Introduction to angles (old)

WEBVTT
Kind: captions
Language: en

00:00:01.130 --> 00:00:01.890
Hello.

00:00:01.890 --> 00:00:05.120
In this series of presentations
I'm going to try to teach you

00:00:05.120 --> 00:00:09.020
everything you need to know
about triangles and angles

00:00:09.020 --> 00:00:10.120
and parallel lines.

00:00:10.120 --> 00:00:13.820
And this is probably the
highest yield information that

00:00:13.820 --> 00:00:16.460
you could ever learn, and
especially in terms of

00:00:16.460 --> 00:00:17.720
standardized tests.

00:00:17.720 --> 00:00:19.980
And then when we've learned all
the rules we'll play something

00:00:19.980 --> 00:00:21.920
that I call the angle game,
which is essentially what the

00:00:21.920 --> 00:00:24.500
SAT makes you do over
and over again.

00:00:24.500 --> 00:00:26.780
So let's start
with some basics.

00:00:26.780 --> 00:00:28.120
Well, you know
what an angle is.

00:00:28.120 --> 00:00:29.550
Well actually, maybe you
don't know what an angle is.

00:00:29.550 --> 00:00:31.040
And I'll tell you
what an angle is.

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If I have two lines-- I'm
going to draw a thicker

00:00:36.400 --> 00:00:37.530
line than that.

00:00:37.530 --> 00:00:47.130
If I have two lines and they
intersect at some point, the

00:00:47.130 --> 00:00:52.360
angle is a measure of exactly
kind of how wide the

00:00:52.360 --> 00:00:55.600
intersection is between
those two lines.

00:00:55.600 --> 00:00:58.470
Let me use a better tool.

00:00:58.470 --> 00:00:59.940
So this is the angle.

00:00:59.940 --> 00:01:04.470
An angle is how wide those
two lines kind of open up.

00:01:04.470 --> 00:01:07.020
And they're measured either in
degrees or radians, and for the

00:01:07.020 --> 00:01:10.890
sake of most of a geometry
class we'll use degrees.

00:01:10.890 --> 00:01:12.620
And when we start doing
trigonometry you'll

00:01:12.620 --> 00:01:13.230
learn radians.

00:01:13.230 --> 00:01:14.525
Or maybe you could
learn it now.

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And you're probably
familiar with this.

00:01:17.670 --> 00:01:20.350
0 degrees would be these two
lines on top of each other.

00:01:20.350 --> 00:01:26.620
This, if I were to just eyeball
it, looks like 45 degrees.

00:01:26.620 --> 00:01:34.920
If I had the lines even
wider apart like that,

00:01:34.920 --> 00:01:36.720
that's 90 degrees.

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And 90 degree lines are also
called perpendicular, because

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they are-- I feel like saying,
because they are

00:01:43.370 --> 00:01:47.080
perpendicular-- but because one
is going completely vertical

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while the other is
going horizontal.

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It's actually amazingly
difficult to find the

00:01:54.170 --> 00:01:55.150
exact right wording.

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But I think you get the idea.

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By definition, perpendicular
lines have a 90 degree

00:01:59.950 --> 00:02:01.070
apart from each other.

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And, you know, you've seen this
all the time in things like

00:02:03.820 --> 00:02:05.880
squares and rectangles.

00:02:05.880 --> 00:02:08.570
If I were to draw a
rectangle like that.

00:02:08.570 --> 00:02:08.940
Right?

00:02:08.940 --> 00:02:12.920
A rectangle is made up of a
bunch of perpendicular lines,

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or lines at 90 degree angles.

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So example.

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These two lines are at
a 90 degree angle.

00:02:17.410 --> 00:02:19.525
The way you draw a 90
degree angle is you draw

00:02:19.525 --> 00:02:21.700
a little box like that.

00:02:21.700 --> 00:02:24.700
That's the same thing as
doing this-- as doing

00:02:24.700 --> 00:02:27.520
a 90 degree angle.

00:02:27.520 --> 00:02:30.050
And you could even
get wider angles.

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So if you go above 90 degrees.

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So let's say I had
lines like this.

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So this would be-- I don't
know, I'm just eyeballing

00:02:44.820 --> 00:02:48.440
it-- 135 degrees or
something like that.

00:02:48.440 --> 00:02:51.130
If you ever want to really
measure the angles, you can use

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something called a protractor.

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That's a tool maybe your
teachers can help you use that.

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So that'd be 135 degrees.

00:02:57.410 --> 00:03:02.190
And then if you had it so wide
that the two lines are actually

00:03:02.190 --> 00:03:10.950
almost forming a line, then
this becomes 180 degrees.

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It's almost like one line.

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Right?

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This is 180 degrees.

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And then you can keep going.

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So if this angle here is 135
degrees, you can actually also

00:03:23.200 --> 00:03:25.030
measure this angle right here.

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Let me do it in a different
color just to add some variety.

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So then this angle right here.

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So the angles in a circle
are there are 360

00:03:36.920 --> 00:03:38.970
degrees in a circle.

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So if this is 135, this magenta
angle would be 360 degrees

00:03:49.610 --> 00:03:53.050
minus 135 degrees.

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And that's equal to what?

00:03:54.380 --> 00:04:02.230
That's 225 degrees, is
this magenta angle.

00:04:02.230 --> 00:04:03.440
And then we could do
other things like that.

00:04:03.440 --> 00:04:05.220
So one, you know that the
degrees in the circle

00:04:05.220 --> 00:04:06.670
are 360 degrees.

00:04:06.670 --> 00:04:07.760
This is important to know.

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Degrees in a circle
are 360 degrees.

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It's also important to know
that if you just go kind of

00:04:13.350 --> 00:04:15.450
halfway around a circle,
like we did here,

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that's 180 degrees.

00:04:16.890 --> 00:04:18.420
Like if you viewed the
pivot point as like,

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let's say, right here.

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I mean it looks like just
one line and it really is.

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But that's 180 degrees.

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And then if you go quarter
way around the circle,

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that's 90 degrees.

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All right?

00:04:31.400 --> 00:04:32.770
Hopefully you're getting
a bit of an intuition

00:04:32.770 --> 00:04:34.380
for what an angle is.

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So now I will teach you
a bunch of very useful

00:04:39.290 --> 00:04:40.675
rules for angles.

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Clear this.

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So let me redraw.

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So if I had a line like this.

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I like using the colors, just
so I think it keeps you from

00:04:55.910 --> 00:04:57.060
getting completely bored.

00:05:02.200 --> 00:05:05.780
And it might not be completely
intuitive what I'm doing, but

00:05:05.780 --> 00:05:07.120
let's add an angle like that.

00:05:10.230 --> 00:05:13.850
And so, let's just say-- you
know, I'm not measuring these

00:05:13.850 --> 00:05:18.140
exactly-- let's say that
this is 30 degrees.

00:05:18.140 --> 00:05:26.300
We know that if we go all the
way around the circle, we know

00:05:26.300 --> 00:05:28.550
that that's 360 degrees.

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Right?

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And that's a very ugly
looking around the circle

00:05:32.330 --> 00:05:34.040
angle that I drew.

00:05:34.040 --> 00:05:39.230
So then we also know
that this angle right

00:05:39.230 --> 00:05:43.370
here is 330 degrees.

00:05:43.370 --> 00:05:43.670
Right?

00:05:43.670 --> 00:05:47.630
Because this angle plus this
magenta angle is going to

00:05:47.630 --> 00:05:48.850
equal the whole circle.

00:05:48.850 --> 00:05:50.620
So this is equal
to 330 degrees.

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So remember that.

00:05:54.930 --> 00:05:57.410
The angles in a circle--
or there are 360

00:05:57.410 --> 00:05:58.470
degrees in a circle.

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I don't know if you remember.

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You probably don't.

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This was probably
before you were born.

00:06:05.460 --> 00:06:07.920
But there used to be a game
called 720, and it was a

00:06:07.920 --> 00:06:09.940
skateboarding game--
it was a video game.

00:06:09.940 --> 00:06:12.870
And the 720 was essentially
you were trying to jump

00:06:12.870 --> 00:06:15.430
your skateboard and
spin around twice.

00:06:15.430 --> 00:06:17.280
And that's 720 degrees.

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If you go around a circle
twice that's 720 degrees.

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If you just jump and
spin around once, you

00:06:23.470 --> 00:06:25.140
went 360 degrees.

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So you've probably heard this
in just popular culture.

00:06:28.690 --> 00:06:29.510
But anyway.

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So 360 degrees in a circle.

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And you could imagine half
a circle is 180 degrees.

00:06:34.730 --> 00:06:39.240
So the other important thing to
realize is, like we said, if

00:06:39.240 --> 00:06:42.410
we go halfway around the
circle it's 180 degrees.

00:06:42.410 --> 00:06:45.890
But if we have two angles that
add up to that-- so let's say.

00:06:49.580 --> 00:06:52.280
I don't know if these lines are
thick enough for you to see.

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Let me draw something thicker.

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It doesn't look ideal,
but you get the idea.

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So if we have this
angle, let's call it x.

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And then this angle is y.

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What do we know about the
relationship between x and y?

00:07:21.280 --> 00:07:26.490
Well, we know that the entire
angle is half of a circle.

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Right?

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So that's 180 degrees.

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That's 180 degrees,
this entire angle.

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So what are angles x and
y going to add up to?

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I'm trying to stay
color consistent.

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x plus y are going to
equal-- I'm color blind,

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I think-- 180 degrees.

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Or you could write y is
equal to 180 minus x.

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Or x is equal to 180 minus y.

00:08:03.680 --> 00:08:08.040
But if x plus y are equal to
180 degrees-- and you can see

00:08:08.040 --> 00:08:10.870
that it makes sense that they
do-- if you add the two angles

00:08:10.870 --> 00:08:13.020
you go halfway around a circle.

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Then that tells us that x and y
are-- and this is a fancy word,

00:08:19.390 --> 00:08:21.930
and it's just good to commit
this to memory-- they are

00:08:21.930 --> 00:08:24.905
supplementary angles.

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That's when you add
to 180 degrees.

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Now what if we had
this situation.

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Oh my God, that was horrible.

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Undo.

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Let's say I had this situation.

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Let's see.

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I draw two perpendicular lines.

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Right?

00:08:57.570 --> 00:09:01.960
So this is going a quarter
way around the circle.

00:09:01.960 --> 00:09:02.170
All right.

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Let's say this entire angle
here-- I'm drawing it really

00:09:07.710 --> 00:09:09.060
big-- that's 90 degrees.

00:09:09.060 --> 00:09:09.290
Right?

00:09:09.290 --> 00:09:10.960
They're perpendicular.

00:09:10.960 --> 00:09:15.210
And now if I had two
angles within that.

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So now if I have two angles
here-- so let's say that this

00:09:20.800 --> 00:09:26.150
is x and this is y-- what
do x and y add up to?

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Well, x plus y is 90.

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And we can say that x and
y are complementary.

00:09:37.220 --> 00:09:41.670
And it's important to not get
confused between the two.

00:09:41.670 --> 00:09:44.610
Just remember complementary
means two angles add up to 90

00:09:44.610 --> 00:09:48.360
degrees, supplementary means
that two angles add

00:09:48.360 --> 00:09:51.160
up to 180 degrees.

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I'm running out of time, so I
will see you in the next video.


2-dimensional projectile motion part 5

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:01.560
Welcome back.

00:00:01.560 --> 00:00:05.990
Well, I just-- hopefully, we
together figured out correctly

00:00:05.990 --> 00:00:09.570
that the total time in the air
is this initial velocity in

00:00:09.570 --> 00:00:12.950
this direction divided by 10.

00:00:12.950 --> 00:00:15.040
That's our total time
in the air.

00:00:15.040 --> 00:00:17.330
And now, how far is the
object going to go?

00:00:17.330 --> 00:00:20.870
Well it's going to go, our
displacement or our distance

00:00:20.870 --> 00:00:24.212
in this x direction is going
to be-- so, let's see, We

00:00:24.212 --> 00:00:27.880
could call it distance in the
x direction is going to be

00:00:27.880 --> 00:00:33.330
equal to vx, this
vx times time.

00:00:33.330 --> 00:00:35.210
Times time in the air, right?

00:00:35.210 --> 00:00:38.100
Times this number.

00:00:38.100 --> 00:00:39.060
And this is the average
velocity

00:00:39.060 --> 00:00:40.340
because it's not changing.

00:00:40.340 --> 00:00:42.080
This is a constant.

00:00:42.080 --> 00:00:45.220
Assuming we have no friction
or no air resistance-- air

00:00:45.220 --> 00:00:47.710
resistance is really just
a form of friction.

00:00:47.710 --> 00:00:50.310
The horizontal component of this
velocity is going to stay

00:00:50.310 --> 00:00:52.870
the same the entire time.

00:00:52.870 --> 00:00:57.670
So the total distance traveled
is going to equal this number.

00:00:57.670 --> 00:00:58.990
Let me switch back to
that purple, so you

00:00:58.990 --> 00:01:00.030
know what I'm doing.

00:01:00.030 --> 00:01:07.870
v sub i square root
of 3 over 2.

00:01:07.870 --> 00:01:12.180
That's this times the
time in the air.

00:01:12.180 --> 00:01:15.310
Well the time in the
air is this.

00:01:15.310 --> 00:01:20.370
Times v sub i over 10.

00:01:20.370 --> 00:01:22.250
So the distance total traveled--
let's see.

00:01:22.250 --> 00:01:24.160
Let's multiply these.

00:01:24.160 --> 00:01:30.540
vi squared times the square
root of 3, all

00:01:30.540 --> 00:01:35.020
of that over 20.

00:01:35.020 --> 00:01:39.580
So now we know as a function
of our initial velocity in

00:01:39.580 --> 00:01:42.510
this direction, how far is this
thing going to travel?

00:01:42.510 --> 00:01:45.900
Well we know that we want it to
travel exactly 100 meters.

00:01:45.900 --> 00:01:48.220
So we just set them equal
to each other.

00:01:48.220 --> 00:01:56.760
We say 100 is equal to vi
squared times the square root

00:01:56.760 --> 00:02:00.670
of 3 over 20.

00:02:00.670 --> 00:02:03.580
And then we can multiply
both sides times

00:02:03.580 --> 00:02:04.460
the inverse of this.

00:02:04.460 --> 00:02:06.460
We could multiply both sides
by 20 and divide by the

00:02:06.460 --> 00:02:07.700
square root of 3.

00:02:07.700 --> 00:02:14.700
So we get v sub i squared is
equal to-- let me see.

00:02:14.700 --> 00:02:18.675
20 times 100 is 2,000, divided
by the square root of 3.

00:02:18.675 --> 00:02:23.630
So it's 2,000 times the
square root of 3.

00:02:23.630 --> 00:02:27.420
And so our initial velocity we
need to shoot this thing at is

00:02:27.420 --> 00:02:28.950
going to be the square
root of this.

00:02:28.950 --> 00:02:35.010
So initial velocity is going to
be the square root of 2,000

00:02:35.010 --> 00:02:36.890
divided by the square
root of 3.

00:02:36.890 --> 00:02:39.290
So let's figure out what that
number is, just to give us a

00:02:39.290 --> 00:02:41.630
sense of whether this
feels right.

00:02:44.370 --> 00:02:48.080
Let me take the square root
of 3 first. 3 square root.

00:02:48.080 --> 00:02:54.920
And let me inverse
that times 2,000.

00:02:54.920 --> 00:03:01.110
That equals 1154.7 and then
I can square root this.

00:03:01.110 --> 00:03:03.365
33.98.

00:03:03.365 --> 00:03:06.320
So almost 34 meters
per second.

00:03:06.320 --> 00:03:16.290
So my initial velocity has to be
33.98 meters per second for

00:03:16.290 --> 00:03:20.860
me-- so my initial velocity
has to be 33.98 meters per

00:03:20.860 --> 00:03:25.990
second to exactly hit this
target 100 meters away.

00:03:25.990 --> 00:03:28.620
I think that is fairly cool.

00:03:28.620 --> 00:03:30.860
And we could also do it to
figure out-- you know, we

00:03:30.860 --> 00:03:33.840
could do it backwards to figure
out if we know exactly

00:03:33.840 --> 00:03:36.570
what our velocity is going to
be, maybe like a slingshot--

00:03:36.570 --> 00:03:38.490
always has the same velocity
and the only thing I can

00:03:38.490 --> 00:03:39.730
change is the angle.

00:03:39.730 --> 00:03:42.700
As long as we know our inverse
trigonometric functions, we

00:03:42.700 --> 00:03:45.040
can actually figure out the
exact angle that we have to

00:03:45.040 --> 00:03:49.520
aim the slingshot or the
catapult or whatever to hit

00:03:49.520 --> 00:03:50.410
the target.

00:03:50.410 --> 00:03:53.110
So I don't want you to go out
and raise your own army and

00:03:53.110 --> 00:03:57.570
then whatever else, but if you
have to, you now know how to

00:03:57.570 --> 00:03:59.170
command artillery.

00:03:59.170 --> 00:04:01.090
With that, I will leave you.

00:04:01.090 --> 00:04:04.090
And hopefully, you will not do
something overly destructive.

00:04:04.090 --> 00:04:05.710
I'll see you soon.


2 dimensional projectile motion part 4

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.400
Welcome back, I'm now going
to do a slightly harder

00:00:03.400 --> 00:00:05.790
2-dimensional projectile
motion problem.

00:00:05.790 --> 00:00:10.210
So let's start by drawing
the ground.

00:00:10.210 --> 00:00:12.480
This is the ground and I'm going
to leave a lot of space,

00:00:12.480 --> 00:00:16.350
so that we can do other things
with this chalk board.

00:00:16.350 --> 00:00:19.530
Just draw the ground
like this.

00:00:19.530 --> 00:00:22.510
And let's say I have something--
say I don't know.

00:00:22.510 --> 00:00:23.870
Let's say it's a slingshot.

00:00:23.870 --> 00:00:27.690
And it's angled up at-- let
me think of a good angle.

00:00:27.690 --> 00:00:34.800
It has an angle of-- say the
initial slingshot's angle, so

00:00:34.800 --> 00:00:48.820
just the angle at which it's
pointed is 30 degrees.

00:00:48.820 --> 00:00:50.590
And I want to hit a target.

00:00:50.590 --> 00:00:50.990
I don't know.

00:00:50.990 --> 00:00:52.870
Let me just make up a number.

00:00:52.870 --> 00:00:55.520
Let me say it is 100
meters away.

00:00:58.580 --> 00:01:00.820
I want to know-- and I'm
on earth, so I'm

00:01:00.820 --> 00:01:01.760
dealing with gravity.

00:01:01.760 --> 00:01:06.300
I want to know, how fast does
the slingshots velocity in

00:01:06.300 --> 00:01:10.390
this direction, in the 30 degree
direction-- I want to

00:01:10.390 --> 00:01:15.080
know what this initial velocity
needs to be for me to

00:01:15.080 --> 00:01:18.450
hit this target exactly.

00:01:18.450 --> 00:01:20.880
So let's see if we could
figure this out.

00:01:20.880 --> 00:01:23.400
Well, we don't know the initial
velocity, but let's

00:01:23.400 --> 00:01:26.310
pretend like we do and maybe
we can break it up into its

00:01:26.310 --> 00:01:29.680
horizontal and vertical
components.

00:01:29.680 --> 00:01:31.800
Let's just do that right here
since we already have the

00:01:31.800 --> 00:01:32.850
vector drawn out.

00:01:32.850 --> 00:01:34.290
Actually, no, let me do it
in the corner up here.

00:01:34.290 --> 00:01:35.880
So the first thing we want to
do is break it up into its

00:01:35.880 --> 00:01:37.170
horizontal and vertical
components.

00:01:37.170 --> 00:01:40.155
So this is our initial vector.

00:01:44.130 --> 00:01:45.750
And it's vi.

00:01:45.750 --> 00:01:48.810
That's our initial.

00:01:48.810 --> 00:01:51.360
Let me call it vi,
initial velocity.

00:01:51.360 --> 00:01:54.120
Its initial velocity in this
direction, in the direction of

00:01:54.120 --> 00:01:55.760
30 degrees to the ground.

00:01:55.760 --> 00:01:59.040
And we want to break that up
into its horizontal and

00:01:59.040 --> 00:01:59.820
vertical velocities.

00:01:59.820 --> 00:02:03.165
Or we could say it's along the
x-axis or the y-- whoops.

00:02:07.750 --> 00:02:10.330
I want to be neat this time.

00:02:10.330 --> 00:02:12.860
So I think it helps your
concentration, maybe.

00:02:12.860 --> 00:02:14.110
I don't know.

00:02:17.560 --> 00:02:21.020
So everything with horizontal
will be purple and everything

00:02:21.020 --> 00:02:24.380
with vertical or along the
y-axis-- oh, whoops.

00:02:24.380 --> 00:02:27.820
I just said that and I changed
my convention.

00:02:27.820 --> 00:02:29.070
I'm malfunctioning.

00:02:31.580 --> 00:02:33.440
That's that.

00:02:33.440 --> 00:02:34.060
That's that.

00:02:34.060 --> 00:02:36.680
And we're ready to
rock and roll.

00:02:36.680 --> 00:02:42.080
And this angle right
here is 30 degrees.

00:02:42.080 --> 00:02:43.790
So instead of saying along the
vertical direction, let me

00:02:43.790 --> 00:02:45.050
change this to along
the y-axis.

00:02:45.050 --> 00:02:47.930
You can think of kind of the
traditional coordinate axes.

00:02:47.930 --> 00:02:51.340
So what is the component of the
velocity along the y-axis?

00:02:51.340 --> 00:02:54.000
Well we learned from the last
presentation, we could just

00:02:54.000 --> 00:02:55.890
use standard trigonometry.

00:02:55.890 --> 00:03:02.780
Opposite the vy, v sub y-- this
is v sub y right here,

00:03:02.780 --> 00:03:04.300
this line right here.

00:03:04.300 --> 00:03:05.440
Which I'm using the
wrong color.

00:03:05.440 --> 00:03:12.410
v sub y over the hypotenuse,
v sub i is equal

00:03:12.410 --> 00:03:16.380
to sine of 30 degrees.

00:03:16.380 --> 00:03:24.740
Or we could say v sub y is equal
to the initial velocity

00:03:24.740 --> 00:03:29.260
times the sine of 30 degrees.

00:03:29.260 --> 00:03:32.710
What's sine of 30 degrees?

00:03:32.710 --> 00:03:35.130
If you remember your 30, 60,
90 triangles, you maybe

00:03:35.130 --> 00:03:35.900
remember this by now.

00:03:35.900 --> 00:03:38.430
But sine of 30 degrees-- worst
case, use a table or a

00:03:38.430 --> 00:03:38.920
calculator.

00:03:38.920 --> 00:03:40.680
But hopefully, you know what
the sine of 30 degrees is.

00:03:40.680 --> 00:03:42.060
It's 1/2.

00:03:42.060 --> 00:03:46.420
So we know that my vertical
velocity-- last time I called

00:03:46.420 --> 00:03:47.140
it v sub v.

00:03:47.140 --> 00:03:49.810
This time I'm going to call it
v sub y because this is kind

00:03:49.810 --> 00:03:51.620
of going along the y-axis.

00:03:51.620 --> 00:03:55.210
My vertical velocity
is equal to 1/2

00:03:55.210 --> 00:03:56.900
times my initial velocity.

00:03:56.900 --> 00:03:59.580
That looks like an l,
I don't know why.

00:03:59.580 --> 00:04:01.660
Did I actually say that an
l, a little while ago?

00:04:01.660 --> 00:04:03.700
Anyway, that's v sub i.

00:04:03.700 --> 00:04:10.110
So it equals 1/2 v sub i.

00:04:14.010 --> 00:04:15.810
And oh, where was I?

00:04:15.810 --> 00:04:18.670
My brain really is doing
something else right now.

00:04:18.670 --> 00:04:21.480
Anyway, and this 1/2 is just
the sine of 30 degrees.

00:04:21.480 --> 00:04:24.190
And what is my horizontal
velocity?

00:04:24.190 --> 00:04:26.490
Let's get all the trigonometry
out of the way.

00:04:26.490 --> 00:04:29.010
Well, my horizontal velocity,
doing it the exact same way

00:04:29.010 --> 00:04:31.920
here, let's call that v sub x.

00:04:31.920 --> 00:04:38.790
v sub x is equal to v sub i, my
initial velocity, times the

00:04:38.790 --> 00:04:42.900
cosine of 30 degrees.

00:04:42.900 --> 00:04:45.760
And if you remember your 30,
60, 90 triangles, you'll

00:04:45.760 --> 00:04:50.270
remember cosine of 30 degrees
is square root of 3 over 2.

00:04:50.270 --> 00:04:59.590
So v sub x is equal to v sub i
times the square root of 3,

00:04:59.590 --> 00:05:01.540
all of that over 2.

00:05:01.540 --> 00:05:03.350
So how do we figure
out how far?

00:05:03.350 --> 00:05:05.020
Let's say we didn't know we
had to get to 100 meters.

00:05:05.020 --> 00:05:08.560
How do you figure out how far
this projectile-- whatever's

00:05:08.560 --> 00:05:11.330
coming out of my slingshot--
is going to go?

00:05:11.330 --> 00:05:14.730
Well, we figure out how long
it's going to stay in the air.

00:05:14.730 --> 00:05:16.580
And then, once we figure out how
long it stays in the air,

00:05:16.580 --> 00:05:19.065
we can just multiply it times
our horizontal velocity and

00:05:19.065 --> 00:05:21.090
it'll tell us how far it goes.

00:05:21.090 --> 00:05:23.130
So how long does this thing
stay in the air?

00:05:23.130 --> 00:05:28.230
For that we use our vertical
velocity or v sub y.

00:05:31.820 --> 00:05:35.970
So we know that the change in
velocity-- let me switch back

00:05:35.970 --> 00:05:37.620
to that other color.

00:05:37.620 --> 00:05:41.910
We know that change in
velocity is equal to

00:05:41.910 --> 00:05:44.540
acceleration times time.

00:05:44.540 --> 00:05:46.200
And what I want to do is figure
out, how long does it

00:05:46.200 --> 00:05:50.470
take for this object to kind of
get to its maximum height?

00:05:50.470 --> 00:05:53.600
So it's decelerating
from this vertical

00:05:53.600 --> 00:05:56.120
velocity, v sub y, to 0.

00:05:56.120 --> 00:05:58.390
And how long is it going
to take to do that?

00:05:58.390 --> 00:05:59.600
Well, let's see.

00:05:59.600 --> 00:06:05.610
The change in velocity is 0
minus v sub y, which is this.

00:06:05.610 --> 00:06:09.130
So it's 0 minus v sub y, which
is my initial vertical

00:06:09.130 --> 00:06:12.000
velocity, the y-axis.

00:06:12.000 --> 00:06:14.730
It's 1/2 v sub i.

00:06:14.730 --> 00:06:17.460
I just wrote this
instead of this.

00:06:17.460 --> 00:06:18.800
And that equals acceleration.

00:06:18.800 --> 00:06:21.800
We're going to say minus 10
meters per second squared

00:06:21.800 --> 00:06:23.140
times time.

00:06:23.140 --> 00:06:26.780
I'm not writing the units for
the sake of efficiency.

00:06:26.780 --> 00:06:27.620
And so, let's see.

00:06:27.620 --> 00:06:30.440
If I divide-- so this
doesn't matter.

00:06:30.440 --> 00:06:31.390
The 0 doesn't matter.

00:06:31.390 --> 00:06:34.220
This and this, if I multiply
both sides by negative 1.

00:06:34.220 --> 00:06:42.590
And then, if I divide both sides
by 10 I get t is equal

00:06:42.590 --> 00:06:48.020
to 1/20 v sub i.

00:06:48.020 --> 00:06:51.550
Or v sub i over 20.

00:06:51.550 --> 00:06:54.080
Is this the total time of the
travel of the object?

00:06:54.080 --> 00:06:54.800
Well, no.

00:06:54.800 --> 00:06:57.890
This is the time that it takes
the object to kind of start at

00:06:57.890 --> 00:07:01.870
a very fast initial
vertical velocity.

00:07:01.870 --> 00:07:04.950
And initially it starts at
this velocity and then it

00:07:04.950 --> 00:07:07.400
keeps decelerating until at some
point along its curve,

00:07:07.400 --> 00:07:11.760
like here where it's vertical
velocity is 0.

00:07:11.760 --> 00:07:14.380
And then it starts decelerating
or accelerating

00:07:14.380 --> 00:07:16.120
downwards and comes here.

00:07:16.120 --> 00:07:19.660
So just remember-- let me draw
the curve, actually.

00:07:19.660 --> 00:07:20.990
Because I think that
might make it--

00:07:20.990 --> 00:07:25.340
So the object is going to
take a path like this.

00:07:25.340 --> 00:07:26.280
Whoops.

00:07:26.280 --> 00:07:29.720
Something like this, right?

00:07:29.720 --> 00:07:33.860
At this point, the object has
its full vertical velocity and

00:07:33.860 --> 00:07:35.820
its full horizontal velocity.

00:07:35.820 --> 00:07:39.770
And at every point as we get
higher and higher, gravity

00:07:39.770 --> 00:07:41.545
slows down its vertical
velocity.

00:07:41.545 --> 00:07:43.760
It slows down v sub y.

00:07:43.760 --> 00:07:46.700
It slows down its vertical
velocity, but the horizontal

00:07:46.700 --> 00:07:48.190
velocity stays the same.

00:07:48.190 --> 00:07:51.530
And at this point, the peak
point, its vertical velocity,

00:07:51.530 --> 00:07:53.600
how fast is it moving
upwards is 0.

00:07:53.600 --> 00:07:56.890
And then its vertical velocity
turns negative and it starts

00:07:56.890 --> 00:08:00.030
accelerating in the negative
direction or decelerating in

00:08:00.030 --> 00:08:01.140
the positive direction
depending on how you

00:08:01.140 --> 00:08:02.290
want to view it.

00:08:02.290 --> 00:08:04.750
So this time tells us literally,
how long did it

00:08:04.750 --> 00:08:07.650
take to go from this initial
vertical velocity to 0?

00:08:07.650 --> 00:08:11.410
So that's how long does it take
to get to this point.

00:08:11.410 --> 00:08:14.050
So the total time that
I'm traveling is

00:08:14.050 --> 00:08:15.000
going to be twice this.

00:08:15.000 --> 00:08:15.670
Right?

00:08:15.670 --> 00:08:20.100
So let's say total time-- time
in the air, let me call it

00:08:20.100 --> 00:08:21.210
time in the air.

00:08:21.210 --> 00:08:24.730
That could be a good--
so time in the air.

00:08:24.730 --> 00:08:27.030
We could call this time up.

00:08:27.030 --> 00:08:28.970
And the time up plus time down,
and they're equal to

00:08:28.970 --> 00:08:30.830
each other, is equal
to time in the air.

00:08:30.830 --> 00:08:35.210
Time in the air is going to
be equal 2 times time up.

00:08:35.210 --> 00:08:40.409
That equals 2 times this, which
is equal to 1/10 v sub i

00:08:40.409 --> 00:08:46.150
or vi over 10.

00:08:46.150 --> 00:08:50.750
So we know now how long this
object will stay in the air as

00:08:50.750 --> 00:08:54.270
a function of its initial
velocity in

00:08:54.270 --> 00:08:56.770
this 30 degree direction.

00:08:56.770 --> 00:08:59.580
Now, we have to just figure
out, how far is this thing

00:08:59.580 --> 00:09:00.510
going to travel?

00:09:00.510 --> 00:09:02.620
And I think I'm running out of
time, so I'm going to continue

00:09:02.620 --> 00:09:03.870
this in the next video.


Proof of the law of cosines

WEBVTT
Kind: captions
Language: en

00:00:01.230 --> 00:00:04.850
In the last video, we had a
word problem where we had-- we

00:00:04.850 --> 00:00:07.130
essentially had to figure out
the sides of a triangle, but

00:00:07.130 --> 00:00:08.650
instead of, you know, just
being able to do the

00:00:08.650 --> 00:00:11.540
Pythagorean theorem and because
it was a right triangle, it was

00:00:11.540 --> 00:00:13.220
just kind of a normal triangle.

00:00:13.220 --> 00:00:14.610
It wasn't a right triangle.

00:00:14.610 --> 00:00:17.460
And we just kind of chugged
through it using SOHCAHTOA and

00:00:17.460 --> 00:00:20.360
just our very simple trig
functions, and we got

00:00:20.360 --> 00:00:21.480
the right answer.

00:00:21.480 --> 00:00:23.300
What I want to do now is to
introduce you to something

00:00:23.300 --> 00:00:26.850
called the law of cosines,
which we essentially proved in

00:00:26.850 --> 00:00:29.100
the last video, but I want to
kind of prove it in a more--

00:00:29.100 --> 00:00:31.360
you know, without the word
problem getting in the way, and

00:00:31.360 --> 00:00:33.600
I want to show you, once you
know the law of cosines, so you

00:00:33.600 --> 00:00:35.810
can then apply it to a problem,
like we did in the past,

00:00:35.810 --> 00:00:37.480
and you'll do it faster.

00:00:37.480 --> 00:00:41.480
I have a bit of a mixed opinion
about it because I'm not a

00:00:41.480 --> 00:00:43.010
big fan of memorizing things.

00:00:43.010 --> 00:00:46.290
You know, when you're 40 years
old, you probably won't have

00:00:46.290 --> 00:00:49.070
the law of cosines still
memorized, but if you have that

00:00:49.070 --> 00:00:51.190
ability to start with the trig
functions and just move

00:00:51.190 --> 00:00:53.560
forward, then you'll
always be set.

00:00:53.560 --> 00:00:55.090
And I'd be impressed if
you're still doing trig

00:00:55.090 --> 00:00:56.560
at 40, but who knows?

00:00:56.560 --> 00:00:58.980
So let's go and let's
see what this law of

00:00:58.980 --> 00:01:00.300
cosines is all about.

00:01:00.300 --> 00:01:03.700
So let's say that I
know this angle theta.

00:01:07.860 --> 00:01:12.475
And let's called this
side-- I don't know, a.

00:01:12.475 --> 00:01:14.620
No, let's call this side b.

00:01:14.620 --> 00:01:17.230
I'm being a little
arbitrary here.

00:01:17.230 --> 00:01:21.640
Actually, let me stay in
the colors of the sides.

00:01:21.640 --> 00:01:28.030
Let's call that b and let's
call this c, and let's

00:01:28.030 --> 00:01:31.150
call this side a.

00:01:31.150 --> 00:01:33.080
So if this is a right triangle,
then we could have used

00:01:33.080 --> 00:01:37.630
the Pythagorean theorem
somehow, but now we can't.

00:01:37.630 --> 00:01:38.310
So what do we do?

00:01:38.310 --> 00:01:42.490
So we know a-- well, let's
assume that we know b, we know

00:01:42.490 --> 00:01:45.120
c, we know theta, and then
we want to solve for a.

00:01:45.120 --> 00:01:49.130
But, in general, as long as you
know three of these, you can

00:01:49.130 --> 00:01:51.560
solve for the fourth once you
know the law of cosines.

00:01:51.560 --> 00:01:52.610
So how can we do it?

00:01:52.610 --> 00:01:54.820
Well, we're going to do
it the exact same way we

00:01:54.820 --> 00:01:57.170
did that last problem.

00:01:57.170 --> 00:02:01.580
We can drop a line
here to make-- oh, my

00:02:01.580 --> 00:02:02.210
God, that's messy.

00:02:02.210 --> 00:02:04.420
I thought I was using
the line tool.

00:02:04.420 --> 00:02:05.165
Edit, undo.

00:02:08.380 --> 00:02:11.320
So I can drop a line like that.

00:02:11.320 --> 00:02:14.150
So I have two right angles.

00:02:14.150 --> 00:02:16.220
And then once I have right
triangles, then now I can start

00:02:16.220 --> 00:02:18.610
to use trig functions and
the Pythagorean theorem,

00:02:18.610 --> 00:02:20.350
et cetera, et cetera.

00:02:20.350 --> 00:02:25.210
So, let's see, this is a right
angle, this is a right angle.

00:02:25.210 --> 00:02:29.750
So what is this side here?

00:02:29.750 --> 00:02:30.770
Let me pick another color.

00:02:30.770 --> 00:02:33.560
I'm probably going to get too
involved with all the colors,

00:02:33.560 --> 00:02:36.380
but it's for your improvement.

00:02:36.380 --> 00:02:37.400
So what is this side here?

00:02:37.400 --> 00:02:40.720
What is the length of that
side, that purple side?

00:02:40.720 --> 00:02:44.700
Well, that purple side is just,
you know, we use SOHCAHTOA.

00:02:44.700 --> 00:02:46.605
I was just going to write
SOHCAHTOA up here.

00:02:50.850 --> 00:02:56.700
So this purple side is adjacent
to theta, and then this blue or

00:02:56.700 --> 00:03:03.910
mauve side b is the hypotenuse
of this right triangle.

00:03:03.910 --> 00:03:06.140
So we know that-- I'm just
going to stick to one color

00:03:06.140 --> 00:03:08.760
because it'll take me forever
if I keep switching colors.

00:03:08.760 --> 00:03:13.790
We know that cosine of theta--
let's call this side, let's

00:03:13.790 --> 00:03:16.660
call this kind of subside--
I don't know, let's

00:03:16.660 --> 00:03:21.200
call this d, side d.

00:03:21.200 --> 00:03:27.850
We know that cosine of theta
is equal to d over b, right?

00:03:27.850 --> 00:03:30.290
And we know b.

00:03:30.290 --> 00:03:36.870
Or that d is equal to what?

00:03:36.870 --> 00:03:42.660
It equals b cosine theta.

00:03:42.660 --> 00:03:47.750
Now, let's call this
side e right here.

00:03:47.750 --> 00:03:48.810
Well, what's e?

00:03:48.810 --> 00:03:52.340
Well, e is this whole c
side-- c side, oh, that's

00:03:52.340 --> 00:03:56.780
interesting-- this whole c side
minus this d side, right?

00:03:56.780 --> 00:04:02.630
So e is equal to c minus d.

00:04:02.630 --> 00:04:09.320
We just solved for d, so
side e is equal to c

00:04:09.320 --> 00:04:11.980
minus b cosine of theta.

00:04:14.800 --> 00:04:15.590
So that's e.

00:04:15.590 --> 00:04:18.900
We got e out of the way.

00:04:18.900 --> 00:04:21.340
Well, what's this magenta
side going to be?

00:04:21.340 --> 00:04:23.690
Well, let's call this magenta--
let's call it m from magenta.

00:04:27.230 --> 00:04:29.340
Well, m is opposite to theta.

00:04:32.800 --> 00:04:33.350
Now, we know it.

00:04:33.350 --> 00:04:36.110
We've solved for c as well, but
we know b, and b is simple.

00:04:36.110 --> 00:04:40.210
So what relationship gives us
m over b, or involves the

00:04:40.210 --> 00:04:41.400
opposite and the hypotenuse?

00:04:41.400 --> 00:04:45.110
Well, that's sine:
opposite over hypotenuse.

00:04:45.110 --> 00:04:49.670
So we know that m over b is
equal to sine of theta.

00:04:49.670 --> 00:04:52.520
We know that-- let
me go over here.

00:04:52.520 --> 00:04:57.180
m over b, right, because this
is the hypotenuse, is equal to

00:04:57.180 --> 00:05:08.550
sine of theta, or that m
is equal to b sine

00:05:08.550 --> 00:05:10.500
of theta, right?

00:05:10.500 --> 00:05:13.470
So we figured out m, we
figured out e, and now

00:05:13.470 --> 00:05:14.690
we want to figure out a.

00:05:14.690 --> 00:05:16.070
And this should
jump out at you.

00:05:16.070 --> 00:05:17.960
We have two sides of
a right triangle.

00:05:17.960 --> 00:05:19.750
We want to figure
out the hypotenuse.

00:05:19.750 --> 00:05:22.410
We can use the
Pythagorean theorem.

00:05:22.410 --> 00:05:28.490
The Pythagorean theorem tells
us a squared is equal to m

00:05:28.490 --> 00:05:31.510
squared plus e squared, right?

00:05:31.510 --> 00:05:33.660
Just the square of
the other two sides.

00:05:33.660 --> 00:05:35.920
Well, what's m squared
plus e squared?

00:05:35.920 --> 00:05:38.970
Let me switch to another
color just to be arbitrary.

00:05:38.970 --> 00:05:42.360
a squared is equal
to m squared.

00:05:42.360 --> 00:05:44.030
m is b sine of theta.

00:05:44.030 --> 00:05:53.590
So it's b sine of theta
squared plus e squared.

00:05:53.590 --> 00:05:55.790
Well, e we figure out is this.

00:05:55.790 --> 00:06:02.820
So it's plus c minus b
cosine theta squared.

00:06:02.820 --> 00:06:04.940
Now, let's just chug
through some algebra.

00:06:04.940 --> 00:06:13.090
So that equals b sine-- b
squared sine squared of theta.

00:06:13.090 --> 00:06:13.970
Sine squared of theta
just means sine of

00:06:13.970 --> 00:06:15.480
theta squared, right?

00:06:15.480 --> 00:06:17.700
Plus, and we just foiled
this out, although I

00:06:17.700 --> 00:06:18.420
don't like using foil.

00:06:18.420 --> 00:06:20.620
I just multiply it out.

00:06:20.620 --> 00:06:33.810
c squared minus 2cb cosine
theta plus b squared

00:06:33.810 --> 00:06:35.420
cosine theta, right?

00:06:35.420 --> 00:06:38.490
I just expanded this out
by multiplying it out.

00:06:38.490 --> 00:06:40.310
And now let's see if we can
do anything interesting.

00:06:40.310 --> 00:06:46.780
Well, if we take this term and
this term, we get-- those two

00:06:46.780 --> 00:06:54.360
terms are b squared sine
squared of theta plus b squared

00:06:54.360 --> 00:06:57.110
cosine-- this should be squared
there, right, because

00:06:57.110 --> 00:06:58.350
we squared it.

00:06:58.350 --> 00:07:03.780
b squared cosine squared of
theta, and then we have plus c

00:07:03.780 --> 00:07:10.350
squared minus 2bc cosine theta.

00:07:10.350 --> 00:07:12.470
Well, what does
this simplify to?

00:07:12.470 --> 00:07:17.735
Well, this is the same thing
as b squared times the

00:07:17.735 --> 00:07:22.470
sine squared theta plus
cosine squared of theta.

00:07:22.470 --> 00:07:27.420
Something should be jumping out
at you, and that's plus c

00:07:27.420 --> 00:07:33.330
squared minus 2bc cosine theta.

00:07:33.330 --> 00:07:36.170
Well, this thing, sine
squared plus cosine

00:07:36.170 --> 00:07:37.910
squared of any angle is 1.

00:07:37.910 --> 00:07:40.010
That's one of the
earlier identities.

00:07:40.010 --> 00:07:41.930
That's the Pythagorean
identity right there.

00:07:41.930 --> 00:07:47.210
So this equals 1, so then
we're left with-- going

00:07:47.210 --> 00:07:48.970
back to my original color.

00:07:48.970 --> 00:07:56.230
We're almost there-- a squared
is equal to-- this term just

00:07:56.230 --> 00:07:58.020
becomes 1, so b squared.

00:07:58.020 --> 00:08:07.160
We're just left with a b
squared plus c squared

00:08:07.160 --> 00:08:16.330
minus 2bc cosine of theta.

00:08:16.330 --> 00:08:20.940
That's pretty neat, and this
is called the law of cosines.

00:08:20.940 --> 00:08:24.410
And it's useful because, you
know, if you know an angle

00:08:24.410 --> 00:08:28.460
and two of the sides of
any triangle, you can now

00:08:28.460 --> 00:08:31.930
solve for the other side.

00:08:31.930 --> 00:08:35.170
Or really, if you want to, if
you know three sides of a

00:08:35.170 --> 00:08:37.810
triangle, you can now solve
for any angle, so that

00:08:37.810 --> 00:08:39.780
also is very useful.

00:08:39.780 --> 00:08:42.150
The only reason why I'm a
little bit, you know, here,

00:08:42.150 --> 00:08:46.050
there, is I don't-- if you are
in trigonometry right now and

00:08:46.050 --> 00:08:48.590
you might have a test, you
should memorize this because

00:08:48.590 --> 00:08:50.490
it'll make you faster, and
you'll get the answer

00:08:50.490 --> 00:08:51.650
right quicker.

00:08:51.650 --> 00:08:55.310
I'm not a big fan of just
memorizing it without knowing

00:08:55.310 --> 00:08:58.600
where it came from, because a
year from now or two years from

00:08:58.600 --> 00:09:01.530
now when you go to college and
it's been four years since you

00:09:01.530 --> 00:09:05.040
took trigonometry, you probably
won't have this memorized.

00:09:05.040 --> 00:09:06.990
And if you face a trig problem
all of a sudden, it's good to

00:09:06.990 --> 00:09:09.030
kind of get there from scratch.

00:09:09.030 --> 00:09:11.820
With that said, this is the law
of cosines, and if you use the

00:09:11.820 --> 00:09:14.450
law of cosines, you could have
done that problem we just did a

00:09:14.450 --> 00:09:17.070
lot faster because we just--
you know, you just have to set

00:09:17.070 --> 00:09:19.620
up the triangle and then just
substitute into this, and you

00:09:19.620 --> 00:09:23.600
could have solved for a in
that ship off-course problem.

00:09:23.600 --> 00:09:25.780
I'll see you in the next video.


Trigonometry word problems (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:01.470
Welcome back.

00:00:01.470 --> 00:00:04.850
I was just chugging through the
hairy math of this problem,

00:00:04.850 --> 00:00:06.300
which I think we're
almost done.

00:00:06.300 --> 00:00:08.950
But as I was saying, we're
just simplifying this.

00:00:08.950 --> 00:00:11.880
We have 25 times sine
squared of 15 plus

00:00:11.880 --> 00:00:13.010
cosine squared of 15.

00:00:13.010 --> 00:00:14.730
I said well, that's
an identity, right?

00:00:14.730 --> 00:00:16.690
Remember, I
undistributed this 25.

00:00:16.690 --> 00:00:20.190
So sine squared plus cosine
squared of the same angle is 1.

00:00:20.190 --> 00:00:23.130
So I can simplify this to --
and I will now change colors

00:00:23.130 --> 00:00:24.810
just to be arbitrary.

00:00:24.810 --> 00:00:35.240
So, 25 plus 100 minus 100
cosine of 15 degrees, and this

00:00:35.240 --> 00:00:38.090
is what m squared is equal to,
and I used the variable m

00:00:38.090 --> 00:00:40.910
because it stands for nothing
relevant to this problem.

00:00:40.910 --> 00:00:49.440
And that equals 125 minus
100 cosine of 15 degrees.

00:00:49.440 --> 00:00:53.320
Now I have to use a calculator
if I want to get a real answer.

00:00:53.320 --> 00:00:59.060
But the possibility for error
is much lower now that I've

00:00:59.060 --> 00:01:00.320
hopefully simplified it.

00:01:00.320 --> 00:01:05.080
So I could use my calculator,
and the cosine of 15 degrees --

00:01:05.080 --> 00:01:07.610
I'll write that down here, you
could just use a calculator or

00:01:07.610 --> 00:01:20.220
table to figure that out --
it is equal to 0.96593.

00:01:20.220 --> 00:01:25.040
So this term would be 125
minus 100 times this, right?

00:01:25.040 --> 00:01:26.430
So what's 100 times this?

00:01:26.430 --> 00:01:32.600
Well that's just minus 96.593,
right, I just did 100 times

00:01:32.600 --> 00:01:34.880
the cosine of 15 degrees.

00:01:34.880 --> 00:01:37.970
And now let me use the
trusty calculator to

00:01:37.970 --> 00:01:39.220
figure out what that is.

00:01:39.220 --> 00:01:56.800
So if I say 125 minus
96.593, I get 28.407.

00:01:56.800 --> 00:02:04.050
So I have m squared
is equal to 28.407.

00:02:04.050 --> 00:02:06.492
And so m is equal to the
square root of that.

00:02:06.492 --> 00:02:08.910
It's going to be 5 point
something, right?

00:02:08.910 --> 00:02:09.970
Square root.

00:02:09.970 --> 00:02:12.220
5.3298.

00:02:12.220 --> 00:02:13.950
So 5.33.

00:02:13.950 --> 00:02:18.690
m is equal to 5.33.

00:02:18.690 --> 00:02:19.570
And we're done.

00:02:19.570 --> 00:02:21.870
The poor guy -- he's got to
figure out the right direction

00:02:21.870 --> 00:02:24.040
to go in and he probably has to
think about the current in all

00:02:24.040 --> 00:02:26.820
of that, but he hasn't asked us
that question so we

00:02:26.820 --> 00:02:28.300
won't answer it.

00:02:28.300 --> 00:02:32.190
But all he has left
to go is 5.33 miles.

00:02:32.190 --> 00:02:36.770
So he only has to go an extra
0.33 miles than he would have,

00:02:36.770 --> 00:02:40.240
had he gone in the right
direction from the get-go.

00:02:40.240 --> 00:02:42.490
Hopefully that
makes some sense.

00:02:42.490 --> 00:02:45.740
If it doesn't, well, not much
I can do, but maybe you

00:02:45.740 --> 00:02:47.960
can re-watch this video.

00:02:47.960 --> 00:02:49.460
Have fun.


Trigonometry word problems (part 1)

WEBVTT
Kind: captions
Language: en

00:00:01.370 --> 00:00:03.880
Now that we know some
trigonometry, let's use

00:00:03.880 --> 00:00:07.270
that trigonometry to
solve some word problems.

00:00:07.270 --> 00:00:09.270
So let's get started
with-- This is exciting.

00:00:09.270 --> 00:00:11.370
This is a navigation problem.

00:00:11.370 --> 00:00:16.270
So I have a ship that needs to
go from point A to point B.

00:00:16.270 --> 00:00:17.140
So let me draw that.

00:00:17.140 --> 00:00:21.440
So I'm going from point
A, which is that

00:00:21.440 --> 00:00:24.400
point, to point B.

00:00:24.400 --> 00:00:27.250
Let me draw a nice, big--
well not so big --triangle.

00:00:27.250 --> 00:00:28.620
Big enough.

00:00:28.620 --> 00:00:30.130
There you go.

00:00:30.130 --> 00:00:32.040
Because I need space
to do my math.

00:00:32.040 --> 00:00:33.460
So let me-- Let's see.

00:00:33.460 --> 00:00:35.750
So this is point A.

00:00:35.750 --> 00:00:38.880
Point A I will draw in mauve.

00:00:38.880 --> 00:00:41.560
Point A to point B.

00:00:41.560 --> 00:00:45.140
So this ship needs to go
from point A to point B.

00:00:45.140 --> 00:00:48.470
And this problem tells us that
the distance between point A

00:00:48.470 --> 00:00:50.455
and point B is 10 kilometers.

00:00:53.460 --> 00:00:56.690
So this guy set sail and,
because of whatever reason.

00:00:56.690 --> 00:00:59.520
He's a bad navigator or
the water is flowing

00:00:59.520 --> 00:01:01.070
faster maybe upwards.

00:01:01.070 --> 00:01:02.670
Maybe the water's flowing
in this direction.

00:01:02.670 --> 00:01:05.340
Let me draw that in blue to
show which direction the

00:01:05.340 --> 00:01:07.610
water-- Say the current is
flowing in that direction.

00:01:07.610 --> 00:01:09.490
And he ends up off track.

00:01:09.490 --> 00:01:11.890
He actually ends up, after
5 kilometers-- So he

00:01:11.890 --> 00:01:15.120
travels for 5 kilometers.

00:01:15.120 --> 00:01:17.220
Let me do that in blue-green.

00:01:17.220 --> 00:01:23.380
He travels for 5 kilometers and
after 5 kilometers, he realizes

00:01:23.380 --> 00:01:25.640
that he has gone off track.

00:01:25.640 --> 00:01:28.550
And somehow-- I don't know
maybe using his astrolabe.

00:01:28.550 --> 00:01:31.730
I don't really know what one
is, but that's what people use

00:01:31.730 --> 00:01:33.980
when they're on ships to figure
out where they are I think.

00:01:33.980 --> 00:01:39.760
--he realizes that he is
15 degrees off course.

00:01:39.760 --> 00:01:43.630
So what that means is that
he's-- Between the path he took

00:01:43.630 --> 00:01:46.020
and the path he should've
took is 15 degrees.

00:01:46.020 --> 00:01:48.540
At least that's my
interpretation of it.

00:01:48.540 --> 00:01:51.690
So this is 15 degrees.

00:01:51.690 --> 00:01:53.630
And, of course, he's traveled
5 kilometers in the

00:01:53.630 --> 00:01:54.280
wrong direction.

00:01:54.280 --> 00:01:57.960
15 degrees in the
wrong direction.

00:01:57.960 --> 00:02:00.750
So what he needs to know
is-- So this is-- Let

00:02:00.750 --> 00:02:01.890
me draw a little ship.

00:02:01.890 --> 00:02:04.670
So this is where
the ship is now.

00:02:04.670 --> 00:02:07.560
That's my ship.

00:02:07.560 --> 00:02:11.120
So now this guy, this poor poor
captain, he said so how far do

00:02:11.120 --> 00:02:13.820
I still have to go
to get to point B?

00:02:13.820 --> 00:02:18.150
So let me-- So he needs to know
how far is-- let me pick a nice

00:02:18.150 --> 00:02:21.150
color --how far is this
distance right here?

00:02:25.640 --> 00:02:29.510
This may just involve
some trigonometry.

00:02:29.510 --> 00:02:31.920
So how can we figure
out this distance?

00:02:31.920 --> 00:02:34.650
Well let's just break
down what we know.

00:02:34.650 --> 00:02:36.000
And that's how I
do trig problems.

00:02:36.000 --> 00:02:37.480
I just keep messing around
with it until I get

00:02:37.480 --> 00:02:38.260
the right answer.

00:02:38.260 --> 00:02:42.950
So let's see if we can tackle
this problem that way.

00:02:42.950 --> 00:02:44.060
So what can we figure out?

00:02:44.060 --> 00:02:45.610
We know this is 5 kilometers.

00:02:45.610 --> 00:02:46.750
This side is 5 kilometers.

00:02:46.750 --> 00:02:48.060
We know that this
is 10 kilometers.

00:02:48.060 --> 00:02:49.380
We know this angle.

00:02:49.380 --> 00:02:53.190
Well one thing I know I can
figure out is-- Let's just make

00:02:53.190 --> 00:02:56.420
some right angles so we can
start using some trigonometry.

00:02:56.420 --> 00:02:58.310
Let me draw a right angle here.

00:02:58.310 --> 00:02:59.550
I just drop it straight down.

00:03:02.310 --> 00:03:05.080
So can I figure out what
this side is right here?

00:03:05.080 --> 00:03:07.110
Let me draw that in
a different color.

00:03:07.110 --> 00:03:08.470
In this brown color.

00:03:08.470 --> 00:03:10.700
Can I figure out what
this side right here is?

00:03:13.790 --> 00:03:16.520
Well this side is
going to be what?

00:03:16.520 --> 00:03:19.010
This is-- If we look at this
angle, this is adjacent to

00:03:19.010 --> 00:03:21.860
the angle and this
is the hypotenuse.

00:03:21.860 --> 00:03:24.410
So if we know the hypotenuse,
we know the angle, we want to

00:03:24.410 --> 00:03:27.190
figure out adjacent, what
trig function should we use?

00:03:27.190 --> 00:03:29.110
Let me write out--
down our SOH CAH TOA.

00:03:34.450 --> 00:03:36.690
So we know the hypotenuse.

00:03:36.690 --> 00:03:38.310
We want to figure
out the adjacent.

00:03:38.310 --> 00:03:40.760
So what involves adjacent
and hypotenuse?

00:03:40.760 --> 00:03:43.620
CAH, or cosine.

00:03:43.620 --> 00:03:47.620
So we know that the cosine of
15 degrees-- so I'm going to

00:03:47.620 --> 00:03:51.780
write over this just not to
waste space --cosine of 15

00:03:51.780 --> 00:03:56.030
degrees is equal to
this brown side.

00:03:56.030 --> 00:03:57.170
So let's just call
this-- I don't know.

00:03:57.170 --> 00:04:01.440
I'm just going to
call it x, right?

00:04:01.440 --> 00:04:08.940
So this is equal to x over the
hypotenuse, over 5, right?

00:04:08.940 --> 00:04:13.560
If we solve for x, we
get x is equal to 5

00:04:13.560 --> 00:04:18.110
cosine of 15 degrees.

00:04:18.110 --> 00:04:19.090
Maybe we made progress.

00:04:19.090 --> 00:04:19.640
Maybe we didn't.

00:04:19.640 --> 00:04:21.010
Let's keep trying.

00:04:21.010 --> 00:04:21.490
All right.

00:04:21.490 --> 00:04:23.140
Now let's see if we could
figure out this side

00:04:23.140 --> 00:04:24.680
of this triangle.

00:04:24.680 --> 00:04:27.640
Well this is, to this angle,
this is the opposite, right?

00:04:27.640 --> 00:04:28.770
This is the opposite.

00:04:28.770 --> 00:04:29.910
And we know the hypotenuse.

00:04:29.910 --> 00:04:32.220
So what trig identity
should we-- trig

00:04:32.220 --> 00:04:33.670
function should we use?

00:04:33.670 --> 00:04:35.970
well if we know the opposite--
Well if we want to figure out

00:04:35.970 --> 00:04:38.130
the opposite and we know
the hypotenuse and if

00:04:38.130 --> 00:04:38.820
we know the angle.

00:04:38.820 --> 00:04:42.050
So what trig function deals
opposite-- I was going

00:04:42.050 --> 00:04:43.570
to say hypotenuse.

00:04:43.570 --> 00:04:44.180
But anyway.

00:04:44.180 --> 00:04:46.640
What trig function deals with
the opposite of the hypotenuse?

00:04:46.640 --> 00:04:49.250
Well that's sine, right?

00:04:49.250 --> 00:04:52.940
So the sine-- We call this y.

00:04:52.940 --> 00:04:59.600
We know that y is equal to--
Well, doing it the same way,

00:04:59.600 --> 00:05:05.050
sine-- 5 sine of 15 degrees.

00:05:05.050 --> 00:05:06.020
I skipped a step, right?

00:05:06.020 --> 00:05:12.660
Because we could say that sine
of 15 degrees is equal to the

00:05:12.660 --> 00:05:18.510
opposite, y, over the
hypotenuse, over 5.

00:05:18.510 --> 00:05:20.490
And then that step
takes us here.

00:05:20.490 --> 00:05:22.200
We just multiply
both sides by 5.

00:05:22.200 --> 00:05:24.960
You get y is equal to
5 sine of 15 degrees.

00:05:24.960 --> 00:05:25.620
So that's pretty cool.

00:05:25.620 --> 00:05:26.500
We know this y.

00:05:26.500 --> 00:05:27.610
We know the x.

00:05:27.610 --> 00:05:29.620
Can we figure out
what this length is?

00:05:29.620 --> 00:05:32.570
Let me draw it in
yet another color.

00:05:32.570 --> 00:05:36.020
Can we figure out what the
length of this side is?

00:05:36.020 --> 00:05:38.050
Well we know from the
prompt that this whole

00:05:38.050 --> 00:05:39.700
length is 10, right?

00:05:39.700 --> 00:05:41.600
We know this whole
length is 10.

00:05:41.600 --> 00:05:44.530
And we know that this little
part of it is x, which

00:05:44.530 --> 00:05:47.690
is, we figured out is 5
cosine of 15 degrees.

00:05:47.690 --> 00:05:50.810
So let's call this z.

00:05:50.810 --> 00:05:59.410
We know that z is equal
to 10 minus x, right?

00:05:59.410 --> 00:06:00.760
Because this whole thing is 10.

00:06:00.760 --> 00:06:01.750
This thing is x.

00:06:01.750 --> 00:06:02.700
So z equals 10 minus x.

00:06:02.700 --> 00:06:04.630
And we already figured
out what x is equal to.

00:06:04.630 --> 00:06:07.820
Z is equal to 10 minus this.

00:06:07.820 --> 00:06:08.570
That's what x is.

00:06:08.570 --> 00:06:09.935
We didn't even have to
use the variable x.

00:06:09.935 --> 00:06:11.590
We could have just
written it like that.

00:06:11.590 --> 00:06:18.850
So 10 minus 5 cosine
of 15 degrees, right?

00:06:18.850 --> 00:06:19.720
That's this side.

00:06:19.720 --> 00:06:23.200
Z is equal to 10 minus 5
cosine of 15 degrees.

00:06:23.200 --> 00:06:24.050
So we know z.

00:06:24.050 --> 00:06:25.150
We know y.

00:06:25.150 --> 00:06:26.470
This is a right triangle.

00:06:26.470 --> 00:06:29.010
And we're looking for this
yellow side, whatever

00:06:29.010 --> 00:06:30.110
we want to call it.

00:06:30.110 --> 00:06:31.750
That's the answer
to the problem.

00:06:31.750 --> 00:06:33.410
Well this is starting
to look easy.

00:06:33.410 --> 00:06:34.990
This is just the
Pythagorean Theorem.

00:06:34.990 --> 00:06:38.420
So let's just-- y squared plus
z squared is going to be equal

00:06:38.420 --> 00:06:40.010
to our-- let's just call this.

00:06:40.010 --> 00:06:40.460
I don't know.

00:06:40.460 --> 00:06:43.640
Let's call this m.

00:06:43.640 --> 00:06:45.550
Picking an arbitrary letter.

00:06:45.550 --> 00:06:46.230
Let's call that m.

00:06:46.230 --> 00:06:50.530
So y squared plus z squared is
going to be equal to m squared.

00:06:50.530 --> 00:06:52.730
So we could just say-- Let's
just write that down.

00:06:52.730 --> 00:06:53.770
I don't know why I picked m.

00:06:53.770 --> 00:06:55.920
Really just to confuse
you, I think.

00:06:55.920 --> 00:07:02.960
So m squared is equal to-- Well
what's y squared? y squared

00:07:02.960 --> 00:07:05.510
is 5 sine of 15 degrees.

00:07:05.510 --> 00:07:10.350
So it equals-- Let me
do it in that color.

00:07:10.350 --> 00:07:20.540
5 sine of 15 degrees squared
plus-- What's z squared?

00:07:20.540 --> 00:07:22.570
z is this, right?

00:07:22.570 --> 00:07:28.830
z squared plus-- I'm running
out of space --10 minus 5

00:07:28.830 --> 00:07:34.290
cosine of 15 degrees squared.

00:07:34.290 --> 00:07:35.826
And now I just have
to simplify this.

00:07:35.826 --> 00:07:38.060
And if you have a calculator
you could do this without

00:07:38.060 --> 00:07:38.860
any simplification.

00:07:38.860 --> 00:07:42.390
But I want to get it
as simple as I can.

00:07:42.390 --> 00:07:45.030
So let me just-- So
what's this squared?

00:07:45.030 --> 00:07:47.310
Well this is equal to-- Let me
draw a line here because I

00:07:47.310 --> 00:07:49.580
don't want to get too messy.

00:07:49.580 --> 00:07:50.590
But I need all of this space.

00:07:53.730 --> 00:07:57.620
So this is equal to-- What
color am I using now?

00:07:57.620 --> 00:07:59.520
This is equal to-- And
remember, this is m squared.

00:07:59.520 --> 00:08:00.930
We're going to have to take the
square root of all of this

00:08:00.930 --> 00:08:03.030
at the end to solve for m.

00:08:03.030 --> 00:08:11.390
This is equal to 25 sine
squared 15 degrees, or sine

00:08:11.390 --> 00:08:12.250
of 15 degrees squared.

00:08:12.250 --> 00:08:13.240
That's just how you write it.

00:08:13.240 --> 00:08:17.130
And then we do a little bit
of FOIL here to expand this.

00:08:17.130 --> 00:08:27.170
So plus 100 minus-- this is
just expanding this expression

00:08:27.170 --> 00:08:39.330
--minus 100 cosine of 15
degrees plus 25 cosine

00:08:39.330 --> 00:08:41.740
squared of 15 degrees.

00:08:41.740 --> 00:08:43.000
All I is I expanded.

00:08:43.000 --> 00:08:45.340
I said, well this whole
expression squared is equal to

00:08:45.340 --> 00:08:49.460
this squared minus 2 times the
two things multiplied out.

00:08:49.460 --> 00:08:51.580
So that's 100 cosine
of 15 degrees.

00:08:51.580 --> 00:08:54.180
And then I just squared this
last term, which is plus

00:08:54.180 --> 00:08:55.950
25 cosine of 15 degrees.

00:08:55.950 --> 00:08:58.350
If that confused you, you
might want to review the

00:08:58.350 --> 00:08:59.890
multiplying expressions.

00:08:59.890 --> 00:09:01.580
So let's see if we can
simplify this further.

00:09:01.580 --> 00:09:05.780
So if we take this term and
this term, we could simplify

00:09:05.780 --> 00:09:14.610
that to 25 times sine
squared of 15 degrees plus

00:09:14.610 --> 00:09:17.290
cosine squared of 15.

00:09:17.290 --> 00:09:20.090
And you could skip all this and
just use a calculator and

00:09:20.090 --> 00:09:21.150
figure out this exact value.

00:09:21.150 --> 00:09:23.590
But I'm just going to keep
working on it just because

00:09:23.590 --> 00:09:25.160
I like to get it as simple
as possible before I

00:09:25.160 --> 00:09:26.630
use the calculator.

00:09:26.630 --> 00:09:28.760
So that term and
that term are that.

00:09:28.760 --> 00:09:31.120
And then it's plus this stuff.

00:09:31.120 --> 00:09:38.450
Plus 100 minus 100 cosine
of 15 degrees, right?

00:09:38.450 --> 00:09:41.020
And then-- What is the sine
squared of 15 plus the

00:09:41.020 --> 00:09:42.320
cosine squared of 15?

00:09:42.320 --> 00:09:44.500
That's one of our most
basic identities, right?

00:09:44.500 --> 00:09:47.050
That's the Pythagorean
Identity.

00:09:47.050 --> 00:09:49.690
Sine squared of x plus
cosine squared of x, or of

00:09:49.690 --> 00:09:51.350
theta, is just 1, right?

00:09:51.350 --> 00:09:53.000
So this term becomes 1.

00:09:53.000 --> 00:09:54.130
And I'm running out of time.

00:09:54.130 --> 00:09:56.420
So I will continue
in the next video.


2-dimensional projectile motion (part 3)

WEBVTT
Kind: captions
Language: en

00:00:01.120 --> 00:00:01.950
Welcome back.

00:00:01.950 --> 00:00:03.680
So we just figured out
how long does this

00:00:03.680 --> 00:00:04.850
thing stay in the air.

00:00:04.850 --> 00:00:06.190
So what can we do with that?

00:00:06.190 --> 00:00:09.510
Let me clear all of this and
redraw it because maybe every

00:00:09.510 --> 00:00:12.890
time I redraw it, it might
further clarify things.

00:00:15.520 --> 00:00:18.710
Let me actually draw things a
little bit fancier this time.

00:00:18.710 --> 00:00:24.430
So I have the ball and it's
moving-- look at that.

00:00:24.430 --> 00:00:27.530
Let me just draw the triangle
ahead of time.

00:00:27.530 --> 00:00:29.960
I'm going to use
fancy-- whoops.

00:00:29.960 --> 00:00:33.070
Actually, let me use
a thicker line.

00:00:33.070 --> 00:00:37.540
So this was the initial
vector.

00:00:37.540 --> 00:00:41.460
Then I wanted to do--
switch to green.

00:00:41.460 --> 00:00:42.670
This is its horizontal.

00:00:42.670 --> 00:00:44.070
I'm going to draw the
arrows in a second.

00:00:44.070 --> 00:00:48.070
I just want to get all the line
tool work out of the way.

00:00:48.070 --> 00:00:49.160
Bam.

00:00:49.160 --> 00:00:55.430
And this is the--
close enough.

00:00:55.430 --> 00:00:58.310
And then, if I use my
paint tool, I could

00:00:58.310 --> 00:01:00.260
draw an arrow there.

00:01:00.260 --> 00:01:01.880
Switch back to green.

00:01:01.880 --> 00:01:03.370
Draw an arrow there.

00:01:03.370 --> 00:01:04.569
Switch back to magenta.

00:01:04.569 --> 00:01:07.110
Draw an arrow there.

00:01:07.110 --> 00:01:14.450
So this was 10 meters per second
at a 45 degree angle.

00:01:14.450 --> 00:01:17.690
We figured out in the last
video that its horizontal

00:01:17.690 --> 00:01:23.930
velocity is equal to
10 cosine of 45.

00:01:23.930 --> 00:01:28.260
Which equals 5 square
root of 2.

00:01:28.260 --> 00:01:32.960
And we also figured out that the
vertical velocity is equal

00:01:32.960 --> 00:01:39.300
to 10 sine of 45 degrees.

00:01:39.300 --> 00:01:43.200
Which also equals 5
square root of 2.

00:01:43.200 --> 00:01:44.910
And we wanted to figure out, how
long does this thing stay

00:01:44.910 --> 00:01:45.780
in the air?

00:01:45.780 --> 00:01:48.840
And we said, well, if we're
going-- the vertical component

00:01:48.840 --> 00:01:52.020
is 5 square root of 2
meters per second.

00:01:52.020 --> 00:01:55.000
So if I'm going 5 square root
of 2 meters per second, the

00:01:55.000 --> 00:01:57.910
ball-- how long does it take for
the ball to go from that

00:01:57.910 --> 00:01:59.820
point to the point that
it's stationary at

00:01:59.820 --> 00:02:01.070
the top of its path?

00:02:04.270 --> 00:02:08.030
It's going at 5 square root
of 2 meters per second.

00:02:08.030 --> 00:02:10.590
And then it goes to 0.

00:02:10.590 --> 00:02:12.905
So how long does it take to
decelerate from 5 square

00:02:12.905 --> 00:02:14.480
root of 2 to 0?

00:02:14.480 --> 00:02:16.630
And there we just said, you
know, change in velocity.

00:02:16.630 --> 00:02:19.930
Well my change in velocity is
minus 5 square root of 2.

00:02:19.930 --> 00:02:23.600
That equals acceleration, which
is minus 10 meters per

00:02:23.600 --> 00:02:25.710
second squared times time.

00:02:25.710 --> 00:02:29.530
We figured out the time when the
ball leaves to kind of its

00:02:29.530 --> 00:02:36.265
highest point is the square
root of 2 seconds.

00:02:36.265 --> 00:02:39.680
And that happens if you just
solve this equation.

00:02:39.680 --> 00:02:42.510
And that's how long the ball
takes to go from the ground to

00:02:42.510 --> 00:02:43.390
its highest point.

00:02:43.390 --> 00:02:46.350
And it's going to take the exact
amount of time for it to

00:02:46.350 --> 00:02:49.500
reaccelerate and
come back down.

00:02:49.500 --> 00:02:52.715
So it takes square root of 2
seconds to go up, it's going

00:02:52.715 --> 00:02:54.740
to take square root of
2 seconds to go down.

00:02:54.740 --> 00:03:03.760
So the total time is equal
to 2 square root of 2.

00:03:03.760 --> 00:03:05.280
This is the total time.

00:03:05.280 --> 00:03:07.980
This is kind of a nice
intermediate thing that we've

00:03:07.980 --> 00:03:09.620
discovered.

00:03:09.620 --> 00:03:11.630
Now, can we use this to
figure out how far

00:03:11.630 --> 00:03:13.340
does that ball travel?

00:03:13.340 --> 00:03:15.760
How far is this sucker
going to go?

00:03:15.760 --> 00:03:18.670
Well, the entire time that--
you know, we were messing

00:03:18.670 --> 00:03:21.895
around with the vertical speed
or the vertical velocity of

00:03:21.895 --> 00:03:22.760
this ball and the vertical

00:03:22.760 --> 00:03:24.000
acceleration, vertical position.

00:03:24.000 --> 00:03:27.040
The whole time the ball was
going up and down, it was also

00:03:27.040 --> 00:03:32.570
going to the right because it
has this horizontal velocity.

00:03:32.570 --> 00:03:34.410
And we're going to assume
that the horizontal

00:03:34.410 --> 00:03:35.860
velocity stays constant.

00:03:35.860 --> 00:03:37.010
And why can we assume that?

00:03:37.010 --> 00:03:39.010
Well one, we're assuming that
there's no air resistance.

00:03:39.010 --> 00:03:41.040
Because air resistance makes
very-- things very

00:03:41.040 --> 00:03:41.580
complicated.

00:03:41.580 --> 00:03:44.370
If this is something like a
penny or a marble, it actually

00:03:44.370 --> 00:03:45.840
does have very little
air resistance.

00:03:45.840 --> 00:03:48.570
So air isn't going
to slow it down.

00:03:48.570 --> 00:03:52.710
And so if there's no air, what
can slow it down in the

00:03:52.710 --> 00:03:54.220
horizontal direction?

00:03:54.220 --> 00:03:55.180
Well, nothing.

00:03:55.180 --> 00:03:57.020
Gravity only acts downwards.

00:03:57.020 --> 00:04:00.740
Gravity doesn't make something
slow down as it moves

00:04:00.740 --> 00:04:01.660
horizontally.

00:04:01.660 --> 00:04:03.840
You could imagine pushing
something on ice.

00:04:03.840 --> 00:04:06.430
And if there's nothing slowing
it down, no friction, no air

00:04:06.430 --> 00:04:08.340
resistance, it'll just
keep on going.

00:04:08.340 --> 00:04:10.550
And actually, the assumptions
we're making would be

00:04:10.550 --> 00:04:12.110
absolutely true on the moon.

00:04:12.110 --> 00:04:13.630
Because the moon has no air.

00:04:13.630 --> 00:04:17.380
But anyway, so the whole time
the ball's going up and down

00:04:17.380 --> 00:04:20.155
it's also moving to the right
at this speed, at 5 square

00:04:20.155 --> 00:04:24.370
root of 2 meters per second.

00:04:24.370 --> 00:04:26.770
And let me ask you a question:
how long is it going to be

00:04:26.770 --> 00:04:28.350
moving to the right?

00:04:28.350 --> 00:04:31.300
Well, it's going to be moving
to the right as long as it's

00:04:31.300 --> 00:04:32.500
up in the air.

00:04:32.500 --> 00:04:34.570
And do we know how long it's
going to be up in the air?

00:04:34.570 --> 00:04:36.270
Well yeah, we just
figured that out.

00:04:36.270 --> 00:04:38.650
We figured out that it stays in
the air for 2 square root

00:04:38.650 --> 00:04:42.250
of 2 seconds.

00:04:42.250 --> 00:04:44.770
So to figure out its horizontal
distance, to figure

00:04:44.770 --> 00:04:48.110
out how long it is along the
ground, so we could say change

00:04:48.110 --> 00:04:49.360
in distance horizontally.

00:04:54.830 --> 00:04:58.040
That just equals my average
velocity, which is just this

00:04:58.040 --> 00:04:59.980
because it doesn't
really change.

00:04:59.980 --> 00:05:04.260
So it equals 5 square root
of 2 meters per second.

00:05:04.260 --> 00:05:06.690
Times the time that
I'm in the air.

00:05:06.690 --> 00:05:08.850
Times 2 square root
of 2 seconds.

00:05:08.850 --> 00:05:10.340
Times this sucker.

00:05:10.340 --> 00:05:12.900
2 square root of 2 seconds.

00:05:12.900 --> 00:05:17.450
So when you multiply
it out-- let's see.

00:05:17.450 --> 00:05:20.515
10 times-- square root of
2 times square root of

00:05:20.515 --> 00:05:22.820
2 is 2 times 2.

00:05:22.820 --> 00:05:29.420
So we go 20 meters.

00:05:29.420 --> 00:05:31.640
That I think is pretty cool.

00:05:31.640 --> 00:05:34.450
All I did is I said well I'm
throwing something at 10

00:05:34.450 --> 00:05:37.010
meters per second at
a 45 degree angle,

00:05:37.010 --> 00:05:37.930
how far did it go?

00:05:37.930 --> 00:05:39.860
And you could figure it out.

00:05:39.860 --> 00:05:43.180
You could figure out that
it went 20 meters.

00:05:43.180 --> 00:05:45.310
And actually, you could do
things the other way.

00:05:45.310 --> 00:05:47.570
And I'll do that in future
problems. You can figure out

00:05:47.570 --> 00:05:51.690
the exact angle you need to
throw something or you need to

00:05:51.690 --> 00:05:55.630
shoot something, if we were
thinking military terms. If

00:05:55.630 --> 00:05:58.070
you were an artillery man and
you had to arm the-- you had

00:05:58.070 --> 00:06:01.550
to figure out what angle to
shoot the big gun at to shoot

00:06:01.550 --> 00:06:03.580
a target-- I don't know, 5
miles away-- not that I

00:06:03.580 --> 00:06:05.900
advocate these type of things--
war is very bad.

00:06:05.900 --> 00:06:07.860
You would be doing essentially
this problem.

00:06:07.860 --> 00:06:09.810
I don't know if people
in the military are

00:06:09.810 --> 00:06:10.470
actually doing it.

00:06:10.470 --> 00:06:12.340
They're probably having a
computer do it for them.

00:06:12.340 --> 00:06:14.310
But it's really not
that hard to do.

00:06:14.310 --> 00:06:16.850
And you know, if we ever lose
our computers and we have to

00:06:16.850 --> 00:06:18.950
do things with bows and arrows,
it's good to be able

00:06:18.950 --> 00:06:22.020
to sit down and do the
math like this.

00:06:22.020 --> 00:06:24.400
So that's all I'm going to do
for-- well, we could do a

00:06:24.400 --> 00:06:25.280
couple of other things too.

00:06:25.280 --> 00:06:27.150
We can also figure out how
high the ball went.

00:06:27.150 --> 00:06:29.920
And how high the ball went is
actually no different than the

00:06:29.920 --> 00:06:32.020
1-dimensional problem.

00:06:32.020 --> 00:06:34.370
We know that this is the
initial velocity

00:06:34.370 --> 00:06:36.050
in its upward direction.

00:06:36.050 --> 00:06:37.790
We know it goes to 0.

00:06:37.790 --> 00:06:40.810
So its average velocity in
the upward direction

00:06:40.810 --> 00:06:41.870
is going to be what?

00:06:41.870 --> 00:06:44.310
Its average velocity in the
upward direction is going to

00:06:44.310 --> 00:06:47.910
be this plus this
divided by 2.

00:06:47.910 --> 00:06:52.480
So the average horizontal
velocity--

00:06:52.480 --> 00:06:54.030
vertical velocity, sorry.

00:06:54.030 --> 00:06:57.090
Vertical velocity average.

00:06:57.090 --> 00:06:58.870
This is funny notation.

00:06:58.870 --> 00:07:00.030
Is just going to be
half of this.

00:07:00.030 --> 00:07:04.000
It's going to be 5 square
root of 2 over 2.

00:07:04.000 --> 00:07:07.230
And it travels that
for how long?

00:07:07.230 --> 00:07:09.420
Well just while it's going up,
it travels that for a square

00:07:09.420 --> 00:07:11.755
root of 2 seconds times the
square root of 2 seconds.

00:07:16.050 --> 00:07:18.660
Distance is the average
velocity times time.

00:07:18.660 --> 00:07:19.750
So this is a sequel.

00:07:19.750 --> 00:07:23.060
This square root of 2 times this
square root of 2 is 2,

00:07:23.060 --> 00:07:24.350
and it cancels out with this.

00:07:24.350 --> 00:07:29.290
So the peak height that we could
say, the height is going

00:07:29.290 --> 00:07:31.390
to be 5 meters.

00:07:31.390 --> 00:07:32.360
So that's pretty neat.

00:07:32.360 --> 00:07:35.500
If I throw something at 10
meters per second at a 45

00:07:35.500 --> 00:07:38.730
degree angle, its maximum height
is going to be 5 meters

00:07:38.730 --> 00:07:40.550
and it's going to
go 20 meters.

00:07:40.550 --> 00:07:43.050
So I could draw that.

00:07:43.050 --> 00:07:46.110
Image, clear image.

00:07:46.110 --> 00:07:48.750
Image, invert colors.

00:07:48.750 --> 00:07:52.310
So let me draw my ground.

00:07:52.310 --> 00:07:54.096
Oh, whoops.

00:07:54.096 --> 00:07:57.420
So I'm going to draw my ground,
it's kind of a yellow

00:07:57.420 --> 00:07:58.820
ochre ground.

00:07:58.820 --> 00:08:09.790
And I'm going to shoot something
at an angle of 45

00:08:09.790 --> 00:08:15.960
degrees at 10 meters
per second.

00:08:15.960 --> 00:08:22.600
And we figured out that it
will go 20 meters and its

00:08:22.600 --> 00:08:25.630
maximum height is going
to be-- let's see.

00:08:25.630 --> 00:08:27.400
This is 10, so I'm
trying to draw

00:08:27.400 --> 00:08:28.620
something kind of to scale.

00:08:28.620 --> 00:08:31.680
So its maximum height
is roughly 5 meters.

00:08:31.680 --> 00:08:33.010
It is exactly 5 meters.

00:08:33.010 --> 00:08:34.400
Roughly is what I'm drawing.

00:08:34.400 --> 00:08:37.789
And so its path is going to
look something like this.

00:08:37.789 --> 00:08:39.799
You know, right from the get-go
it's at that angle, but

00:08:39.799 --> 00:08:41.650
the angle is going to get
shallower and shallower.

00:08:44.360 --> 00:08:46.530
Then it gets to 5 meters
and it comes down.

00:08:46.530 --> 00:08:50.270
And then it hits the ground.

00:08:50.270 --> 00:08:50.800
So there you go.

00:08:50.800 --> 00:08:53.090
We just figured out how far
does that thing go and how

00:08:53.090 --> 00:08:53.590
high does it go.

00:08:53.590 --> 00:08:55.290
What else could you want?

00:08:55.290 --> 00:08:58.020
So in the next video I'm going
to do this more generally, so

00:08:58.020 --> 00:09:00.610
that we can kind of-- we
can work backwards.

00:09:00.610 --> 00:09:03.846
If we know how far something
went and how long it was in

00:09:03.846 --> 00:09:05.440
the air, maybe we could
figure out its angle.

00:09:05.440 --> 00:09:07.640
Or maybe we could actually
figure out what the optimal

00:09:07.640 --> 00:09:10.240
angle is to throw something
for distance.

00:09:10.240 --> 00:09:12.410
I'll see you in the
next video.


2 dimensional projectile motion (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:01.460
Welcome back.

00:00:01.460 --> 00:00:03.180
So where I left off
we said that I was

00:00:03.180 --> 00:00:04.690
going to throw a ball.

00:00:04.690 --> 00:00:10.700
And I threw it at an angle of
45 degrees to the ground.

00:00:10.700 --> 00:00:13.460
Let me draw the ground
in green as

00:00:13.460 --> 00:00:16.070
hopefully the ground is.

00:00:16.070 --> 00:00:19.620
That looks more like green
water, but I think you get

00:00:19.620 --> 00:00:20.500
what I'm saying.

00:00:20.500 --> 00:00:23.150
So I throw a ball at a 45 degree
angle and I throw it

00:00:23.150 --> 00:00:24.940
at-- what did I say?

00:00:24.940 --> 00:00:26.930
I think it was 10 meters
per second.

00:00:30.210 --> 00:00:33.710
I want to figure out how far
this ball will go or how high

00:00:33.710 --> 00:00:34.910
does-- not how high.

00:00:34.910 --> 00:00:36.220
Well, I could figure
that out as well.

00:00:36.220 --> 00:00:38.480
Or how long does the ball
stay in the air.

00:00:38.480 --> 00:00:41.820
And what we said is we can
break this vector.

00:00:41.820 --> 00:00:46.610
This vector has magnitude 10
in the 45 degree direction.

00:00:46.610 --> 00:00:51.030
We can break into its vertical
and horizontal components.

00:00:51.030 --> 00:00:54.600
So it's horizontal components,
this horizontal component

00:00:54.600 --> 00:00:56.330
would be this vector.

00:00:56.330 --> 00:00:59.900
And its vertical component
would be this vector.

00:00:59.900 --> 00:01:00.660
Whoops.

00:01:00.660 --> 00:01:03.440
The other way around.

00:01:03.440 --> 00:01:05.230
Vertical and horizontal are
always the two things that I

00:01:05.230 --> 00:01:06.340
found most confusing.

00:01:06.340 --> 00:01:09.330
Its vertical component would
be this vector and its

00:01:09.330 --> 00:01:11.420
horizontal component would
be this vector.

00:01:11.420 --> 00:01:13.410
Sorry for confusing it.

00:01:13.410 --> 00:01:15.820
And so we could treat it just
like the sides of a right

00:01:15.820 --> 00:01:19.650
triangle because as we know, the
vertical is perpendicular

00:01:19.650 --> 00:01:20.890
to the horizontal.

00:01:20.890 --> 00:01:23.550
And what I find very fascinating
by that is we're

00:01:23.550 --> 00:01:27.370
dealing with the velocities and
we can still treat them

00:01:27.370 --> 00:01:29.150
like they're the sides
of a right triangle.

00:01:29.150 --> 00:01:32.750
And that's why vectors in
general are so useful.

00:01:32.750 --> 00:01:34.060
And as we do more problems,
hopefully you

00:01:34.060 --> 00:01:35.700
get more of an intuition.

00:01:35.700 --> 00:01:39.090
And we said from the previous
problem that-- well, not the

00:01:39.090 --> 00:01:41.800
previous problem, the previous
presentation-- the length of

00:01:41.800 --> 00:01:44.700
this side, if you just use your
basic trigonometry would

00:01:44.700 --> 00:01:51.360
be-- the vertical component
of my velocity, so

00:01:51.360 --> 00:01:52.630
that's v sub v.

00:01:52.630 --> 00:01:56.370
So v sub vertical, which is an
unconventional way to write

00:01:56.370 --> 00:02:02.420
it-- would be 10 sine
of 45 degrees.

00:02:02.420 --> 00:02:03.310
And that equals what?

00:02:03.310 --> 00:02:04.600
What's sine of 45 degrees?

00:02:04.600 --> 00:02:05.730
That's square root
of 2 over 2.

00:02:05.730 --> 00:02:11.360
So it's 10 times square root
of 2 over 2, which equals 5

00:02:11.360 --> 00:02:14.840
square root of 2.

00:02:14.840 --> 00:02:18.920
And the horizontal component
of my velocity is

00:02:18.920 --> 00:02:22.900
10 cosine 45 degrees.

00:02:22.900 --> 00:02:26.520
To figure out this and this
we just used soh cah toa.

00:02:26.520 --> 00:02:29.670
We just said that the vertical
velocity over the hypotenuse

00:02:29.670 --> 00:02:31.690
is equal to sine
of 45 degrees.

00:02:31.690 --> 00:02:33.920
And then we just rearranged
it to get this.

00:02:33.920 --> 00:02:36.900
And we said that the horizontal
velocity over the

00:02:36.900 --> 00:02:40.970
hypotenuse is equal to the
cosine of 45 degrees and then

00:02:40.970 --> 00:02:42.310
we solved it to get this.

00:02:42.310 --> 00:02:45.250
We solved for the horizontal
velocity.

00:02:45.250 --> 00:02:48.620
And this also equals
5 square root of 2.

00:02:48.620 --> 00:02:50.720
So what can we do now?

00:02:50.720 --> 00:02:52.750
This is what I find to be the
really interesting part.

00:02:52.750 --> 00:02:56.520
If we wanted to figure
out how long does the

00:02:56.520 --> 00:02:57.820
ball stay in the air?

00:02:57.820 --> 00:03:00.160
The only thing that's keeping
the ball from staying in the

00:03:00.160 --> 00:03:02.400
air forever is gravity.

00:03:02.400 --> 00:03:07.160
Gravity is what's bringing the
ball back down to the ground.

00:03:07.160 --> 00:03:10.690
And gravity only works in
the vertical direction.

00:03:10.690 --> 00:03:13.430
Gravity only works-- I don't
like this color.

00:03:13.430 --> 00:03:15.700
Gravity only works in the
vertical direction, always

00:03:15.700 --> 00:03:21.340
pulling directionally down at 10
meters per second squared.

00:03:21.340 --> 00:03:23.930
So what we can do is we can
break this problem up into two

00:03:23.930 --> 00:03:24.870
problems.

00:03:24.870 --> 00:03:28.760
We can say well this ball is
going to stay up in the air

00:03:28.760 --> 00:03:36.060
just as long as a ball that was
thrown straight up with

00:03:36.060 --> 00:03:37.310
the same vertical velocity.

00:03:40.010 --> 00:03:43.940
This ball is going to stay up in
the air for the same amount

00:03:43.940 --> 00:03:48.065
of time as something thrown
straight up at 5 square root

00:03:48.065 --> 00:03:51.050
of 2 meters per second.

00:03:51.050 --> 00:03:52.140
And I want you to think
about that.

00:03:52.140 --> 00:03:54.263
Because that's actually-- at
least, when I first learned

00:03:54.263 --> 00:03:55.310
it, it wasn't intuitive.

00:03:55.310 --> 00:03:56.220
And the more you think
about it, it

00:03:56.220 --> 00:03:57.740
kind of becomes intuitive.

00:03:57.740 --> 00:04:02.590
And then you might want
to experiment with it.

00:04:02.590 --> 00:04:07.330
A ball that's thrown at an
angle will stay up-- well

00:04:07.330 --> 00:04:10.140
let's say a ball that's thrown
at an angle that gets as high

00:04:10.140 --> 00:04:13.220
as like-- let's say it gets
as high as 500 meters.

00:04:13.220 --> 00:04:14.530
Let's say I throw a ball
and this is the

00:04:14.530 --> 00:04:16.360
motion of the ball.

00:04:16.360 --> 00:04:17.829
So let's say this is the
path of the ball.

00:04:17.829 --> 00:04:20.899
The ball looks like this.

00:04:20.899 --> 00:04:23.980
And let's say the highest point
that the ball traveled,

00:04:23.980 --> 00:04:26.120
let's say this is 500 meters.

00:04:26.120 --> 00:04:28.850
That would be a very--
probably more like a

00:04:28.850 --> 00:04:30.770
cannonball or something.

00:04:30.770 --> 00:04:32.620
I can't throw a ball
500 meters high.

00:04:32.620 --> 00:04:34.950
So this is the ball that's
actually thrown at an angle.

00:04:34.950 --> 00:04:37.750
That travels some distance
along the ground.

00:04:37.750 --> 00:04:40.180
What I'm saying is that this
ball is going to stay in the

00:04:40.180 --> 00:04:43.740
ground-- is going to stay in the
air the exact same amount

00:04:43.740 --> 00:04:51.350
of time as a ball
that does this.

00:04:51.350 --> 00:04:55.990
And this is 500 meters.

00:04:55.990 --> 00:04:57.240
That's interesting.

00:05:02.080 --> 00:05:05.200
The difference between these two
balls-- and I'm probably,

00:05:05.200 --> 00:05:06.970
I should probably use
different colors.

00:05:06.970 --> 00:05:09.200
The difference between this
ball, which is thrown at a

00:05:09.200 --> 00:05:14.720
steeper angle, then this ball,
which is thrown at kind of a--

00:05:14.720 --> 00:05:17.920
it looks almost like a 45 degree
angle, is that this

00:05:17.920 --> 00:05:21.530
ball's horizontal velocity,
its velocity in this

00:05:21.530 --> 00:05:23.230
direction was lower.

00:05:23.230 --> 00:05:26.110
This ball had a much better
horizontal velocity.

00:05:26.110 --> 00:05:28.820
But the time that they stay in
the air-- and once again,

00:05:28.820 --> 00:05:31.020
we're assuming nothing about
air resistance or anything.

00:05:31.020 --> 00:05:35.370
The time they stay in the
air is exactly the same.

00:05:35.370 --> 00:05:36.120
That's really interesting.

00:05:36.120 --> 00:05:40.980
Another way to think about it
is, if I'm on a cliff and I

00:05:40.980 --> 00:05:44.920
just drop a ball and it just
goes straight down on a 500

00:05:44.920 --> 00:05:48.290
meter cliff, that ball is going
to take the same amount

00:05:48.290 --> 00:05:50.380
of time to hit the ground.

00:05:50.380 --> 00:05:52.230
It's going to take the same
amount of time to hit the

00:05:52.230 --> 00:05:56.710
ground as if another ball that
was moving-- let's say it was

00:05:56.710 --> 00:05:59.070
moving, it only had
horizontal motion.

00:05:59.070 --> 00:06:03.000
So the second ball
goes like this.

00:06:03.000 --> 00:06:04.360
So it moves along
the horizontal

00:06:04.360 --> 00:06:05.450
axis the whole times.

00:06:05.450 --> 00:06:06.630
So it's moving to the
right the whole

00:06:06.630 --> 00:06:08.470
time while it's falling.

00:06:08.470 --> 00:06:10.410
But both of these balls--
let me draw the

00:06:10.410 --> 00:06:12.700
second ball in magenta.

00:06:12.700 --> 00:06:15.680
This second magenta ball is
going to take the same amount

00:06:15.680 --> 00:06:17.100
of time to hit the ground.

00:06:17.100 --> 00:06:20.550
And that's because its vertical
velocity at any given

00:06:20.550 --> 00:06:23.570
time, its velocity in the
downward direction is going to

00:06:23.570 --> 00:06:25.120
be the exact same.

00:06:25.120 --> 00:06:27.830
And that's because gravity
is only pulling in

00:06:27.830 --> 00:06:29.040
the downward direction.

00:06:29.040 --> 00:06:31.280
I want you think about that a
lot because that's kind of a

00:06:31.280 --> 00:06:35.200
core intuition of vectors and
physics, is that you can break

00:06:35.200 --> 00:06:38.240
out the velocity into horizontal
and vertical

00:06:38.240 --> 00:06:38.990
components.

00:06:38.990 --> 00:06:41.770
And that once you do, kind of
the things that are affecting

00:06:41.770 --> 00:06:44.730
it in the vertical component
are identical regardless of

00:06:44.730 --> 00:06:46.520
what's happening in
the horizontal.

00:06:46.520 --> 00:06:47.740
So that might have
been helpful.

00:06:47.740 --> 00:06:49.380
It might have confused
you further.

00:06:49.380 --> 00:06:52.230
But let's just continue
on with the problem.

00:06:52.230 --> 00:06:55.700
So let me put this a little
bit of the side.

00:06:55.700 --> 00:06:57.590
I took too much space.

00:06:57.590 --> 00:07:00.950
So my vertical velocity is going
to be 5 square root of 2

00:07:00.950 --> 00:07:02.990
meters per second up.

00:07:02.990 --> 00:07:05.200
And we're going to stay with the
convention positive is up

00:07:05.200 --> 00:07:09.200
and negative is down, just like
in the coordinate axis.

00:07:09.200 --> 00:07:10.570
So how long is it going
to stay in the air?

00:07:10.570 --> 00:07:13.480
If I threw a ball straight up
at 5 square root of 2 meters

00:07:13.480 --> 00:07:16.230
per second, how long is it
going to stay in the air?

00:07:16.230 --> 00:07:18.630
Well, the way I think about it,
the easiest way to think

00:07:18.630 --> 00:07:22.990
about it is the same amount of
time it takes to go up is the

00:07:22.990 --> 00:07:25.560
same amount of time it's going
to take it to go down.

00:07:25.560 --> 00:07:28.310
We learned that in a couple
of videos before.

00:07:28.310 --> 00:07:30.600
So how long is it going
to take it to go up?

00:07:30.600 --> 00:07:33.220
Well to go up, it's going to
start at this speed, 5

00:07:33.220 --> 00:07:34.400
square root of 2.

00:07:34.400 --> 00:07:36.740
And then at the top, when it's
at the top, it's going to be

00:07:36.740 --> 00:07:40.710
going-- it's going
to have 0 speed.

00:07:40.710 --> 00:07:44.560
So it's going to decelerate from
5 square root of 2 to 0

00:07:44.560 --> 00:07:46.720
in some amount of time.

00:07:46.720 --> 00:07:50.740
And here we could just use a
very simple equation: change

00:07:50.740 --> 00:07:51.950
in velocity.

00:07:51.950 --> 00:07:55.960
Change is velocity is equal to
acceleration times time.

00:07:55.960 --> 00:07:57.320
Well what's the change
in velocity?

00:07:57.320 --> 00:08:03.530
It's the final velocity, which
is 0, minus my initial

00:08:03.530 --> 00:08:07.440
velocity, minus 5 square root
of 2 is equal to the

00:08:07.440 --> 00:08:08.250
acceleration.

00:08:08.250 --> 00:08:10.310
That's 10 meters per
second squared.

00:08:10.310 --> 00:08:12.520
And forgive me for not putting
the units in right now because

00:08:12.520 --> 00:08:13.830
I'm running out of space.

00:08:13.830 --> 00:08:16.060
Times time.

00:08:16.060 --> 00:08:19.840
So if we solve for t, we get--
the negatives cancel out.

00:08:19.840 --> 00:08:21.220
The 0 doesn't matter.

00:08:21.220 --> 00:08:27.520
So we get t is equal to 10 over
5 square root of 2, which

00:08:27.520 --> 00:08:32.650
equals 2 over the square
root of 2 seconds.

00:08:32.650 --> 00:08:34.860
t equals 2 over the square
root of 2 seconds.

00:08:34.860 --> 00:08:36.970
And of course, if we wanted to
rationalize this, we could

00:08:36.970 --> 00:08:38.929
multiply the top and
the bottom by the

00:08:38.929 --> 00:08:42.169
square root of 2.

00:08:42.169 --> 00:08:43.770
Well actually, you don't
have to do that.

00:08:43.770 --> 00:08:45.590
What's 2 divided by the
square root of 2?

00:08:45.590 --> 00:08:48.750
Well that's just equal to the
square root of 2, right?

00:08:48.750 --> 00:08:50.270
Because the square root of 2
times the square root of 2 is

00:08:50.270 --> 00:08:51.290
of course, 2.

00:08:51.290 --> 00:08:53.570
So it takes a square
root of 2 seconds.

00:08:53.570 --> 00:08:56.020
But is that the total time that
the ball's in the air?

00:08:56.020 --> 00:08:58.390
No, that's the amount of time it
takes for the ball to go to

00:08:58.390 --> 00:08:59.780
its highest point.

00:08:59.780 --> 00:09:01.530
It takes square root
of 2 seconds to go

00:09:01.530 --> 00:09:03.010
to its highest point.

00:09:03.010 --> 00:09:05.080
But then it's going to take the
exact amount of time to

00:09:05.080 --> 00:09:06.500
come back down.

00:09:06.500 --> 00:09:11.790
So the total time in the air
is going to be-- and I know

00:09:11.790 --> 00:09:13.960
I'm doing this very messy.

00:09:13.960 --> 00:09:16.760
The total time in the air
is 2 square root of 2.

00:09:16.760 --> 00:09:18.700
Just two times this.

00:09:18.700 --> 00:09:20.990
Because it takes the same amount
of time to go up as it

00:09:20.990 --> 00:09:22.150
does to come down.

00:09:22.150 --> 00:09:24.580
And it might be confusing, but
all I did is I figured out the

00:09:24.580 --> 00:09:26.960
vertical component of the
velocity using some basic

00:09:26.960 --> 00:09:28.070
trigonometry.

00:09:28.070 --> 00:09:32.030
And I said well, it's going to
decelerate with gravity and

00:09:32.030 --> 00:09:35.330
over that time, when it's going
up, however long it

00:09:35.330 --> 00:09:37.640
takes is going to be going from
that initial-- that 5

00:09:37.640 --> 00:09:39.820
square root of 2 meters
per second to 0.

00:09:39.820 --> 00:09:43.100
So how long does it take to
decelerate from 5 square root

00:09:43.100 --> 00:09:46.880
of 2 to 0 given that I'm
decelerating with gravity?

00:09:46.880 --> 00:09:50.165
Given that I'm decelerating at
10 meters per second squared.

00:09:50.165 --> 00:09:54.020
And it takes us 2 square
root of 2 seconds.

00:09:54.020 --> 00:09:56.480
I will continue this
in the next video.