610 --> 00:02:59.640
If you do that last problem that
we just did using this

00:02:59.640 --> 00:03:02.680
example, you'll get
the same answer.

00:03:02.680 --> 00:03:06.230
I said we're accelerating with
gravity, so a is equal to 10

00:03:06.230 --> 00:03:11.490
meters per second squared,
then time is equal to 2

00:03:11.490 --> 00:03:17.600
seconds, and then initial
velocity is equal to 0, and

00:03:17.600 --> 00:03:19.110
let me make some space here.

00:03:22.360 --> 00:03:25.420
The initial velocity is 0-- so
this term just cancels out--

00:03:25.420 --> 00:03:29.570
plus acceleration, 10
meters per second

00:03:29.570 --> 00:03:32.170
squared times time squared.

00:03:46.830 --> 00:03:53.620
You have 10 meters per second
squared times time squared,

00:03:53.620 --> 00:03:56.750
time is two seconds,
so it's 4.

00:03:56.750 --> 00:04:00.210
Since we've squared the number,
we should also square

00:04:00.210 --> 00:04:04.780
the units, so it's 4 seconds
squared, and all of

00:04:04.780 --> 00:04:07.180
that is over 2.

00:04:07.180 --> 00:04:11.620
Like we learned before-- 10
times 4 divided 2 is 20, and

00:04:11.620 --> 00:04:13.800
we have seconds squared the
denominator and we have

00:04:13.800 --> 00:04:15.430
seconds squared in the
numerator here.

00:04:15.430 --> 00:04:18.353
They cancel out, and we're
just left with meters.

00:04:21.329 --> 00:04:23.690
Actually, I'm going to leave it
there for now, and in the

00:04:23.690 --> 00:04:27.630
next presentation, I'll
explore some of these

00:04:27.630 --> 00:04:29.150
mechanics even further.

00:04:29.150 --> 00:04:30.840
I'll see you soon.


Introduction to motion (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.980 --> 00:00:01.800
All right.

00:00:01.800 --> 00:00:05.090
Where I left off in the last
presentation I was dropping a

00:00:05.090 --> 00:00:08.990
penny from the top of a
building-- once again, you

00:00:08.990 --> 00:00:10.850
should not do, because you
can kill somebody.

00:00:14.060 --> 00:00:20.380
Here's the building, and here's
the bad person who's

00:00:20.380 --> 00:00:22.560
going to drop something.

00:00:22.560 --> 00:00:27.480
Let's say they just hold it
out, and the penny drops.

00:00:27.480 --> 00:00:29.420
The penny is going to accelerate
at the rate of

00:00:29.420 --> 00:00:32.030
gravity, so it's going to
accelerate downwards at 10

00:00:32.030 --> 00:00:34.070
meters per second squared.

00:00:34.070 --> 00:00:36.180
Let's start with an interesting
question.

00:00:36.180 --> 00:00:41.730
After after two seconds-- and
lets say they drop it right at

00:00:41.730 --> 00:00:53.800
t equals 0-- so after two
seconds how fast is it going?

00:00:53.800 --> 00:00:57.435
Time is equal to two seconds--
we could even say this change

00:00:57.435 --> 00:00:59.450
in time, but we're assuming that
we're starting at time

00:00:59.450 --> 00:01:02.970
equals 0, so and change in
time the same thing.

00:01:02.970 --> 00:01:05.193
If time is equal to two seconds,
how fast is it going

00:01:05.193 --> 00:01:10.030
to be going after two seconds?

00:01:10.030 --> 00:01:11.240
Let's use that formula.

00:01:11.240 --> 00:01:16.610
Velocity is equal to
acceleration-- acceleration is

00:01:16.610 --> 00:01:19.060
the acceleration of gravity,
and that's 10 meters per

00:01:19.060 --> 00:01:24.910
second squared-- so velocity
will be 10 meters per second

00:01:24.910 --> 00:01:29.940
squared times time, which
is times two seconds.

00:01:37.710 --> 00:01:41.310
We can multiply the numbers,
and you get 20.

00:01:41.310 --> 00:01:44.520
Just like the numbers, you can
treat the units almost like

00:01:44.520 --> 00:01:48.230
variables-- the seconds is the
same thing as this s, so this

00:01:48.230 --> 00:01:50.120
s is going to the numerator, and
then you have an s squared

00:01:50.120 --> 00:01:53.120
in the denominator.

00:01:53.120 --> 00:01:56.190
This s will cancel out with one
of the two s's that are

00:01:56.190 --> 00:01:59.600
multiplied down here, so we'll
end up-- actually, let me

00:01:59.600 --> 00:02:04.520
write it down-- it'll be 10
meter seconds per second

00:02:04.520 --> 00:02:07.900
squared, and that's the
same thing as 20.

00:02:07.900 --> 00:02:12.430
That cancels out, this makes
that 1, and so that equals 20

00:02:12.430 --> 00:02:14.520
meters per second.

00:02:14.520 --> 00:02:16.610
Hopefully, you're starting to
get a little intuition of why

00:02:16.610 --> 00:02:20.290
acceleration's units are meters
per second squared.

00:02:20.290 --> 00:02:25.820
After two seconds, we're going
20 meters per second.

00:02:25.820 --> 00:02:29.090
Let me ask you a slightly more
difficult problem that might

00:02:29.090 --> 00:02:30.860
have not been obvious to you.

00:02:30.860 --> 00:02:35.540
After two seconds, how far
has the penny gone?

00:02:35.540 --> 00:02:38.700
This is interesting.

00:02:38.700 --> 00:02:42.250
We have this formula here:
distance is equal to velocity

00:02:42.250 --> 00:02:48.410
times time, but the velocity is
changing the entire time.

00:02:48.410 --> 00:02:52.640
We know after two seconds that
the velocity is 20 meters per

00:02:52.640 --> 00:02:55.530
second, so we could call this
the final velocity-- we'll

00:02:55.530 --> 00:02:57.340
called v sub f.

00:02:57.340 --> 00:02:59.870
That's just a fancy way of
saying final velocity.

00:02:59.870 --> 00:03:01.880
Right when we start
at t equals 0,

00:03:01.880 --> 00:03:04.480
what was the velocity?

00:03:04.480 --> 00:03:08.870
Right when it started, the
initial velocity-- v sub i,

00:03:08.870 --> 00:03:14.230
for initial-- is equal to
0 meters per second.

00:03:17.610 --> 00:03:20.290
Can we use this formula?

00:03:20.290 --> 00:03:23.390
You might think of a way
to already do it.

00:03:23.390 --> 00:03:26.990
Since the acceleration is
constant, and you can only do

00:03:26.990 --> 00:03:30.130
this when the acceleration is
constant-- most of what you'll

00:03:30.130 --> 00:03:32.152
encounter in a first year
physics course, the

00:03:32.152 --> 00:03:34.480
acceleration will be constant,
and especially when you're

00:03:34.480 --> 00:03:36.160
dealing with gravity, the
acceleration will be

00:03:36.160 --> 00:03:40.770
constant-- you can actually take
the average velocity to

00:03:40.770 --> 00:03:43.270
figure out the distance.

00:03:43.270 --> 00:03:46.370
So what was the average velocity
over the two seconds?

00:03:50.400 --> 00:03:53.550
My final velocity was 20 meters
per second, and my

00:03:53.550 --> 00:03:55.650
initial velocity was 0
meters per second.

00:03:55.650 --> 00:03:59.560
Obviously, I went continuously
over those two seconds from 0

00:03:59.560 --> 00:04:03.940
to 20, so my average velocity--
actually, I've

00:04:03.940 --> 00:04:06.620
never seen it done this way
before, but let's just call it

00:04:06.620 --> 00:04:10.910
average velocity-- is equal to
the final velocity plus the

00:04:10.910 --> 00:04:13.300
initial velocity divided by 2.

00:04:13.300 --> 00:04:16.490
I just took the average of the
initial and the final, which

00:04:16.490 --> 00:04:23.830
is 20 plus 0-- which is 20--
divided by 2, which is equal

00:04:23.830 --> 00:04:30.820
to 10 meters per second.

00:04:30.820 --> 00:04:33.850
Right when I let go of the rock,
the ball, or the penny,

00:04:33.850 --> 00:04:36.780
whatever I'm dropping, the thing
is stationary, and so

00:04:36.780 --> 00:04:38.400
it's 0 meters per second.

00:04:38.400 --> 00:04:42.110
After two seconds-- we used this
acceleration formula--

00:04:42.110 --> 00:04:43.840
after two seconds,
it accelerated to

00:04:43.840 --> 00:04:46.290
20 meters per second.

00:04:46.290 --> 00:04:49.780
Over the course of those two
seconds, its average velocity

00:04:49.780 --> 00:04:52.340
was 10 meters per second.

00:04:52.340 --> 00:04:56.600
We can now use that average
velocity in this

00:04:56.600 --> 00:04:58.680
formula right here.

00:04:58.680 --> 00:05:02.620
The average velocity, distance
equals average velocity times

00:05:02.620 --> 00:05:05.530
time-- you can make a mental
footnote, so it's average

00:05:05.530 --> 00:05:08.370
velocity times time when the
velocity is changing and

00:05:08.370 --> 00:05:10.570
acceleration is constant, which
is most of what you'll

00:05:10.570 --> 00:05:13.650
see in most projectile
motion problems.

00:05:13.650 --> 00:05:17.180
Now we could say distance is
equal to the average velocity

00:05:17.180 --> 00:05:24.590
times time, which equals 10
meters per second times two

00:05:24.590 --> 00:05:27.650
seconds-- once again,
the s's cancel out--

00:05:27.650 --> 00:05:30.070
so we're at 20 meters.

00:05:30.070 --> 00:05:33.030
After two seconds, not only is
my velocity 20 meters per

00:05:33.030 --> 00:05:36.110
second down-- once again, if I
said speed, it would just be

00:05:36.110 --> 00:05:43.930
20 meters per second-- but my
distance is the ball, or the

00:05:43.930 --> 00:05:46.270
rock, assuming no
air resistance,

00:05:46.270 --> 00:05:51.900
has dropped 20 meters.

00:05:51.900 --> 00:05:55.020
Hopefully, that makes a little
bit of intuition for you.

00:05:57.550 --> 00:06:00.760
If you are taking physics--
which you don't have to view

00:06:00.760 --> 00:06:03.110
these videos, that's the idea--
I wanted to show you

00:06:03.110 --> 00:06:07.300
that this is actually exactly
like one of the formulas that

00:06:07.300 --> 00:06:09.860
you'll see in your
physics class.

00:06:09.860 --> 00:06:11.605
It's kind of a shame, but people
tend to just memorize

00:06:11.605 --> 00:06:15.040
it in physics without-- when
they're learning projectile

00:06:15.040 --> 00:06:19.330
motion without really
appreciating that it just

00:06:19.330 --> 00:06:21.900
comes from distance is equal
to velocity times time.

00:06:26.400 --> 00:06:27.900
Before, I said velocity
is equal to

00:06:27.900 --> 00:06:29.340
acceleration times to time.

00:06:29.340 --> 00:06:31.440
Let me just expand that a little
bit, because I assume

00:06:31.440 --> 00:06:33.420
that my initial velocity is 0.

00:06:33.420 --> 00:06:37.270
Let me just say that the final
velocity is equal to the

00:06:37.270 --> 00:06:40.870
initial velocity, because you
could already be going 10

00:06:40.870 --> 00:06:43.360
meters per second, and then
you're going to accelerate.

00:06:43.360 --> 00:06:46.150
Final velocity is equal to the
initial velocity-- this is an

00:06:46.150 --> 00:06:50.110
i-- plus acceleration
times time.

00:06:52.840 --> 00:06:57.650
We said that the distance-- we
could rewrite this as the

00:06:57.650 --> 00:07:01.360
distance is equal to the average
velocity times time.

00:07:03.920 --> 00:07:05.915
I just realized how funny
that character looks.

00:07:09.350 --> 00:07:11.840
So, the final velocity is equal
to the initial velocity

00:07:11.840 --> 00:07:15.090
plus acceleration times time,
and the distance is equal to

00:07:15.090 --> 00:07:18.910
the average velocity
times time.

00:07:18.910 --> 00:07:20.820
Let's see if we can use these
two formulas, which we

00:07:20.820 --> 00:07:22.810
essentially just applied in
the previous example-- we

00:07:22.810 --> 00:07:27.300
didn't do it exactly so
formally-- to come up with a

00:07:27.300 --> 00:07:32.130
formula for distance, given
acceleration and time.

00:07:32.130 --> 00:07:37.230
We know that the average
velocity-- oh, I switched

00:07:37.230 --> 00:07:43.210
colors-- the average velocity is
equal to the final velocity

00:07:43.210 --> 00:07:46.180
plus the initial velocity
divided by 2.

00:07:51.720 --> 00:07:53.880
What is the final velocity?

00:07:53.880 --> 00:08:00.650
The final velocity is equal to
this: substitute, and so the

00:08:00.650 --> 00:08:09.200
initial velocity plus
acceleration times time plus

00:08:09.200 --> 00:08:13.470
the initial velocity-- my i's
are getting blurred, they're

00:08:13.470 --> 00:08:15.590
not showing up-- these are all
i's for initial velocity.

00:08:15.590 --> 00:08:17.730
They look like a 2, but I
think you get the idea--

00:08:17.730 --> 00:08:19.000
that's all initial velocity.

00:08:19.000 --> 00:08:22.300
All of that is over 2.

00:08:22.300 --> 00:08:24.910
The average velocity is equal
to the initial velocity plus

00:08:24.910 --> 00:08:29.115
acceleration times time, plus
the initial velocity, all of

00:08:29.115 --> 00:08:30.710
that divided by 2.

00:08:30.710 --> 00:08:37.429
That just equals 2 times initial
velocity-- that looks

00:08:37.429 --> 00:08:41.929
like an i now-- plus
acceleration times time

00:08:41.929 --> 00:08:47.740
divided by 2, and that equals
the initial velocity plus

00:08:47.740 --> 00:08:51.410
acceleration times time
divided by 2.

00:08:51.410 --> 00:08:53.720
This might be intuitive for you,
as well, that the average

00:08:53.720 --> 00:08:58.590
velocity is equal to your
initial velocity plus-- this

00:08:58.590 --> 00:09:04.240
is essentially the difference
between how much you're

00:09:04.240 --> 00:09:07.430
accelerating over that time and
speed is going to be that

00:09:07.430 --> 00:09:09.780
divided by 2, because we're
taking the average.

00:09:09.780 --> 00:09:11.570
If what I just said confused
you, don't worry about it; you

00:09:11.570 --> 00:09:14.400
could just backtrack into
what we said before.

00:09:14.400 --> 00:09:17.940
Think about a lot of these
formulas yourself, and plug in

00:09:17.940 --> 00:09:20.900
numbers, and I think it'll
start to make more sense.

00:09:20.900 --> 00:09:23.520
We figured out that the average
velocity is equal to

00:09:23.520 --> 00:09:26.670
the initial velocity plus
acceleration times time.

00:09:26.670 --> 00:09:29.430
We can just substitute
that back into

00:09:29.430 --> 00:09:32.280
this original equation.

00:09:32.280 --> 00:09:34.620
Once again, I realized I'm
running out of time, so I'll

00:09:34.620 --> 00:09:36.420
see you shortly.


Introduction to motion

WEBVTT
Kind: captions
Language: en

00:00:01.460 --> 00:00:03.730
Hello, and welcome.

00:00:03.730 --> 00:00:07.530
This will be the first in the
series of lectures on physics.

00:00:07.530 --> 00:00:11.120
My goal is really to give you
an intuitive feeling of what

00:00:11.120 --> 00:00:13.340
physics is all about, because
especially on the mechanics

00:00:13.340 --> 00:00:16.530
side of things-- projectile
motion, force, and momentum--

00:00:16.530 --> 00:00:19.170
it's actually pretty
intuitive.

00:00:19.170 --> 00:00:21.980
Let me know if I'm not giving
you an intuitive sense.

00:00:21.980 --> 00:00:25.170
Let's just start with probably
the most basic formula in

00:00:25.170 --> 00:00:27.890
physics, and a lot of you have
already seen this: it's

00:00:27.890 --> 00:00:34.690
distance is equal to velocity
times time.

00:00:34.690 --> 00:00:36.090
You might have seen it in
different forms-- you might

00:00:36.090 --> 00:00:38.560
have seen it written as the
distance is equal to rate

00:00:38.560 --> 00:00:42.850
times time, distance is equal
to speed times time, or

00:00:42.850 --> 00:00:44.910
instead of a d, if you're
doing it in math class,

00:00:44.910 --> 00:00:46.830
sometimes you'll write an s for
distance, because they use

00:00:46.830 --> 00:00:49.190
d for derivatives-- but it's
all the same thing.

00:00:49.190 --> 00:00:52.140
It just says the distance you
travel is equal to the speed

00:00:52.140 --> 00:00:54.570
or the velocity you
travel times time.

00:00:54.570 --> 00:00:57.490
Before I do a couple of quick
example problems just to hit

00:00:57.490 --> 00:00:59.780
this point home, I just want to
make a slight distinction

00:00:59.780 --> 00:01:01.100
between velocity and speed.

00:01:03.920 --> 00:01:07.300
When people use it in everyday
language, it tends to be used

00:01:07.300 --> 00:01:10.330
interchangeably-- it's how fast
are you going, velocity

00:01:10.330 --> 00:01:11.810
versus speed.

00:01:11.810 --> 00:01:14.590
Technically, there's a
difference: velocity is a

00:01:14.590 --> 00:01:21.700
vector of measurement, and
speed is a scalar.

00:01:21.700 --> 00:01:24.740
I probably have already confused
you, but all you have

00:01:24.740 --> 00:01:30.550
to know is a vector has a
magnitude and a direction.

00:01:30.550 --> 00:01:33.580
If I were to give a velocity,
I really shouldn't just say

00:01:33.580 --> 00:01:34.740
five miles per hour.

00:01:34.740 --> 00:01:37.930
I should say five miles
per hour north.

00:01:37.930 --> 00:01:40.630
Or, instead of saying five miles
per hour north, I could

00:01:40.630 --> 00:01:43.140
say negative five miles
per hour south.

00:01:43.140 --> 00:01:43.390
Right?

00:01:43.390 --> 00:01:45.740
Those would be the same thing.

00:01:45.740 --> 00:01:50.320
Speed, or a scalar, only has
magnitude, so speed would say

00:01:50.320 --> 00:01:52.710
five miles per hour, but
I don't know what

00:01:52.710 --> 00:01:53.770
direction I'm going in.

00:01:53.770 --> 00:01:55.990
I could be going forward,
backward, left, right, north,

00:01:55.990 --> 00:01:58.030
south, up, down, who knows?

00:01:58.030 --> 00:02:02.130
That's the technical difference
between a vector

00:02:02.130 --> 00:02:05.180
and a scalar and between
velocity and speed.

00:02:05.180 --> 00:02:08.210
It might not seem so obvious,
and probably on these few

00:02:08.210 --> 00:02:11.150
example problems we're doing
right now, the distinction

00:02:11.150 --> 00:02:13.720
probably could be used
interchangeably.

00:02:13.720 --> 00:02:17.420
Later on, as we progress to
fancier problems, I think

00:02:17.420 --> 00:02:21.020
you'll see that velocity is a
more useful notion, because

00:02:21.020 --> 00:02:23.010
there's the idea of a
negative velocity.

00:02:23.010 --> 00:02:25.440
You can go in one direction,
which is positive, and then

00:02:25.440 --> 00:02:27.800
you can turn around
and go negative.

00:02:27.800 --> 00:02:30.710
With speed, there's no
direction involved.

00:02:30.710 --> 00:02:32.860
With that said-- I don't want
to dwell on that too much,

00:02:32.860 --> 00:02:34.380
because I don't want to make
you think that this is

00:02:34.380 --> 00:02:38.580
difficult-- let's do a couple
of really quick problems.

00:02:38.580 --> 00:02:44.830
Let's say I went 50 meters,
so distance is equal to 50

00:02:44.830 --> 00:02:47.230
meters-- I go 50 meters.

00:02:47.230 --> 00:02:52.640
Time is equal to-- this is
should really be distance,

00:02:52.640 --> 00:02:55.270
because it actually changes in
distance, but for the problems

00:02:55.270 --> 00:02:57.706
we'll do, it doesn't make a
difference-- let's say the

00:02:57.706 --> 00:03:01.630
time is-- and this could be
change in time, as well, the

00:03:01.630 --> 00:03:06.990
time is 10 seconds.

00:03:06.990 --> 00:03:12.480
So if we use that formula, we
have 50 meters is equal to

00:03:12.480 --> 00:03:17.240
velocity times 10 seconds.

00:03:17.240 --> 00:03:19.560
This is a pretty simple
algebra equation.

00:03:19.560 --> 00:03:23.700
If we want to solve for
velocity, we just divide both

00:03:23.700 --> 00:03:26.740
sides of this situation
by 10 seconds.

00:03:26.740 --> 00:03:28.570
I'm doing this because I
actually want to show you that

00:03:28.570 --> 00:03:30.550
when you divide the numbers,
you should also divide the

00:03:30.550 --> 00:03:32.620
units with them, and then
you always end up

00:03:32.620 --> 00:03:33.920
with the right units.

00:03:33.920 --> 00:03:36.850
Let's multiply both sides
of this equation

00:03:36.850 --> 00:03:38.960
by 1 over 10 seconds.

00:03:38.960 --> 00:03:47.750
I get 1 over 10 seconds times 50
meters is equal to velocity

00:03:47.750 --> 00:03:51.470
times-- I'll write it all out,
later on, I'll skip some

00:03:51.470 --> 00:03:55.800
steps, and this obviously
cancels out, because that's

00:03:55.800 --> 00:03:59.965
why we did in the first place--
and then the 50 and

00:03:59.965 --> 00:04:03.230
the 10 cancel and
this becomes 5.

00:04:03.230 --> 00:04:09.240
We're left with 5, meter in the
numerator, and the seconds

00:04:09.240 --> 00:04:11.380
in the denominator.

00:04:11.380 --> 00:04:14.910
I could write it as sec, or just
an s-- so 5 meters per

00:04:14.910 --> 00:04:17.540
second is our velocity, and
you could have done that.

00:04:17.540 --> 00:04:21.350
The one thing I just wanted to
highlight is that the units,

00:04:21.350 --> 00:04:24.020
you can manipulate with the
numbers, and then you always

00:04:24.020 --> 00:04:25.380
get the right answer.

00:04:25.380 --> 00:04:27.700
It might have been obvious in
this case-- you didn't have to

00:04:27.700 --> 00:04:29.930
do it this way-- but once again,
later on, when we start

00:04:29.930 --> 00:04:35.480
doing power and work and energy,
which is actually the

00:04:35.480 --> 00:04:37.640
same thing as work, but once we
start doing those things,

00:04:37.640 --> 00:04:40.670
then the units might not
seem so obvious.

00:04:40.670 --> 00:04:45.590
It's good to be able to deal
with the units this way.

00:04:45.590 --> 00:04:50.020
We could solve-- if I said that
the velocity is equal to

00:04:50.020 --> 00:04:59.460
7 meters per second, and that
time is equal to 5 seconds--

00:04:59.460 --> 00:05:00.620
how far did I go?

00:05:00.620 --> 00:05:03.770
I could use that formula again:
distance is equal to

00:05:03.770 --> 00:05:09.070
velocity, which is 7 meters per
second, times time, which

00:05:09.070 --> 00:05:11.820
is 5 seconds.

00:05:11.820 --> 00:05:15.550
Once again, not only can we
multiply the numbers-- 7 times

00:05:15.550 --> 00:05:19.600
5 is 35-- but we can multiply
the units, so we have meters

00:05:19.600 --> 00:05:22.360
over second times second.

00:05:22.360 --> 00:05:24.200
You can almost treat them like
variables, but they're not--

00:05:24.200 --> 00:05:25.570
they're units.

00:05:25.570 --> 00:05:29.180
Meters over seconds times
seconds-- the second in the

00:05:29.180 --> 00:05:31.300
numerator and the second in the
denominator cancel out,

00:05:31.300 --> 00:05:33.670
and so you're left
with 35 meters.

00:05:33.670 --> 00:05:36.550
There-- not only do you have the
right number, you have the

00:05:36.550 --> 00:05:37.780
right units.

00:05:37.780 --> 00:05:41.020
Actually, this is going to be
super useful when you have to

00:05:41.020 --> 00:05:46.390
convert from centimeters to
meters, and hours to seconds,

00:05:46.390 --> 00:05:47.910
and all of that-- maybe we'll
do a couple of examples.

00:05:47.910 --> 00:05:51.060
Actually, a while ago, I
actually made a separate video

00:05:51.060 --> 00:05:55.350
on unit conversion, and that's
going to come and really handy

00:05:55.350 --> 00:05:57.540
when we do the physics.

00:05:57.540 --> 00:06:00.100
With that out of the way, let's
make things a little bit

00:06:00.100 --> 00:06:01.350
more complicated.

00:06:05.680 --> 00:06:09.550
Most of what you've probably
experienced so far-- distance

00:06:09.550 --> 00:06:14.420
is equal to velocity times time,
or rate times time-- is

00:06:14.420 --> 00:06:16.590
where velocity is constant.

00:06:16.590 --> 00:06:19.080
You're going 30 meters per
second, you're always going

00:06:19.080 --> 00:06:21.480
the 30 meters per second, and
you'll stay going 30 meters

00:06:21.480 --> 00:06:22.200
per second.

00:06:22.200 --> 00:06:28.500
But we know from moving,
generally, that your velocity

00:06:28.500 --> 00:06:30.700
isn't-- sometimes, you're
stationary, then sometimes

00:06:30.700 --> 00:06:33.460
you're moving, and in order to
start stationary, and then get

00:06:33.460 --> 00:06:37.760
moving, your velocity
has to change.

00:06:37.760 --> 00:06:40.860
How could we describe a
change in velocity?

00:06:40.860 --> 00:06:43.120
Once again, I don't think I'm
teaching you anything

00:06:43.120 --> 00:06:44.580
fundamentally new.

00:06:44.580 --> 00:06:47.090
You know what it is--
it's acceleration.

00:06:47.090 --> 00:06:52.300
So, velocity is acceleration
times time.

00:06:52.300 --> 00:06:57.690
There's a pretty good analogy
here: just as distance is

00:06:57.690 --> 00:06:59.340
velocity times time,
and velocity is

00:06:59.340 --> 00:07:00.650
acceleration times time.

00:07:00.650 --> 00:07:05.920
Or, you view it as the change in
distance over the change in

00:07:05.920 --> 00:07:12.020
time is velocity, while the
change in velocity versus the

00:07:12.020 --> 00:07:16.940
change in time is equal
to acceleration.

00:07:16.940 --> 00:07:19.220
My phone's ringing-- let me
answer that a little bit

00:07:19.220 --> 00:07:23.440
later, because once again,
you are more important.

00:07:23.440 --> 00:07:25.000
So where was I?

00:07:25.000 --> 00:07:28.060
I was saying how they're very
similar-- at least, there's

00:07:28.060 --> 00:07:30.550
kind of an analogy here.

00:07:30.550 --> 00:07:32.500
What can we do with
this notion?

00:07:32.500 --> 00:07:36.460
I'm going to do a bunch of what
I think you'll find is

00:07:36.460 --> 00:07:39.120
pretty useful-- they're called
projectile problems, and

00:07:39.120 --> 00:07:42.120
projectile problems involve the
acceleration of gravity.

00:07:42.120 --> 00:07:47.620
We could do other acceleration
problems involving the

00:07:47.620 --> 00:07:49.470
acceleration of cars,
and actually, I

00:07:49.470 --> 00:07:51.040
probably will do that.

00:07:51.040 --> 00:07:58.050
The acceleration of gravity is
actually 9.8 meters per second

00:07:58.050 --> 00:08:02.400
downwards-- once again,
acceleration is actually not

00:08:02.400 --> 00:08:04.530
downwards, but towards the
center of the earth.

00:08:04.530 --> 00:08:08.060
Acceleration is also a vector
quantity, but for the sake of

00:08:08.060 --> 00:08:13.260
our computations, we'll just
say 10 meters per second.

00:08:13.260 --> 00:08:17.850
Acceleration is equal to g-- g
is normally the variable used

00:08:17.850 --> 00:08:21.160
when people talk about the
acceleration of gravity, and

00:08:21.160 --> 00:08:27.290
let's say that equals 10 meters
per second squared.

00:08:27.290 --> 00:08:31.410
I know what you're thinking,
that this is kind of a strange

00:08:31.410 --> 00:08:34.270
set of units-- meters per second
squared, and it's hard

00:08:34.270 --> 00:08:35.059
to visualize.

00:08:35.059 --> 00:08:38.610
That's acceleration, and I think
once you see it used in

00:08:38.610 --> 00:08:41.070
some of these formulas, it'll
start to make a little sense

00:08:41.070 --> 00:08:46.770
in terms of how these
units work out.

00:08:46.770 --> 00:08:49.080
Let's start with a fairly
simple problem.

00:08:49.080 --> 00:08:53.640
Let's say I drop a rock from-- I
don't recommend you do this,

00:08:53.640 --> 00:08:57.170
you could kill somebody-- not
a rock, a penny, from the

00:08:57.170 --> 00:09:01.500
Empire State Building, and I'm
assuming no air resistance.

00:09:01.500 --> 00:09:03.920
Actually, I just looked at the
clock, and I realize that I'm

00:09:03.920 --> 00:09:05.030
running out of time.

00:09:05.030 --> 00:09:06.830
I'm going to actually start
this problem in the next

00:09:06.830 --> 00:09:07.760
presentation.

00:09:07.760 --> 00:09:09.370
I'll see you soon.


Trig identities part 3 (part 5 if you watch the proofs)

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.400
Welcome back, because I
was hitting against the

00:00:03.400 --> 00:00:04.760
YouTube 10-minute limit.

00:00:04.760 --> 00:00:06.110
But all I was saying
is, we said, you know,

00:00:06.110 --> 00:00:07.260
cosine of minus a.

00:00:07.260 --> 00:00:12.500
So I drew a right triangle with
a, and then I showed you minus

00:00:12.500 --> 00:00:15.160
a, and I said, well, all of the
lengths are going to be the

00:00:15.160 --> 00:00:18.520
same, but now the direction
of-- and we're kind of assuming

00:00:18.520 --> 00:00:19.670
this is all on the unit circle.

00:00:19.670 --> 00:00:23.410
If you don't remember the unit
circle, maybe you'll want to

00:00:23.410 --> 00:00:26.000
rewatch the videos
we have on that.

00:00:26.000 --> 00:00:28.890
But I'm just showing you that
the cosine of minus a is equal

00:00:28.890 --> 00:00:32.140
to this side over the
hypotenuse, and this hypotenuse

00:00:32.140 --> 00:00:34.350
is the same as this
hypotenuse, right?

00:00:34.350 --> 00:00:38.250
So cosine of minus a is
adjacent over this hypotenuse,

00:00:38.250 --> 00:00:42.290
while cosine of a is adjacent
over this hypotenuse.

00:00:42.290 --> 00:00:44.660
But it's the same thing, so
we know that cosine of minus

00:00:44.660 --> 00:00:47.640
a is equal to cosine of a.

00:00:47.640 --> 00:00:50.430
Actually, by definition, that
makes it a-- I don't want to

00:00:50.430 --> 00:00:52.600
confuse you too much, but that
makes cosine an even function,

00:00:52.600 --> 00:00:53.510
and I'll show you more.

00:00:53.510 --> 00:00:55.160
Actually, I should do a
whole presentation on

00:00:55.160 --> 00:00:57.030
even and odd functions.

00:00:57.030 --> 00:00:59.490
Now, let's see what
sine of minus a is.

00:00:59.490 --> 00:01:06.030
Sine of minus a is equal
to-- so this is minus a.

00:01:06.030 --> 00:01:11.100
So it's this side, so it's the
minus length of-- let's call

00:01:11.100 --> 00:01:15.480
this x, let's call this y,
and let's call this, well,

00:01:15.480 --> 00:01:17.030
let's leave that h, right?

00:01:17.030 --> 00:01:20.880
If that is x, this is y, this
length is y, then this length

00:01:20.880 --> 00:01:22.880
right here is minus x, right?

00:01:22.880 --> 00:01:25.890
So the sine of minus
a is minus x/h.

00:01:30.670 --> 00:01:33.390
What's the sine of a?

00:01:33.390 --> 00:01:38.740
Sine of a is equal to-- this is
a-- opposite or hypotenuse,

00:01:38.740 --> 00:01:43.540
x over h, right?

00:01:43.540 --> 00:01:50.030
So sine of minus a is equal to
minus 1 times x over h, right?

00:01:50.030 --> 00:01:52.960
Or this is just the same thing
as-- I mean, we could multiply

00:01:52.960 --> 00:01:56.350
both sides of this by minus
1, minus x over h, right?

00:01:56.350 --> 00:02:00.670
So sine of minus a is
equal to minus sine of a.

00:02:04.100 --> 00:02:07.700
So let me clear this out
and rewrite this identity.

00:02:07.700 --> 00:02:10.030
And as you can see, all I'm
doing is I'm just playing

00:02:10.030 --> 00:02:14.010
around with triangles and
showing you that, you know,

00:02:14.010 --> 00:02:17.430
just using the basic SOHCAHTOA,
you can actually discover a

00:02:17.430 --> 00:02:19.590
whole set of trigonometric
identities.

00:02:22.300 --> 00:02:23.300
So let's clear that.

00:02:23.300 --> 00:02:26.100
And, you know, it is
useful to memorize.

00:02:26.100 --> 00:02:29.680
I normally don't advocate
memorizing, but it's helpful

00:02:29.680 --> 00:02:30.890
just to do things quickly.

00:02:30.890 --> 00:02:33.940
But I'd also advocate being
able to prove it to yourself,

00:02:33.940 --> 00:02:37.150
so if you ever forget it, and
you don't have a cheat sheet

00:02:37.150 --> 00:02:40.170
available, you can prove it,
and if you ever have to teach

00:02:40.170 --> 00:02:45.130
it, then you'll be able to
explain the underlying

00:02:45.130 --> 00:02:46.020
themes a little bit better.

00:02:46.020 --> 00:02:47.490
So let's clear this.

00:02:47.490 --> 00:02:49.895
Let's see if we can discover
some more trig identities.

00:02:52.710 --> 00:02:58.890
So we know that-- so let's
see, if we have sine--

00:02:58.890 --> 00:03:01.370
what's sine of a plus pi/2?

00:03:04.960 --> 00:03:05.980
a plus pi/2.

00:03:05.980 --> 00:03:09.600
Well, we could use our handy
sine of a plus b identity,

00:03:09.600 --> 00:03:12.250
which we've already proved,
so we can use it now.

00:03:12.250 --> 00:03:15.580
So that tells us that it's the
sine of a-- that equals the

00:03:15.580 --> 00:03:34.810
sine of a times the cosine of
pi/2 plus the sine of pi/2.

00:03:34.810 --> 00:03:36.270
And we're in
radians, of course.

00:03:36.270 --> 00:03:38.570
This could have been 90
degrees instead, if we

00:03:38.570 --> 00:03:40.140
wanted to be in degrees.

00:03:40.140 --> 00:03:47.310
sine of pi/2 times the
cosine of a, right?

00:03:47.310 --> 00:03:52.890
Well, this equals the sine of
-- what's cosine sign of pi/2?

00:03:52.890 --> 00:03:55.040
Or cosine of 90 degrees?

00:03:55.040 --> 00:03:57.160
Well, that's when we're on
the unit circle, we're

00:03:57.160 --> 00:03:58.350
pointing straight up.

00:03:58.350 --> 00:04:00.450
And so the x-coordinate is 0.

00:04:00.450 --> 00:04:02.490
I could draw it out, but I
think-- you might want to draw

00:04:02.490 --> 00:04:04.310
the unit circle and figure it
out for yourself, or if you

00:04:04.310 --> 00:04:06.840
don't, do it on a calculator,
but you will learn

00:04:06.840 --> 00:04:07.620
that it is 0.

00:04:07.620 --> 00:04:09.620
The cosine of pi/2 is 0.

00:04:09.620 --> 00:04:12.690
Plus sine of pi/2, for the same
reason, we're pointing straight

00:04:12.690 --> 00:04:15.470
up on the unit circle, so the
y-coordinate, or the sine

00:04:15.470 --> 00:04:19.710
coordinate, is 1, right on the
unit-- is essentially at the

00:04:19.710 --> 00:04:22.400
point 0, 1 on the unit circle.

00:04:22.400 --> 00:04:29.300
So sine of pi/2 is 1, and
then times cosine of a.

00:04:29.300 --> 00:04:31.820
So sine of a times 0 is 0.

00:04:31.820 --> 00:04:34.250
1 times cosine of a
is just cosine of a.

00:04:34.250 --> 00:04:37.010
So we have a new,
useful trig identity.

00:04:40.060 --> 00:04:48.100
Sine a plus pi/2 is
equal to cosine of a.

00:04:48.100 --> 00:04:49.240
Fascinating!

00:04:49.240 --> 00:04:53.880
So really, this is just telling
us that cosine of a is the same

00:04:53.880 --> 00:04:57.100
thing as sine of a shifted.

00:04:57.100 --> 00:04:59.030
So if we were to think of this
graphically, if we were to

00:04:59.030 --> 00:05:03.070
think of, you know, if we were
to draw the graph, if you shift

00:05:03.070 --> 00:05:09.150
the sine graph to the left by
pi/2, you get the cosine graph.

00:05:09.150 --> 00:05:10.960
And if you haven't learned
about shifting yet,

00:05:10.960 --> 00:05:11.670
don't worry about that.

00:05:11.670 --> 00:05:13.930
Or you might want to actually
graph the two, and I

00:05:13.930 --> 00:05:16.860
think you'll get a sense
of what I'm saying.

00:05:16.860 --> 00:05:19.900
So let's do-- I don't know.

00:05:19.900 --> 00:05:23.550
And another way to rewrite this
exact same thing is the sine of

00:05:23.550 --> 00:05:36.090
a is equal to the cosine
of a minus pi/2, right?

00:05:36.090 --> 00:05:40.640
Let's say I said that b
is a plus pi/2, right?

00:05:40.640 --> 00:05:45.570
Let's say I said that b is
equal to a plus pi/2, then we

00:05:45.570 --> 00:05:50.770
can say that this is b, and
then this would b minus pi/2.

00:05:50.770 --> 00:05:52.770
I'm just switching
around variables.

00:05:52.770 --> 00:05:55.520
I'm doing this in a much more
loosey-goosey fashion than I

00:05:55.520 --> 00:05:57.840
normally do a lot of videos,
but I want to show you that a

00:05:57.840 --> 00:06:00.560
lot of this trigonometry can
just be-- you know, it's

00:06:00.560 --> 00:06:02.960
just kind of discovery.

00:06:02.960 --> 00:06:09.490
What's sine of a minus b?

00:06:09.490 --> 00:06:11.400
Well, that looks like a
new one, doesn't it?

00:06:11.400 --> 00:06:13.930
Well, let's try to
figure it out.

00:06:13.930 --> 00:06:26.080
Well, that equals sine of a
cosine of minus b plus sine

00:06:26.080 --> 00:06:31.940
of minus b times the
cosine of a, right?

00:06:31.940 --> 00:06:34.850
Well, what do we know about
the cosine of minus b?

00:06:34.850 --> 00:06:36.970
Before I cleared the screen,
we just figured out that the

00:06:36.970 --> 00:06:40.660
cosine of minus b, since it's
an even function, is the same

00:06:40.660 --> 00:06:42.410
thing as the cosine of b.

00:06:42.410 --> 00:06:44.810
So we can rewrite that
as that equals the

00:06:44.810 --> 00:06:49.900
sine of a cosine of b.

00:06:49.900 --> 00:06:52.600
And then what's the
sine of minus b?

00:06:52.600 --> 00:06:56.380
Well, that's the same thing
as the minus sine of b.

00:06:56.380 --> 00:07:00.120
That's what we just proved,
that the sine of minus

00:07:00.120 --> 00:07:03.070
b, that this is equal
to minus sine of b.

00:07:03.070 --> 00:07:05.340
You could draw the triangle and
the unit circle, if you don't

00:07:05.340 --> 00:07:07.060
believe me, but we
just did that.

00:07:07.060 --> 00:07:11.860
So we can say that that
is equal to minus sine

00:07:11.860 --> 00:07:14.270
of b cosine of a.

00:07:14.270 --> 00:07:15.010
Interesting!

00:07:15.010 --> 00:07:16.580
I encourage you do the
same thing with the

00:07:16.580 --> 00:07:18.120
cosine of a minus b.

00:07:18.120 --> 00:07:21.650
These are all just, you know,
we're using one or two or three

00:07:21.650 --> 00:07:25.480
trig identities together and
trying to come up

00:07:25.480 --> 00:07:26.290
with new things.

00:07:26.290 --> 00:07:29.250
And I think at this point,
we've literally gone over

00:07:29.250 --> 00:07:32.260
everything, that almost every
trig identity you've seen in

00:07:32.260 --> 00:07:36.100
your book, you should be
able to get there somehow,

00:07:36.100 --> 00:07:37.270
just by keep on playing.

00:07:37.270 --> 00:07:39.990
And obviously, all of these
identities, you can invert the

00:07:39.990 --> 00:07:44.130
sines and the cosines and the
tangents, and you can get

00:07:44.130 --> 00:07:47.840
identities for secant and
cotangent and cosecant

00:07:47.840 --> 00:07:50.750
and keep playing around.

00:07:50.750 --> 00:07:51.620
And I encourage you to do so.

00:07:51.620 --> 00:07:53.120
And do it graphically.

00:07:53.120 --> 00:07:54.860
Draw the triangles.

00:07:54.860 --> 00:07:56.490
It's also interesting to
sometimes actually draw the

00:07:56.490 --> 00:08:00.420
graph on the x-y plane of, say,
you know, cosine of x plus

00:08:00.420 --> 00:08:04.680
pi/2, or sine of x plus
pi/2, or sine of x.

00:08:04.680 --> 00:08:07.410
And I think in the future, I'll
do a video where I really

00:08:07.410 --> 00:08:08.930
do explore all of that.

00:08:08.930 --> 00:08:11.370
Well, I hope I haven't
thoroughly confused you.

00:08:11.370 --> 00:08:16.170
I wanted to just kind of show
you that a lot of trig-- it

00:08:16.170 --> 00:08:18.990
all comes from SOHCAHTOA and
playing around with SOHCAHTOA

00:08:18.990 --> 00:08:22.390
and triangles, and you can
pretty much get-- you can

00:08:22.390 --> 00:08:24.740
pretty much solve for
everything you learn

00:08:24.740 --> 00:08:26.240
in trigonometry.

00:08:26.240 --> 00:08:27.510
And if you don't have
SOHCAHTOA, at least the unit

00:08:27.510 --> 00:08:29.795
circle definition, which is
actually better, because

00:08:29.795 --> 00:08:30.580
it's more extensive.

00:08:30.580 --> 00:08:32.230
But anyway, that's all for now.

00:08:32.230 --> 00:08:33.732
See you soon.


Trig identities part 2 (part 4 if you watch the proofs)

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:01.680
Welcome back.

00:00:01.680 --> 00:00:04.210
I'm now going to do a bit of a
review of everything we've

00:00:04.210 --> 00:00:06.080
learned so far about maybe even
trigonometry and

00:00:06.080 --> 00:00:07.350
trig identities.

00:00:07.350 --> 00:00:10.450
And then we'll see if we can
come up with-- maybe use what

00:00:10.450 --> 00:00:13.050
we already know to come up with
a couple more trig identities.

00:00:13.050 --> 00:00:18.940
So we know that from
SOH-CAH-TOA we know that sine

00:00:18.940 --> 00:00:26.070
of theta is equal to the
opposite over the hypotenuse.

00:00:26.070 --> 00:00:28.270
Let me draw a triangle here.

00:00:28.270 --> 00:00:31.560
If I were to draw a
triangle here-- whoops.

00:00:31.560 --> 00:00:33.380
Oh, there you go.

00:00:33.380 --> 00:00:35.780
OK, so this is theta.

00:00:35.780 --> 00:00:37.180
This is the opposite.

00:00:37.180 --> 00:00:38.410
This is the adjacent.

00:00:38.410 --> 00:00:40.400
This is hypotenuse.

00:00:40.400 --> 00:00:42.460
Then sine of theta is equal
to opposite over hypotenuse.

00:00:42.460 --> 00:00:46.040
Cosine of theta-- this is basic
review, hopefully at this

00:00:46.040 --> 00:00:49.170
point-- is the adjacent
over the hypotenuse.

00:00:49.170 --> 00:00:54.470
The tangent of theta is equal
to the opposite over the

00:00:54.470 --> 00:00:59.770
adjacent, which is also equal
to the sine of theta over

00:00:59.770 --> 00:01:00.590
the cosine of theta.

00:01:00.590 --> 00:01:03.380
And we showed this in a
couple of videos ago.

00:01:03.380 --> 00:01:08.160
And then, these are kind of
almost definitional, but the

00:01:08.160 --> 00:01:17.840
cosecant of theta is equal
to the hypotenuse over the

00:01:17.840 --> 00:01:21.260
opposite, which is the same
thing as 1 over sine of theta.

00:01:21.260 --> 00:01:23.900
You can just memorize this.

00:01:23.900 --> 00:01:26.120
I mean, I kind of find is
silly that there is such

00:01:26.120 --> 00:01:27.100
a thing as cosecant.

00:01:27.100 --> 00:01:29.330
I guess it's just for
convenience because you know

00:01:29.330 --> 00:01:31.410
everyone knows it's just
1 over sine of theta.

00:01:31.410 --> 00:01:34.830
And same thing for secant.

00:01:34.830 --> 00:01:37.140
Secant of theta-- it's
really for convenience.

00:01:37.140 --> 00:01:39.990
Instead of having to say, in
the case of secant-- oh, that's

00:01:39.990 --> 00:01:42.910
1 you know-- if you end up with
the equation 1 over cosine of

00:01:42.910 --> 00:01:44.200
theta you can just say,
oh, that's just the

00:01:44.200 --> 00:01:45.550
secant of theta.

00:01:45.550 --> 00:01:48.573
I think it actually has some
obvious properties and if you

00:01:48.573 --> 00:01:50.450
were to draw unit circle
and all of that too.

00:01:50.450 --> 00:01:54.760
But anyway, so that's equal
to the hypotenuse over the

00:01:54.760 --> 00:01:59.400
adjacent, which is equal to
1 over cosine of theta.

00:01:59.400 --> 00:02:05.160
And then, of course, cotangent
of theta is equal to the

00:02:05.160 --> 00:02:08.430
adjacent over the opposite,
which is equal to

00:02:08.430 --> 00:02:12.260
1 over tan theta.

00:02:12.260 --> 00:02:16.010
And of course, that's
also equal to cosine of

00:02:16.010 --> 00:02:19.430
theta over sine theta.

00:02:19.430 --> 00:02:22.310
It's just the opposite of
the tangent of theta.

00:02:22.310 --> 00:02:27.470
Or that's the same
thing as what?

00:02:27.470 --> 00:02:32.230
That's the same thing as
the secant-- no, no, no.

00:02:32.230 --> 00:02:36.632
It's the same thing as the
cosecant-- no, no no.

00:02:36.632 --> 00:02:38.600
Let me make sure I
get this right.

00:02:38.600 --> 00:02:45.700
It's the same thing as the-- I
just want to get the inverses.

00:02:45.700 --> 00:02:47.050
Well, let's prove
what it is actually.

00:02:47.050 --> 00:02:48.110
I always confuse myself.

00:02:48.110 --> 00:02:59.060
So this is the same thing as 1
over the secant of theta over

00:02:59.060 --> 00:03:01.730
1 over the cosecant of theta.

00:03:04.770 --> 00:03:06.370
Secant of theta,
cosecant theta.

00:03:06.370 --> 00:03:12.700
And then that equals the
cosecant of theta over

00:03:12.700 --> 00:03:14.190
the secant of theta.

00:03:14.190 --> 00:03:16.110
I wouldn't waste your
time memorizing.

00:03:16.110 --> 00:03:17.930
So we know that a cotangent
of theta is equal to

00:03:17.930 --> 00:03:19.240
1 over tangent theta.

00:03:19.240 --> 00:03:21.300
Is equal to the cosine
over the sine.

00:03:21.300 --> 00:03:24.040
And it also equals the
cosecant over the secant.

00:03:24.040 --> 00:03:26.690
And I wouldn't worry about
really memorizing this.

00:03:26.690 --> 00:03:27.840
You could derive
it if you had to.

00:03:27.840 --> 00:03:29.460
As you could tell, I
really didn't have

00:03:29.460 --> 00:03:30.850
this memorized either.

00:03:30.850 --> 00:03:38.380
And we also learned in previous
videos that the sine squared

00:03:38.380 --> 00:03:43.440
theta plus the cosine squared
of theta is equal to 1.

00:03:43.440 --> 00:03:45.530
And that just comes from
the pythagorean theorem.

00:03:45.530 --> 00:03:48.170
And if you play around with
this a little bit you'd also

00:03:48.170 --> 00:03:55.950
get that the tangent squared
theta plus 1 is equal to

00:03:55.950 --> 00:04:00.380
the secant squared theta.

00:04:00.380 --> 00:04:02.110
You actually go from here to
here if you just divide both

00:04:02.110 --> 00:04:05.010
sides of this equation
by cosine squared.

00:04:05.010 --> 00:04:05.860
So we know that.

00:04:05.860 --> 00:04:11.020
And then if you've watched the
last two proof videos I made,

00:04:11.020 --> 00:04:20.180
we also know that the sine of--
let's say a plus b-- is equal

00:04:20.180 --> 00:04:31.670
to the sine of a times
the cosine of b.

00:04:31.670 --> 00:04:34.890
Plus-- let me erase some
of this because I don't

00:04:34.890 --> 00:04:38.590
think that that is an
important trig identity.

00:04:38.590 --> 00:04:39.730
You can derive it on your own.

00:04:39.730 --> 00:04:43.570
I just wanted to show you that
you could figure it out.

00:04:43.570 --> 00:04:45.780
I'm using too much space.

00:04:45.780 --> 00:04:48.320
OK, now I have space.

00:04:48.320 --> 00:04:50.330
Let me find that blue color
I was using and make

00:04:50.330 --> 00:04:51.650
sure my pen is small.

00:04:51.650 --> 00:04:52.015
OK.

00:04:52.015 --> 00:04:56.040
So it's the sine of a times
the cosine of b plus the sine

00:04:56.040 --> 00:05:02.430
of b times the cosine of a.

00:05:02.430 --> 00:05:03.690
And you might want to
just memorize it.

00:05:03.690 --> 00:05:05.370
This actually becomes really
useful when you actually start

00:05:05.370 --> 00:05:08.090
doing calculus because you have
to solve derivatives and

00:05:08.090 --> 00:05:09.720
integrals that you might
have to know the identity.

00:05:09.720 --> 00:05:11.870
And it's not that
hard to memorize.

00:05:11.870 --> 00:05:15.080
It's the sine of one of them
times the cosine of one of them

00:05:15.080 --> 00:05:16.370
plus the other way around.

00:05:16.370 --> 00:05:17.130
That's all this is.

00:05:17.130 --> 00:05:23.200
And then we also learned that
the cosine of a plus b-- it's

00:05:23.200 --> 00:05:27.960
the cosine of both of them
minus the sine of both them.

00:05:27.960 --> 00:05:36.890
So that is equal to the cosine
of a times the cosine of b.

00:05:36.890 --> 00:05:39.400
And I proved this in another
video, hopefully did it

00:05:39.400 --> 00:05:41.180
to your satisfaction.

00:05:41.180 --> 00:05:48.160
Minus the sine of a
times the sine of b.

00:05:51.990 --> 00:05:54.110
These are pretty useful because
from these can we can come up

00:05:54.110 --> 00:05:57.040
with a bunch of other
trig identities.

00:05:57.040 --> 00:05:59.280
For example, what
is the sine of 2a?

00:06:05.900 --> 00:06:08.360
Well, that's just the same
thing as the sine of a plus a.

00:06:11.670 --> 00:06:15.010
And if we use this trig
identity up here, that is equal

00:06:15.010 --> 00:06:27.850
to sine of a cosine of a plus
the sine of a, cosine of a.

00:06:27.850 --> 00:06:31.560
I just used this sine of a plus
b identity up here and well,

00:06:31.560 --> 00:06:32.982
want a and b are both a.

00:06:32.982 --> 00:06:34.110
Now what does this equal?

00:06:34.110 --> 00:06:35.990
Well, this is two terms
that are just both sine

00:06:35.990 --> 00:06:36.810
of a, cosine of a.

00:06:36.810 --> 00:06:43.300
So that just equals 2
sine of a, cosine of a.

00:06:43.300 --> 00:06:47.030
So we now have derived another
trigonometric identity that

00:06:47.030 --> 00:06:51.650
might be in the inside cover of
your trig, or actually,

00:06:51.650 --> 00:06:54.270
your calculus book.

00:06:54.270 --> 00:06:55.940
All of these actually,
I could draw a square

00:06:55.940 --> 00:06:56.540
around all of them.

00:06:56.540 --> 00:07:00.110
Let's do another one.

00:07:00.110 --> 00:07:03.070
Once you have a bit of a
library of trig identities you

00:07:03.070 --> 00:07:05.680
can really just keep playing
around and seeing what else you

00:07:05.680 --> 00:07:07.930
can-- and I encourage
you to do so.

00:07:07.930 --> 00:07:10.890
And you'd be amazed how
many other trig identities

00:07:10.890 --> 00:07:12.610
you could come up with.

00:07:12.610 --> 00:07:15.490
For example, let's
do cosine of 2a.

00:07:15.490 --> 00:07:25.410
Cosine of 2a is equal
to cosine of a plus a.

00:07:25.410 --> 00:07:27.530
And cosine of a plus
a, what did we say?

00:07:27.530 --> 00:07:33.220
It's the cosine of both of the
terms times each other minus

00:07:33.220 --> 00:07:34.160
the sine of both of the terms.

00:07:34.160 --> 00:07:39.750
So that equals cosine of
a, cosine of a, right?

00:07:39.750 --> 00:07:46.020
Cosine of a times cosine of a
minus sine of a, sine of a.

00:07:46.020 --> 00:07:49.080
This identity was the cosine
of a plus b identity.

00:07:49.080 --> 00:07:50.260
Minus sine of a.

00:07:50.260 --> 00:07:51.030
So what is this?

00:07:51.030 --> 00:08:03.280
This is equal to cosine squared
a minus sine squared a.

00:08:03.280 --> 00:08:04.720
That's interesting.

00:08:04.720 --> 00:08:05.310
We could play around.

00:08:05.310 --> 00:08:07.830
This is interesting because
this is the form a

00:08:07.830 --> 00:08:09.880
squared minus b squared.

00:08:09.880 --> 00:08:13.770
So that's also the same thing
as a plus b times a minus b.

00:08:13.770 --> 00:08:22.120
So that's the same thing as
cosine of a plus sine of a

00:08:22.120 --> 00:08:26.950
times cosine of a
minus sine of a.

00:08:26.950 --> 00:08:27.360
I don't know.

00:08:27.360 --> 00:08:29.250
This isn't really a trig
identity, but I'm just showing

00:08:29.250 --> 00:08:30.310
you could play with things.

00:08:30.310 --> 00:08:34.650
Cosine of 2a is equal to cosine
of a plus sine of a times

00:08:34.650 --> 00:08:37.290
cosine of a minus sine of a.

00:08:37.290 --> 00:08:41.400
So the sum of the cosine
and sine of a then

00:08:41.400 --> 00:08:42.000
times the difference.

00:08:42.000 --> 00:08:42.760
That's just interesting.

00:08:42.760 --> 00:08:44.870
I'm just showing you that
what's fun about trigonometry

00:08:44.870 --> 00:08:46.740
is you can kind of keep
playing around with it.

00:08:46.740 --> 00:08:49.010
And actually, that's probably--
that is how all of the trig

00:08:49.010 --> 00:08:51.680
identities were discovered.

00:08:51.680 --> 00:08:57.830
So let's say that we have-- we
want to figure out what cosine

00:08:57.830 --> 00:09:01.940
of let's say, negative a is.

00:09:01.940 --> 00:09:02.945
Well, let me draw
a right triangle.

00:09:06.260 --> 00:09:06.645
Whoops.

00:09:06.645 --> 00:09:08.265
That's almost a right triangle.

00:09:08.265 --> 00:09:10.850
Now let's say this angle is a.

00:09:10.850 --> 00:09:13.450
So negative a, unit circle
would look something

00:09:13.450 --> 00:09:16.240
like this, right?

00:09:16.240 --> 00:09:17.250
Negative a.

00:09:17.250 --> 00:09:22.750
So cosine of a, if we say that
this side is the adjacent

00:09:22.750 --> 00:09:24.260
side, this is the hypotenuse.

00:09:24.260 --> 00:09:26.280
This would still be the
hypotenuse, right?

00:09:26.280 --> 00:09:27.710
And this is the opposite.

00:09:27.710 --> 00:09:30.350
This is the negative opposite.

00:09:30.350 --> 00:09:34.500
So cosine of minus a
is equal to what?

00:09:34.500 --> 00:09:38.260
This is minus a, so it's the
adjacent over the hypotenuse.

00:09:38.260 --> 00:09:41.860
So it equals the adjacent
over the hypotenuse,

00:09:41.860 --> 00:09:43.470
which we just say is h.

00:09:43.470 --> 00:09:47.070
But that's the same thing
as cosine of a, right?

00:09:47.070 --> 00:09:51.130
Because cosine of a is also the
adjacent over the hypotenuse.

00:09:53.870 --> 00:09:54.960
Oh, I'm almost out of time.

00:09:54.960 --> 00:09:57.050
Let me switch to a new video.