Normal force and contact force

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.610
Let's say that I have a
huge, maybe frozen over lake,

00:00:04.610 --> 00:00:06.080
or maybe it's a big pond.

00:00:06.080 --> 00:00:09.750
So I have a huge surface of
ice over here-- my best attempt

00:00:09.750 --> 00:00:11.830
to draw a flat surface
of ice-- and I'm

00:00:11.830 --> 00:00:14.710
going to put two
blocks of ice here.

00:00:14.710 --> 00:00:17.650
So I'm going to put
one block of ice

00:00:17.650 --> 00:00:21.814
just like this, one block
of ice right over here.

00:00:21.814 --> 00:00:23.980
And then I'm going to put
another block of ice right

00:00:23.980 --> 00:00:26.220
over here.

00:00:26.220 --> 00:00:30.490
And then another block
of ice right over here.

00:00:30.490 --> 00:00:33.510
And these blocks of
ice are identical.

00:00:33.510 --> 00:00:35.230
They're both 5 kilograms.

00:00:35.230 --> 00:00:38.280
They are both 5 kilograms--
let me write this down.

00:00:38.280 --> 00:00:42.490
So they are both 5 kilograms.

00:00:42.490 --> 00:00:46.310
Or both of their masses, I
should say, are 5 kilograms.

00:00:46.310 --> 00:00:48.700
And the only difference
between the two

00:00:48.700 --> 00:00:51.620
is that relative to
the pond, this one

00:00:51.620 --> 00:00:56.120
is stationary-- this
one is stationary--

00:00:56.120 --> 00:00:59.310
and this one is moving
with a constant velocity--

00:00:59.310 --> 00:01:02.745
constant velocity.

00:01:02.745 --> 00:01:06.460
Constant velocity in the
right-wards direction.

00:01:06.460 --> 00:01:08.950
And let's say that
its constant velocity

00:01:08.950 --> 00:01:13.890
is at 5 meters per second--
5 meters per second.

00:01:13.890 --> 00:01:17.164
And the whole reason why I made
blocks of ice on top of ice

00:01:17.164 --> 00:01:19.330
is that we're going to
assume, at least for the sake

00:01:19.330 --> 00:01:23.540
of this video, that
friction is negligible.

00:01:23.540 --> 00:01:26.360
Now what does Newton's
First Law of Motion

00:01:26.360 --> 00:01:29.590
tell us about something that
is either not in motion--

00:01:29.590 --> 00:01:32.440
or you could view this as
a constant velocity of 0--

00:01:32.440 --> 00:01:35.314
or something that has
a constant velocity?

00:01:35.314 --> 00:01:36.730
Well Newton's First
Law says, well

00:01:36.730 --> 00:01:41.180
look, they're going to keep
their constant velocity

00:01:41.180 --> 00:01:43.930
or stay stationary, which is
the constant velocity of 0,

00:01:43.930 --> 00:01:47.110
unless there is some
unbalance, unless there

00:01:47.110 --> 00:01:49.774
is some net force
acting on an object.

00:01:49.774 --> 00:01:51.190
So let's just think
about it here.

00:01:51.190 --> 00:01:52.850
In either of these
situations, there

00:01:52.850 --> 00:01:56.140
must not be any unbalanced
force acting on them.

00:01:56.140 --> 00:01:58.532
Or their must not
be any net force.

00:01:58.532 --> 00:01:59.990
But if you think
about it, if we're

00:01:59.990 --> 00:02:02.280
assuming that these
things are on Earth,

00:02:02.280 --> 00:02:05.830
there is a net force
acting on both of them.

00:02:05.830 --> 00:02:08.780
Both of them are at the
surface of the Earth,

00:02:08.780 --> 00:02:10.949
and they both have
mass, so there

00:02:10.949 --> 00:02:14.460
will be the force of
gravity acting downwards

00:02:14.460 --> 00:02:15.900
on both of them.

00:02:15.900 --> 00:02:20.140
There is going to be the
downward force of gravity

00:02:20.140 --> 00:02:22.630
on both of these blocks of ice.

00:02:22.630 --> 00:02:25.810
And that downward force of
gravity, the force of gravity,

00:02:25.810 --> 00:02:29.540
is going to be equal to
the gravitational field

00:02:29.540 --> 00:02:32.910
near the surface of the
Earth, times-- which

00:02:32.910 --> 00:02:37.220
is a vector-- times
the mass of the object.

00:02:37.220 --> 00:02:41.050
So times 5 kilograms.

00:02:41.050 --> 00:02:47.040
This right over here is 9.8
meters per second squared.

00:02:47.040 --> 00:02:49.290
So you multiply that times 5.

00:02:49.290 --> 00:02:53.000
You get 49 kilogram meter
per second squared, which

00:02:53.000 --> 00:02:56.080
is the same thing as 49 newtons.

00:02:56.080 --> 00:02:58.670
So this is a little bit
of a conundrum here.

00:02:58.670 --> 00:03:02.490
Newton's First Law
says, an object at rest

00:03:02.490 --> 00:03:04.250
will stay at rest, or
an object in motion

00:03:04.250 --> 00:03:06.610
will stay in motion, unless
there is some unbalanced,

00:03:06.610 --> 00:03:08.580
or unless there
is some net force.

00:03:08.580 --> 00:03:10.340
But based on what
we've drawn right here,

00:03:10.340 --> 00:03:13.320
it looks like there's
some type of a net force.

00:03:13.320 --> 00:03:16.780
It looks like I have 49 newtons
of force pulling this thing

00:03:16.780 --> 00:03:17.740
downwards.

00:03:17.740 --> 00:03:20.900
But you say, no, no no, Sal.

00:03:20.900 --> 00:03:23.860
Obviously this thing won't
start accelerating downwards

00:03:23.860 --> 00:03:25.200
because there's ice here.

00:03:25.200 --> 00:03:32.640
Its resting on a big
pool of frozen water.

00:03:32.640 --> 00:03:35.930
And so my answer to you is,
well, if that's your answer,

00:03:35.930 --> 00:03:42.350
then what is the resulting
force that cancels out

00:03:42.350 --> 00:03:44.990
with gravity to keep
these blocks of ice,

00:03:44.990 --> 00:03:47.600
either one of them,
from plummeting down

00:03:47.600 --> 00:03:51.320
to the core of the Earth?

00:03:51.320 --> 00:03:53.610
From essentially
going into free fall,

00:03:53.610 --> 00:03:56.240
or accelerating towards
the center of the Earth?

00:03:56.240 --> 00:04:01.210
And you say, well, I guess if
these things would be falling,

00:04:01.210 --> 00:04:04.400
if not for the ice,
the ice must be

00:04:04.400 --> 00:04:07.420
providing the
counteracting force.

00:04:07.420 --> 00:04:09.630
And you are absolutely correct.

00:04:09.630 --> 00:04:13.390
The ice is providing
the counteracting force

00:04:13.390 --> 00:04:15.750
in the opposite direction.

00:04:15.750 --> 00:04:17.839
So the exact magnitude
of force, and it

00:04:17.839 --> 00:04:20.220
is in the opposite direction.

00:04:20.220 --> 00:04:23.730
And so if the force of gravity
on each of these blocks of ice

00:04:23.730 --> 00:04:27.690
are 49 newtons downwards
it is completely

00:04:27.690 --> 00:04:32.150
netted off by the force of
the ice on the block upwards.

00:04:32.150 --> 00:04:37.560
And that will be a force 49
newtons upwards in either case.

00:04:37.560 --> 00:04:39.320
And now, hopefully,
it makes sense

00:04:39.320 --> 00:04:41.650
that Newton's First
Law still holds.

00:04:41.650 --> 00:04:45.100
We have no net force on this
in the vertical direction,

00:04:45.100 --> 00:04:47.810
actually no net force on
this in either direction.

00:04:47.810 --> 00:04:51.740
That's why this guy
has a 0 velocity

00:04:51.740 --> 00:04:53.140
in the horizontal direction.

00:04:53.140 --> 00:04:56.010
This guy has a constant velocity
in the horizontal direction.

00:04:56.010 --> 00:04:57.570
And neither of them
are accelerating

00:04:57.570 --> 00:04:58.970
in the vertical direction.

00:04:58.970 --> 00:05:01.520
Because you have the force
of the ice on the block,

00:05:01.520 --> 00:05:03.470
the ice is supporting
the block, that's

00:05:03.470 --> 00:05:05.920
completely
counteracting gravity.

00:05:05.920 --> 00:05:09.740
And this force, in this example,
is called the normal force.

00:05:09.740 --> 00:05:14.570
This is the normal force--
it's 49 newtons upwards.

00:05:14.570 --> 00:05:17.040
This right here is
the normal force.

00:05:17.040 --> 00:05:20.150
And we'll talk more about the
normal force in future videos.

00:05:20.150 --> 00:05:22.610
The normal force
is the force, when

00:05:22.610 --> 00:05:24.380
anything is resting
on any surface that's

00:05:24.380 --> 00:05:25.630
perpendicular to that surface.

00:05:25.630 --> 00:05:27.463
And it's going to start
to matter a lot when

00:05:27.463 --> 00:05:29.692
we start thinking about
friction and all the rest.

00:05:29.692 --> 00:05:32.150
So what we'll see in future
videos, when you have something

00:05:32.150 --> 00:05:35.220
on an incline, and let's say
I have a block on an incline

00:05:35.220 --> 00:05:36.120
like this.

00:05:36.120 --> 00:05:38.440
The normal force
from the, I guess

00:05:38.440 --> 00:05:40.080
you could say, this
wedge on the block,

00:05:40.080 --> 00:05:44.364
is going to be perpendicular
to the surface.

00:05:44.364 --> 00:05:46.530
And if you really think
about what's happening here,

00:05:46.530 --> 00:05:48.760
it's fundamentally an
electromagnetic force.

00:05:48.760 --> 00:05:56.080
Because if you really zoomed
in on the molecules of the ice

00:05:56.080 --> 00:05:58.590
right over here, even better
the atoms of the ice here.

00:05:58.590 --> 00:06:01.840
And you really zoomed in on
the atoms or the molecules

00:06:01.840 --> 00:06:06.320
of the ice up here, what's
keeping this top block of ice

00:06:06.320 --> 00:06:09.030
from falling down
is that in order

00:06:09.030 --> 00:06:13.480
for it to go through its
molecules would have to kind

00:06:13.480 --> 00:06:16.070
of compress against, or I guess
it would have to get closer

00:06:16.070 --> 00:06:19.420
to, the water molecules
or the individual atoms

00:06:19.420 --> 00:06:21.530
in this ice down here.

00:06:21.530 --> 00:06:23.720
And the atoms, let me
draw it on an atomic level

00:06:23.720 --> 00:06:24.890
right over here.

00:06:24.890 --> 00:06:33.300
So maybe, let me draw one
of this guy's molecules.

00:06:33.300 --> 00:06:40.044
So you have an oxygen
with 2 hydrogens

00:06:40.044 --> 00:06:41.710
and it forms this big
lattice structure.

00:06:41.710 --> 00:06:45.980
And we can talk about more of
that in the chemistry playlist.

00:06:45.980 --> 00:06:48.880
And let's talk about this ice
as one of these molecules.

00:06:48.880 --> 00:06:53.500
So maybe it looks
something like this.

00:06:53.500 --> 00:06:55.810
And it has its 2 hydrogens

00:06:55.810 --> 00:06:58.710
And so what's keeping these guys
from getting compressed, what's

00:06:58.710 --> 00:07:02.200
keeping this block of ice
from going down further,

00:07:02.200 --> 00:07:05.330
is the repulsion between the
electrons in this molecule

00:07:05.330 --> 00:07:06.950
and the electrons
in that molecule.

00:07:06.950 --> 00:07:10.570
So on a macro level we view
this is kind of a contact force.

00:07:10.570 --> 00:07:13.110
But on a microscopic
level, on an atomic level,

00:07:13.110 --> 00:07:17.870
it's really just electromagnetic
repulsion at work.


Unit vectors and engineering notation

WEBVTT
Kind: captions
Language: en

00:00:00.409 --> 00:00:02.200
What I want to do in
this video is show you

00:00:02.200 --> 00:00:06.380
a way to represent a
vector by its components.

00:00:06.380 --> 00:00:08.900
And this is sometimes
called engineering notation

00:00:08.900 --> 00:00:09.790
for vectors.

00:00:09.790 --> 00:00:11.670
But it's super
useful, because it

00:00:11.670 --> 00:00:14.339
allows us to keep track of
the components of the vector

00:00:14.339 --> 00:00:15.880
and it makes it a
little bit tangible

00:00:15.880 --> 00:00:18.380
when we talk about the
individual components.

00:00:18.380 --> 00:00:20.460
So let's break down this
vector right over here.

00:00:20.460 --> 00:00:22.168
I'm just assuming it's
a velocity vector.

00:00:22.168 --> 00:00:24.780
Vector v. Its magnitude
is 10 meters per second.

00:00:24.780 --> 00:00:27.150
And it's pointed in a
direction 30 degrees

00:00:27.150 --> 00:00:30.270
above the horizontal.

00:00:30.270 --> 00:00:32.900
So we've broken down these
vectors in the past before.

00:00:32.900 --> 00:00:36.430
The vertical component
right over here.

00:00:36.430 --> 00:00:39.820
Its magnitude would
be-- so the magnitude

00:00:39.820 --> 00:00:42.380
of the vertical component,
right over here,

00:00:42.380 --> 00:00:45.870
is going to be 10
sine of 30 degrees.

00:00:45.870 --> 00:00:49.730
It's going to be 10 meters per
second times the sine of 30

00:00:49.730 --> 00:00:50.230
degrees.

00:00:52.750 --> 00:00:55.780
This comes from the basic
trigonometry from sohcahtoa.

00:00:55.780 --> 00:00:58.720
And I covered that in more
detail in previous videos.

00:00:58.720 --> 00:01:00.960
Sine of 30 degrees is 1/2.

00:01:00.960 --> 00:01:04.569
So this is going to be 5,
or 5 meters per second.

00:01:04.569 --> 00:01:08.480
10 times 1/2 is 5
meters per second.

00:01:08.480 --> 00:01:11.340
So that's the magnitude
of its vertical component.

00:01:11.340 --> 00:01:13.760
And in the last few
videos, I kind of,

00:01:13.760 --> 00:01:17.130
in a less tangible way of
specifying the vertical vector,

00:01:17.130 --> 00:01:21.122
I often use this notation, which
isn't as tangible as I like.

00:01:21.122 --> 00:01:22.580
And that's why I'm
going to make it

00:01:22.580 --> 00:01:23.996
a little bit better
in this video.

00:01:23.996 --> 00:01:29.500
I said that that vector
itself is 5 meters per second.

00:01:29.500 --> 00:01:32.270
But what I told you is that
the direction is implicitly

00:01:32.270 --> 00:01:35.720
given because this
is a vertical vector.

00:01:35.720 --> 00:01:38.680
And I told you in previous
videos that if it's positive,

00:01:38.680 --> 00:01:41.892
it means up and if it's
negative, it means down.

00:01:41.892 --> 00:01:44.350
So I kind of have to give you
this context here so that you

00:01:44.350 --> 00:01:47.280
can appreciate that this is a
vector that just the sine of it

00:01:47.280 --> 00:01:48.710
is giving you its direction.

00:01:48.710 --> 00:01:51.180
But I have to keep telling
you this is a vertical vector.

00:01:51.180 --> 00:01:53.486
So it wasn't that tangible.

00:01:53.486 --> 00:01:54.860
And so we had the
same issue when

00:01:54.860 --> 00:01:59.200
we talked about the
horizontal vectors.

00:01:59.200 --> 00:02:03.270
So this horizontal
vector right over here,

00:02:03.270 --> 00:02:06.530
the magnitude of this
horizontal vector

00:02:06.530 --> 00:02:09.609
is going to be 10
cosine of 30 degrees.

00:02:09.609 --> 00:02:11.150
And once again, it
comes straight out

00:02:11.150 --> 00:02:12.620
of basic trigonometry.

00:02:12.620 --> 00:02:15.990
10 cosine of 30 degrees.

00:02:15.990 --> 00:02:21.370
And so cosine of 30 degrees is
the square root of 3 over 2.

00:02:21.370 --> 00:02:24.270
Multiply it by 10,
you get 5 square roots

00:02:24.270 --> 00:02:28.860
of 3 meters per second.

00:02:28.860 --> 00:02:31.964
And once again, in
previous videos,

00:02:31.964 --> 00:02:34.130
I used this notation
sometimes, where I was actually

00:02:34.130 --> 00:02:36.580
saying that the vector
is 5 square roots of 3

00:02:36.580 --> 00:02:37.930
meters per second.

00:02:37.930 --> 00:02:40.961
But in order to ensure that
this wasn't just a magnitude,

00:02:40.961 --> 00:02:43.210
I kept having to tell you
in the horizontal direction.

00:02:43.210 --> 00:02:45.410
If it's positive it's
going to the right,

00:02:45.410 --> 00:02:47.980
and if it's negative,
it's going to the left.

00:02:47.980 --> 00:02:50.050
What I want to do
in this video, is

00:02:50.050 --> 00:02:51.700
give us a convention
so that I don't

00:02:51.700 --> 00:02:53.610
have to keep doing
this for the direction.

00:02:53.610 --> 00:02:56.600
And it makes it all a
little bit more tangible.

00:02:56.600 --> 00:03:00.180
And so what we do
is we introduced

00:03:00.180 --> 00:03:01.890
the idea of unit vectors.

00:03:04.750 --> 00:03:08.980
So by definition, we'll
introduce the vector i i.

00:03:08.980 --> 00:03:10.400
Sometimes it's called i hat.

00:03:10.400 --> 00:03:13.160
And I'll draw it like here.

00:03:13.160 --> 00:03:15.350
I'll make it a
little bit smaller.

00:03:15.350 --> 00:03:17.530
So the vector i hat.

00:03:17.530 --> 00:03:20.490
So that right there is a
picture of the vector i hat.

00:03:20.490 --> 00:03:22.570
And we've put a little
hat on top of the i

00:03:22.570 --> 00:03:24.630
to show that it
is a unit vector.

00:03:24.630 --> 00:03:26.960
And what a unit
vector is-- so i hat

00:03:26.960 --> 00:03:29.330
goes in the positive
x direction.

00:03:29.330 --> 00:03:31.310
That's just how it's defined.

00:03:31.310 --> 00:03:36.130
And the unit vector tells
us that its magnitude is 1.

00:03:36.130 --> 00:03:41.510
So the magnitude of the
vector i hat is equal to 1.

00:03:41.510 --> 00:03:44.790
And its direction is in
the positive x direction.

00:03:44.790 --> 00:03:49.760
So if we really
wanted to specify

00:03:49.760 --> 00:03:52.980
this kind of x component
vector in a better way,

00:03:52.980 --> 00:03:58.640
we really should call it
5 square roots of 3 times

00:03:58.640 --> 00:03:59.410
this unit vector.

00:03:59.410 --> 00:04:02.260
Because this green
vector over here

00:04:02.260 --> 00:04:04.242
is going to be 5
square roots of 3 times

00:04:04.242 --> 00:04:06.450
this vector right over here,
because this vector just

00:04:06.450 --> 00:04:07.850
has length 1.

00:04:07.850 --> 00:04:11.587
So it's 5 square roots of
3 times the unit vector.

00:04:11.587 --> 00:04:13.170
And what I like about
this is that now

00:04:13.170 --> 00:04:14.628
I don't have to
tell you, remember,

00:04:14.628 --> 00:04:16.540
this is a horizontal vector.

00:04:16.540 --> 00:04:18.790
Positive is to the right,
negative is to the left.

00:04:18.790 --> 00:04:19.690
It's implicit here.

00:04:19.690 --> 00:04:23.021
Because clearly if this
is a positive value,

00:04:23.021 --> 00:04:24.770
it's going to be a
positive multiple of i.

00:04:24.770 --> 00:04:26.060
It's going to go to the right.

00:04:26.060 --> 00:04:28.470
If it's a negative value,
it flips around the vector

00:04:28.470 --> 00:04:29.940
and then it goes to the left.

00:04:29.940 --> 00:04:36.170
So this is actually a better way
of specifying the x component

00:04:36.170 --> 00:04:36.750
vector.

00:04:36.750 --> 00:04:39.950
Or if I broke it down, this
vector v into its x component,

00:04:39.950 --> 00:04:42.070
this is a better way of
specifying that vector.

00:04:42.070 --> 00:04:44.200
Same thing for the y direction.

00:04:44.200 --> 00:04:46.085
We can define a unit vector.

00:04:46.085 --> 00:04:49.460
And let me pick a color
that I have not used yet.

00:04:49.460 --> 00:04:52.190
Let me find this
pink I haven't used.

00:04:52.190 --> 00:04:55.480
We can find a unit vector
that goes straight up in the y

00:04:55.480 --> 00:04:58.300
direction called unit vector j.

00:04:58.300 --> 00:05:02.910
And once again, the magnitude
of unit vector j is equal to 1.

00:05:02.910 --> 00:05:05.870
This little hat on
top of it tells us--

00:05:05.870 --> 00:05:09.430
or sometimes it's called a
caret character-- that tells us

00:05:09.430 --> 00:05:11.760
that it is a vector,
but it is a unit vector.

00:05:11.760 --> 00:05:13.670
It has a magnitude of 1.

00:05:13.670 --> 00:05:19.130
And by definition, the vector
j goes and has a magnitude of 1

00:05:19.130 --> 00:05:21.430
in the positive y direction.

00:05:21.430 --> 00:05:23.610
So the y component
of this vector,

00:05:23.610 --> 00:05:25.360
instead of saying it's
5 meters per second

00:05:25.360 --> 00:05:27.651
in the upwards direction or
instead of saying that it's

00:05:27.651 --> 00:05:30.420
implicitly upwards because
it's a vertical vector

00:05:30.420 --> 00:05:32.760
or it's a vertical
component and it's positive,

00:05:32.760 --> 00:05:35.920
we can now be a little bit
more specific about it.

00:05:35.920 --> 00:05:40.815
We can say that it is
equal to 5 times j.

00:05:44.090 --> 00:05:46.180
Because you see, this
magenta vector, it's

00:05:46.180 --> 00:05:52.109
going the exact same direction
as j, it's just 5 times longer.

00:05:52.109 --> 00:05:53.650
I don't know if it's
exactly 5 times.

00:05:53.650 --> 00:05:55.358
I'm just trying to
estimate it right now.

00:05:55.358 --> 00:05:56.640
It's just 5 times longer.

00:05:56.640 --> 00:05:58.370
Now what's really
cool about this

00:05:58.370 --> 00:06:02.270
is besides just being able to
express the components as now

00:06:02.270 --> 00:06:05.190
multiples of explicit
vectors, instead of just being

00:06:05.190 --> 00:06:07.360
able to do that--
which we did do,

00:06:07.360 --> 00:06:08.870
we're representing
the components

00:06:08.870 --> 00:06:11.040
as explicit vectors--
we also know

00:06:11.040 --> 00:06:14.470
that the vector v is the
sum of its components.

00:06:14.470 --> 00:06:17.990
If you start with this
green vector right here

00:06:17.990 --> 00:06:21.010
and you add this vertical
component right over here,

00:06:21.010 --> 00:06:23.260
you have head to tails.

00:06:23.260 --> 00:06:25.200
You get the blue vector.

00:06:25.200 --> 00:06:27.280
And so we can actually
use the components

00:06:27.280 --> 00:06:28.900
to represent the vector itself.

00:06:28.900 --> 00:06:32.140
We don't always have
to draw it like this.

00:06:32.140 --> 00:06:37.200
So we can write that
vector v is equal to-- let

00:06:37.200 --> 00:06:39.510
me write it this way-- it's
equal to its x component

00:06:39.510 --> 00:06:42.640
vector plus the y
component vector.

00:06:45.340 --> 00:06:47.170
And we can write
that, the x component

00:06:47.170 --> 00:06:52.530
vector is 5 square
roots of 3 times i.

00:06:55.600 --> 00:06:59.400
And then it's going to
be plus the y component,

00:06:59.400 --> 00:07:07.850
the vertical component, which
is 5j, which is 5 times j.

00:07:07.850 --> 00:07:10.980
And so what's really neat here
is now you could specify any

00:07:10.980 --> 00:07:14.020
vector in two dimensions
by some combination

00:07:14.020 --> 00:07:18.490
of i's and j's or some scaled
up combinations of i's and j's.

00:07:18.490 --> 00:07:21.946
And if you want to go
into three dimensions,

00:07:21.946 --> 00:07:24.070
and you often will, especially
as the physics class

00:07:24.070 --> 00:07:26.010
moves on through
the year, you can

00:07:26.010 --> 00:07:30.002
introduce a vector in
the positive z direction,

00:07:30.002 --> 00:07:31.460
depending on how
you want to do it.

00:07:31.460 --> 00:07:33.420
Although z is
normally up and down.

00:07:33.420 --> 00:07:35.110
But whatever the
next dimension is,

00:07:35.110 --> 00:07:39.471
you can define a vector k that
goes into that third dimension.

00:07:39.471 --> 00:07:41.470
Here I'll do it in a kind
of unconventional way.

00:07:41.470 --> 00:07:43.087
I'll make k go in
that direction.

00:07:43.087 --> 00:07:45.045
Although the standard
convention when you do it

00:07:45.045 --> 00:07:48.390
in three dimensions is that k
is the up and down dimension.

00:07:48.390 --> 00:07:50.490
But this by itself is
already pretty neat

00:07:50.490 --> 00:07:54.710
because we can now represent any
vector through its components

00:07:54.710 --> 00:07:58.380
and it's also going to
make the math much easier.


Ideal Math and Science Class Time

WEBVTT
Kind: captions
Language: en

00:00:00.652 --> 00:00:05.113
(children talking)

00:00:05.113 --> 00:00:07.654
Salman: We do a lot of
virtual things on Khan Academy

00:00:07.654 --> 00:00:10.111
and We Teach Science and
the goal of the summer camp

00:00:10.111 --> 00:00:13.862
really is to give the students
a real physical experience

00:00:13.862 --> 00:00:16.529
that really connects a lot of
the concepts that you might learn

00:00:16.529 --> 00:00:18.653
on something like a Khan Academy video.

00:00:18.653 --> 00:00:23.070
So we do things like real
engineering, we build Lego robots

00:00:23.070 --> 00:00:25.445
that the students actually
write software for

00:00:25.445 --> 00:00:29.486
that autonomously sumo wrestle,
we do market simulations,

00:00:29.486 --> 00:00:34.318
we perform experiments but this all
involves indoor and outdoor activities.

00:00:34.318 --> 00:00:36.228
We just do a bunch of
stuff that really connects

00:00:36.228 --> 00:00:39.011
a lot of the abstract conceptual
things that you might learn

00:00:39.011 --> 00:00:41.923
on say a Khan Academy video and then
you bring it into the classroom.

00:00:41.923 --> 00:00:44.589
So in the market simulations,
students learn about probability,

00:00:44.589 --> 00:00:47.506
they learn about gain theory,
they learn about trading.

00:00:47.506 --> 00:00:50.398
In the robotics they learn about
engineering, they learn about math,

00:00:50.398 --> 00:00:53.173
they do a lot of trigonometry
and geometry so we really try

00:00:53.173 --> 00:00:56.090
to make it all very
tangible for the students.

00:00:56.090 --> 00:00:59.253
Aragon: So this is mental boot camp 2010.

00:00:59.253 --> 00:01:03.151
It's a one week course held in
Portola Valley in the Bay area

00:01:03.151 --> 00:01:08.151
for about 20 to 25 students ranging
from 6th grade to roughly 9th grade.

00:01:08.151 --> 00:01:12.442
Spend a week exploring math and science
and technology in the real world.

00:01:12.442 --> 00:01:14.942
What you're seeing today
is a big part of our course

00:01:14.942 --> 00:01:19.363
which is the robotics portion and
we're using the Lego Mindstorm NXT's

00:01:19.363 --> 00:01:22.529
and we've given the kids a
challenge for the week to make

00:01:22.529 --> 00:01:27.897
a sumo wrestling robot and they're
designed to fight on these wooden tables

00:01:27.897 --> 00:01:30.980
that you see, one-on-one, and the idea is

00:01:30.980 --> 00:01:33.695
to push your fellow robot off the edge.

00:01:33.695 --> 00:01:36.195
So they're using a variety
of tools within the robots,

00:01:36.195 --> 00:01:39.146
there are sensors to detect
when you've fallen off the edge,

00:01:39.146 --> 00:01:43.267
there are other things like the
motors that drive them around

00:01:43.267 --> 00:01:47.089
and then they've got actuators to hammer
on each other and push each other off.

00:01:47.089 --> 00:01:48.649
So that's the main thrust of the week.

00:01:48.649 --> 00:01:51.898
We're spending half of
the time doing robotics

00:01:51.898 --> 00:01:54.273
and then the other half
we're doing board games,

00:01:54.273 --> 00:01:56.190
we're doing the math behind board games,

00:01:56.190 --> 00:01:59.689
probability and statistics, game theories.

00:01:59.689 --> 00:02:03.439
The kids are also learning
about markets and for example,

00:02:03.439 --> 00:02:05.484
trading markets and how they work.

00:02:05.484 --> 00:02:06.658
Rahul: Hi, I'm Rahul.

00:02:06.658 --> 00:02:07.524
Lance: And I'm Lance.

00:02:07.524 --> 00:02:16.179
Rahul: And this week we're building
a Lego NXT robot for our project.

00:02:16.179 --> 00:02:20.652
Lance: The goal is to push the
opponent's robot off the table

00:02:20.652 --> 00:02:22.862
before your robot falls off.

00:02:22.862 --> 00:02:28.277
Rahul: We can do that by
making ultrasonic sensor-

00:02:28.277 --> 00:02:30.363
Lance: It goes and tries to look for it.

00:02:30.363 --> 00:02:32.318
Rahul: Yeah and there's a
light sensor to make sure

00:02:32.318 --> 00:02:37.693
it doesn't fall off the table and also
here we have a motion track sensor

00:02:37.693 --> 00:02:40.777
so that when it backs up it
doesn't fall off the table

00:02:40.777 --> 00:02:45.693
because the ultrasonic sensor's facing
that way and so is the light sensor,

00:02:45.693 --> 00:02:48.245
that's why we stuck a
track sensor back here.

00:02:48.245 --> 00:02:53.526
We also have some wheels
and these are our motors.

00:02:53.526 --> 00:02:56.994
Lance: This isn't working
very well but it's a whacker.

00:02:56.994 --> 00:03:03.411
It just kind of whacks the opponent's
robots to try and disassemble it

00:03:03.411 --> 00:03:07.359
or push it or knock it
off balance or something.

00:03:07.359 --> 00:03:10.328
Rahul: And here we can try to motion.

00:03:10.328 --> 00:03:13.160
Lance: I'm just going to run the program.

00:03:13.160 --> 00:03:17.411
(robot sounds)

00:03:17.411 --> 00:03:34.109
(children laughing)

00:03:34.109 --> 00:03:35.917
Voiceover: Your robot still won.


Correction to total final velocity for projectile

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:02.240
In the last video,
I realized that I

00:00:02.240 --> 00:00:05.866
made a mistake near the end,
but it's not a good one,

00:00:05.866 --> 00:00:07.490
because I copied down
the wrong number.

00:00:07.490 --> 00:00:09.323
And then that led to
the rest of the problem

00:00:09.323 --> 00:00:10.660
not having the right values.

00:00:10.660 --> 00:00:12.910
We figured out that
the vertical component

00:00:12.910 --> 00:00:14.970
of our velocity
earlier in the video

00:00:14.970 --> 00:00:20.360
is negative 26.03
meters per second.

00:00:20.360 --> 00:00:22.775
But then something
happened in my brain,

00:00:22.775 --> 00:00:25.450
and it became negative 29.03.

00:00:25.450 --> 00:00:27.930
So this is not negative 29.03.

00:00:27.930 --> 00:00:32.479
This is negative 26.03
meters per second.

00:00:32.479 --> 00:00:38.502
So the total velocity, we should
have 26.03 squared over here.

00:00:38.502 --> 00:00:40.210
And then if we evaluate
that-- and I just

00:00:40.210 --> 00:00:44.249
did it on my calculator--
26.03 squared plus 5.61.

00:00:44.249 --> 00:00:45.540
Actually, I did it wrong again.

00:00:45.540 --> 00:00:46.360
So it's 5.21.

00:00:46.360 --> 00:00:48.340
Let me redo it.

00:00:48.340 --> 00:00:55.670
So 26.03 squared plus 5.21
squared under the radical

00:00:55.670 --> 00:00:58.240
gives us 26.55.

00:00:58.240 --> 00:01:04.769
So this right over here should
be 26.55 meters per second.

00:01:04.769 --> 00:01:11.950
So that is the magnitude
of our final velocity.

00:01:11.950 --> 00:01:15.580
So that is the magnitude of
our total final velocity.

00:01:15.580 --> 00:01:20.980
And the angle, since this
is not 29.03, this is 26.03,

00:01:20.980 --> 00:01:30.110
we can say that the tangent
of this angle theta-- this

00:01:30.110 --> 00:01:32.540
is this value right
over here-- is

00:01:32.540 --> 00:01:36.490
equal to the length of this
vector right over here,

00:01:36.490 --> 00:01:41.220
which is the opposite side 26.03
over the length of this vector

00:01:41.220 --> 00:01:43.910
here, which is 5.21.

00:01:43.910 --> 00:01:47.820
Or we could say, if we
take the inverse tangent

00:01:47.820 --> 00:01:50.360
of both sides of
this, that the angle

00:01:50.360 --> 00:01:53.640
is equal to the inverse
tangent or the arctangent,

00:01:53.640 --> 00:02:00.370
of 26.03 over 5.21.

00:02:00.370 --> 00:02:06.080
We have the inverse
tangent of 26.03

00:02:06.080 --> 00:02:14.660
divided by 5.21 gives
us roughly 78.7 degrees.

00:02:14.660 --> 00:02:20.180
So the angle here
is 78.7 degrees.

00:02:20.180 --> 00:02:22.510
And it's below the horizontal.

00:02:22.510 --> 00:02:27.080
So we can say that the vector,
the final velocity vector,

00:02:27.080 --> 00:02:31.500
has a magnitude of
26.55 meters per second

00:02:31.500 --> 00:02:36.004
at a direction of 78.7
degrees below the horizontal.

00:02:36.004 --> 00:02:37.420
So I hope I didn't
confuse you too

00:02:37.420 --> 00:02:40.950
much with that mistake
at the end of that video.


Projectile on an incline

WEBVTT
Kind: captions
Language: en

00:00:00.314 --> 00:00:01.730
What I want to do
in this video is

00:00:01.730 --> 00:00:05.000
tackle a problem that would
be considered pretty difficult

00:00:05.000 --> 00:00:07.192
for most first-year
physics students.

00:00:07.192 --> 00:00:08.650
And you, frankly,
probably wouldn't

00:00:08.650 --> 00:00:10.320
be expected to solve
a problem like this

00:00:10.320 --> 00:00:12.140
in most first-year
physics class.

00:00:12.140 --> 00:00:14.000
Or if you're in an
advanced or honors class

00:00:14.000 --> 00:00:16.208
you might be expected or it
might be a bonus problem.

00:00:16.208 --> 00:00:18.224
But it's an interesting
type of problem.

00:00:18.224 --> 00:00:19.640
Because what we're
going to do is,

00:00:19.640 --> 00:00:24.010
we're going to launch a
projectile on an incline.

00:00:24.010 --> 00:00:26.225
So maybe we're on
the side of the hill.

00:00:26.225 --> 00:00:26.850
So it's a hill.

00:00:26.850 --> 00:00:28.730
Let me do it in green.

00:00:28.730 --> 00:00:31.700
So let's say we're on the
side of the hill like this.

00:00:31.700 --> 00:00:34.250
And let's say that we know
the inclination of the hill.

00:00:34.250 --> 00:00:42.050
The hill's inclination is 30
degrees off the horizontal.

00:00:42.050 --> 00:00:44.470
So this is the horizontal
right over here.

00:00:44.470 --> 00:00:46.790
So that is the
inclination of the hill.

00:00:46.790 --> 00:00:51.450
And then we're going
to launch a projectile

00:00:51.450 --> 00:00:54.060
at 10 meters per second.

00:00:54.060 --> 00:00:56.510
We're going to launch it
at 10 meters per second.

00:00:56.510 --> 00:01:00.690
And the angle with the
hill is 15 degrees.

00:01:00.690 --> 00:01:03.700
So at a 15 degree
angle with the hill.

00:01:03.700 --> 00:01:05.580
And the reason why
this is more difficult

00:01:05.580 --> 00:01:08.395
than the traditional projectile
motion problems is, well, we

00:01:08.395 --> 00:01:09.270
could think about it.

00:01:09.270 --> 00:01:11.080
The projectile is
going to be launched

00:01:11.080 --> 00:01:14.330
and it is going to eventually
land at some point on the hill.

00:01:14.330 --> 00:01:16.770
But we can't do
the simple figure

00:01:16.770 --> 00:01:19.390
out how long it's in the air
using its vertical velocity

00:01:19.390 --> 00:01:21.880
because we don't know what
the vertical displacement

00:01:21.880 --> 00:01:24.450
for this thing is
going to be, unless we

00:01:24.450 --> 00:01:28.040
know how far down
the hill it lands.

00:01:28.040 --> 00:01:30.410
Because the further
down the hill

00:01:30.410 --> 00:01:33.450
it lands, the higher the
vertical displacement.

00:01:33.450 --> 00:01:36.180
So we have to think
about both the horizontal

00:01:36.180 --> 00:01:38.700
and the vertical displacement
at the same time.

00:01:38.700 --> 00:01:40.410
And as we walk
through this, you'll

00:01:40.410 --> 00:01:43.320
see how that can be done.

00:01:43.320 --> 00:01:46.702
So I guess the first thing
that we really would always

00:01:46.702 --> 00:01:48.160
want to do whenever
you want to try

00:01:48.160 --> 00:01:52.060
to solve this type of problem is
break up our velocity into both

00:01:52.060 --> 00:01:54.970
the horizontal and
vertical components.

00:01:54.970 --> 00:02:00.820
So the vertical
component of our velocity

00:02:00.820 --> 00:02:05.120
is going to be the magnitude
of our total velocity, 10

00:02:05.120 --> 00:02:08.840
meters per second, times--
and be very careful here--

00:02:08.840 --> 00:02:13.350
not the sine of 15 degrees,
but the sine of the angle

00:02:13.350 --> 00:02:14.540
with the horizontal.

00:02:14.540 --> 00:02:17.430
So times the sine of 45 degrees.

00:02:22.056 --> 00:02:24.430
And I go into it in a lot more
detail in previous videos.

00:02:24.430 --> 00:02:26.760
For the sake of time,
I won't go into it.

00:02:26.760 --> 00:02:29.500
But it really just
comes from sohcahtoa.

00:02:29.500 --> 00:02:32.010
If we were to draw the
vertical component,

00:02:32.010 --> 00:02:34.440
it would look like this.

00:02:34.440 --> 00:02:35.600
This is the angle.

00:02:35.600 --> 00:02:38.230
The sine of 45 degrees
is equal to the opposite

00:02:38.230 --> 00:02:40.470
over the hypotenuse or
the hypotenuse times

00:02:40.470 --> 00:02:44.530
the sine of 45 degrees is equal
to the vertical component.

00:02:44.530 --> 00:02:45.950
That's where it's coming from.

00:02:45.950 --> 00:02:47.700
Let me get rid of some
of the stuff I just

00:02:47.700 --> 00:02:49.930
drew just so it makes
it a little bit cleaner.

00:02:49.930 --> 00:02:56.220
And the horizontal component
of our velocity is going to be,

00:02:56.220 --> 00:03:01.990
by the same logic, 10
cosine of 45 degrees.

00:03:01.990 --> 00:03:04.390
Now let's think about what
the horizontal displacement

00:03:04.390 --> 00:03:05.180
is going to be.

00:03:07.940 --> 00:03:10.220
And I'm just going to go
straight to the formula

00:03:10.220 --> 00:03:13.000
that we've derived in
the last few videos.

00:03:13.000 --> 00:03:16.470
The horizontal
displacement is going

00:03:16.470 --> 00:03:19.220
to be our initial-- sorry.

00:03:19.220 --> 00:03:20.710
Let's do the vertical
displacement.

00:03:20.710 --> 00:03:22.475
The vertical
displacement-- I could

00:03:22.475 --> 00:03:24.350
have done the horizontal
displacement first--

00:03:24.350 --> 00:03:26.900
but the vertical
displacement is going

00:03:26.900 --> 00:03:29.030
to be our initial
vertical velocity, which

00:03:29.030 --> 00:03:32.575
we know as 10 sine
of 45 degrees.

00:03:32.575 --> 00:03:34.700
And by the way, we could
just solve that right now.

00:03:34.700 --> 00:03:37.020
What is the sine of 45 degrees?

00:03:37.020 --> 00:03:40.270
Sine of 45 degrees is
square root of 2/2.

00:03:40.270 --> 00:03:45.100
Cosine of 45 degrees is
also square root of 2/2.

00:03:45.100 --> 00:03:48.340
So both of these values,
10 times square root of 2/2

00:03:48.340 --> 00:03:50.452
is 5 square roots of 2.

00:03:50.452 --> 00:03:51.910
So this whole thing
right over here

00:03:51.910 --> 00:03:57.000
is 5 square roots of
2 meters per second.

00:03:57.000 --> 00:03:59.160
That's our vertical velocity.

00:03:59.160 --> 00:04:03.810
And our horizontal
velocity is also

00:04:03.810 --> 00:04:06.520
5 square roots of 2
meters per second.

00:04:06.520 --> 00:04:08.260
So that simplifies
things a little bit.

00:04:08.260 --> 00:04:09.650
But anyway, we
were talking about

00:04:09.650 --> 00:04:11.420
our vertical displacement.

00:04:11.420 --> 00:04:13.550
Our vertical
displacement is going

00:04:13.550 --> 00:04:17.899
to be our initial vertical
velocity, 5 square roots of 2,

00:04:17.899 --> 00:04:27.785
times our change in time
plus the acceleration.

00:04:27.785 --> 00:04:29.410
Well, we know what
the acceleration is.

00:04:29.410 --> 00:04:31.670
It's negative 9.8 meters
per second squared.

00:04:31.670 --> 00:04:34.450
So let me write minus 9.8.

00:04:34.450 --> 00:04:38.560
I'm not going to write the
units here so that we say space.

00:04:38.560 --> 00:04:42.650
Times our change
in time squared.

00:04:42.650 --> 00:04:44.030
All of that over 2.

00:04:44.030 --> 00:04:46.530
We derive this in several
videos, especially the last few

00:04:46.530 --> 00:04:49.160
where we do these two
dimensional projectile motions.

00:04:49.160 --> 00:04:50.720
So this gives us
our displacement

00:04:50.720 --> 00:04:51.969
in the y direction.

00:04:51.969 --> 00:04:53.510
And I can simplify
this a little bit.

00:04:53.510 --> 00:04:55.950
Our displacement in
the vertical direction

00:04:55.950 --> 00:05:02.590
is equal to 5 square
roots of 2 times

00:05:02.590 --> 00:05:09.050
delta t times-- let me do it in
the same-- times delta t change

00:05:09.050 --> 00:05:16.020
in time minus 4.9 times
our change in time squared.

00:05:18.910 --> 00:05:22.320
So we know we have this
constraint right over here.

00:05:22.320 --> 00:05:25.530
So this gives us our
vertical displacement

00:05:25.530 --> 00:05:27.200
as a function of time.

00:05:27.200 --> 00:05:29.320
Let's think about our
horizontal displacement

00:05:29.320 --> 00:05:30.890
as a function of time.

00:05:30.890 --> 00:05:33.880
Our horizontal
displacement is going

00:05:33.880 --> 00:05:36.880
to be equal to our
horizontal velocity, which

00:05:36.880 --> 00:05:41.350
is 5 square roots of 2
times our change in time.

00:05:44.360 --> 00:05:47.000
Now what can we do next?

00:05:47.000 --> 00:05:48.690
Well, we have to have
some relationship

00:05:48.690 --> 00:05:51.050
between our horizontal
displacement

00:05:51.050 --> 00:05:52.870
and our vertical displacement.

00:05:52.870 --> 00:05:54.530
And that relationship
is going to be

00:05:54.530 --> 00:05:56.830
given to us by this incline.

00:05:56.830 --> 00:05:58.330
So wherever we
land-- let's say this

00:05:58.330 --> 00:06:00.405
is where we eventually
do end up landing.

00:06:03.817 --> 00:06:05.400
Well, let's think
about our horizontal

00:06:05.400 --> 00:06:07.130
and our vertical
displacements and what

00:06:07.130 --> 00:06:09.020
their relationship
would have to be.

00:06:09.020 --> 00:06:11.660
So if this is where
we land, then this

00:06:11.660 --> 00:06:16.970
would be our-- let me do it
in the same colors-- that

00:06:16.970 --> 00:06:19.490
right there would be my
vertical displacement.

00:06:19.490 --> 00:06:21.630
I would move that far up.

00:06:21.630 --> 00:06:27.040
And then our
horizontal displacement

00:06:27.040 --> 00:06:30.540
will be this right over
here, will be that length

00:06:30.540 --> 00:06:31.270
right over there.

00:06:31.270 --> 00:06:34.080
So that is our
horizontal displacement.

00:06:34.080 --> 00:06:36.020
So what is the
relationship between

00:06:36.020 --> 00:06:38.850
our vertical displacement and
our horizontal displacement?

00:06:38.850 --> 00:06:41.690
And we know that this angle
right over here is 30 degrees.

00:06:41.690 --> 00:06:45.030
So we can use some
basic trigonometry.

00:06:45.030 --> 00:06:47.460
We have a right triangle.

00:06:47.460 --> 00:06:49.630
We know the opposite
side from the angle.

00:06:49.630 --> 00:06:51.350
We know the adjacent side.

00:06:51.350 --> 00:06:54.570
And the trig function that uses
the opposite and the adjacent

00:06:54.570 --> 00:06:56.300
is the tangent function.

00:06:56.300 --> 00:07:00.560
So we get the
tangent of 30 degrees

00:07:00.560 --> 00:07:05.950
is going to be equal
to the magnitude

00:07:05.950 --> 00:07:09.794
of our vertical displacement
over the magnitude

00:07:09.794 --> 00:07:11.085
of our horizontal displacement.

00:07:15.940 --> 00:07:17.930
And the tangent of
30 degrees, that's

00:07:17.930 --> 00:07:20.304
the same thing as
the sine of 30 over--

00:07:20.304 --> 00:07:21.470
let me just do it over here.

00:07:21.470 --> 00:07:25.090
So the tangent of 30
degrees is the same thing

00:07:25.090 --> 00:07:30.780
as the sine of 30 degrees
over the cosine of 30 degrees.

00:07:30.780 --> 00:07:33.730
Let me do this a
little bit neater.

00:07:33.730 --> 00:07:36.570
And the sine of
30 degrees is 1/2.

00:07:36.570 --> 00:07:38.240
This is equal to 1/2.

00:07:38.240 --> 00:07:41.820
And the cosine of 30 degrees
is the square root of 3/2.

00:07:41.820 --> 00:07:47.200
So this is equal to 1/2 times
2 over the square root of 3,

00:07:47.200 --> 00:07:50.110
which is equal to 1 over
the square root of 3.

00:07:50.110 --> 00:07:52.210
So we get the magnitude
of our vertical component

00:07:52.210 --> 00:07:55.090
over the magnitude of our
horizontal component--

00:07:55.090 --> 00:07:59.160
is our horizontal
component right over here--

00:07:59.160 --> 00:08:03.430
is equal to 1 over
the square root of 3.

00:08:03.430 --> 00:08:05.600
What's useful about
this is it gives us

00:08:05.600 --> 00:08:07.552
a relationship
between our horizontal

00:08:07.552 --> 00:08:09.510
and vertical component,
or between our vertical

00:08:09.510 --> 00:08:11.250
and our horizontal components.

00:08:11.250 --> 00:08:14.260
And we can use this
constraint right over here

00:08:14.260 --> 00:08:17.690
to then solve for
one of these two.

00:08:17.690 --> 00:08:19.520
And let me show you
how we'll do it.

00:08:19.520 --> 00:08:23.016
So let's just
explicitly write this.

00:08:23.016 --> 00:08:24.920
Well, let's do it this way.

00:08:24.920 --> 00:08:26.902
Let us cross multiply
here, which is really

00:08:26.902 --> 00:08:29.360
the same thing as multiplying
both sides by the square root

00:08:29.360 --> 00:08:32.820
of 3 and the magnitude of
our horizontal component.

00:08:32.820 --> 00:08:38.089
We get the square root
of 3 times the magnitude.

00:08:40.600 --> 00:08:43.850
And both of these are
going to be positive.

00:08:43.850 --> 00:08:45.490
Well, let me just
write it this way.

00:08:45.490 --> 00:08:48.290
Times the magnitude of
our vertical component

00:08:48.290 --> 00:08:51.060
is going to be equal
to the magnitude

00:08:51.060 --> 00:08:52.355
of our horizontal component.

00:08:55.760 --> 00:08:59.049
So right, just like that.

00:08:59.049 --> 00:09:01.590
So we now have a relationship
between the length of these two

00:09:01.590 --> 00:09:02.510
vectors.

00:09:02.510 --> 00:09:05.090
And we can use this
relationship to substitute back

00:09:05.090 --> 00:09:09.660
into the constraints
that we already have.

00:09:09.660 --> 00:09:12.390
So the second constraint
right over here,

00:09:12.390 --> 00:09:14.180
let's use this information.

00:09:14.180 --> 00:09:17.190
The second constraint says
that our horizontal component

00:09:17.190 --> 00:09:19.600
of our displacement is
equal to 5 square roots

00:09:19.600 --> 00:09:21.810
of 2 times our change in time.

00:09:21.810 --> 00:09:24.120
Or another way of
thinking about it,

00:09:24.120 --> 00:09:26.960
if we divide both sides
by 5 square roots of 2,

00:09:26.960 --> 00:09:29.930
we get our change
in time is equal

00:09:29.930 --> 00:09:39.260
to the horizontal component
of our displacement divided

00:09:39.260 --> 00:09:41.430
by 5 square roots of 2.

00:09:41.430 --> 00:09:43.590
But we also know that
the horizontal component

00:09:43.590 --> 00:09:45.240
of our displacement
is the square root

00:09:45.240 --> 00:09:49.520
of 3 times the vertical
component of our displacement.

00:09:49.520 --> 00:09:52.440
Here I explicitly wrote
the magnitude notation.

00:09:52.440 --> 00:09:55.460
When we start dealing with
either just the vertical

00:09:55.460 --> 00:09:58.000
or the horizontal
component, I can just

00:09:58.000 --> 00:09:59.940
write it like this,
because it's either

00:09:59.940 --> 00:10:01.970
going to be a positive
or a negative value.

00:10:01.970 --> 00:10:04.580
And that specifies both the
magnitude and the direction.

00:10:04.580 --> 00:10:06.830
So what I'm going to do right
over here, and obviously

00:10:06.830 --> 00:10:08.996
the way I've drawn it right
over here, both of these

00:10:08.996 --> 00:10:10.370
are going to be positive values.

00:10:10.370 --> 00:10:12.536
It's upwards displacement
in the vertical direction,

00:10:12.536 --> 00:10:14.330
so that's positive
by our convention.

00:10:14.330 --> 00:10:15.620
And we're moving to the right.

00:10:15.620 --> 00:10:19.120
So that's positive,
also, by our convention.

00:10:19.120 --> 00:10:24.050
So I can rewrite this
over here as being

00:10:24.050 --> 00:10:32.500
equal to square root of 3 times
our vertical displacement.

00:10:35.040 --> 00:10:37.220
And all of that's over
5 square roots of 2.

00:10:39.730 --> 00:10:41.910
Now, the whole
reason why I did this

00:10:41.910 --> 00:10:47.110
is, this expression right here
contains this information,

00:10:47.110 --> 00:10:50.060
contains the ratio between
our vertical displacement

00:10:50.060 --> 00:10:51.590
and our horizontal displacement.

00:10:51.590 --> 00:10:54.360
And it also contains
the information of,

00:10:54.360 --> 00:10:57.730
how does the horizontal
displacement, how does that

00:10:57.730 --> 00:10:59.580
change as a function of time?

00:10:59.580 --> 00:11:04.190
So our time needs
to be equal to this.

00:11:04.190 --> 00:11:06.060
So this is our
time as a function

00:11:06.060 --> 00:11:07.700
of our vertical
displacement now,

00:11:07.700 --> 00:11:10.440
not time as a function of
our horizontal displacement.

00:11:10.440 --> 00:11:12.820
And what we can do is, we
can use this constraint

00:11:12.820 --> 00:11:17.510
with our original
vertical displacement

00:11:17.510 --> 00:11:19.680
as a function of
time to then solve

00:11:19.680 --> 00:11:21.380
for our vertical displacements.

00:11:21.380 --> 00:11:22.150
So let's do that.

00:11:22.150 --> 00:11:24.530
Let's substitute this
business right here

00:11:24.530 --> 00:11:29.190
for delta t in this top
equation right over here.

00:11:29.190 --> 00:11:31.630
So if we do that-- and
I'll write it big--

00:11:31.630 --> 00:11:38.980
we get our vertical
displacement, right over there,

00:11:38.980 --> 00:11:41.880
is equal to 5 square
roots of 2 times delta t.

00:11:41.880 --> 00:11:44.140
So it's 5 square roots of 2.

00:11:44.140 --> 00:11:46.330
Delta t is all of this
business over here.

00:11:46.330 --> 00:11:48.480
So 5 square roots
of 2 times delta t.

00:11:48.480 --> 00:11:52.936
Delta t is square root of 3
times our vertical component.

00:11:52.936 --> 00:11:54.310
All of that, really
the magnitude

00:11:54.310 --> 00:11:58.510
of our vertical component,
over 5 square roots of 2.

00:11:58.510 --> 00:12:00.579
So that's that right there.

00:12:00.579 --> 00:12:02.870
Or actually, we could look
at this one right over here.

00:12:02.870 --> 00:12:04.078
We could use this constraint.

00:12:04.078 --> 00:12:05.430
This is just simplified.

00:12:05.430 --> 00:12:11.150
So then we have minus 4.9
times delta t squared.

00:12:11.150 --> 00:12:13.700
So delta t squared is
this quantity squared.

00:12:13.700 --> 00:12:14.710
I'll just write it out.

00:12:14.710 --> 00:12:16.310
I don't want to
skip too many steps.

00:12:16.310 --> 00:12:19.060
So delta t once again is
the square root of 3 times

00:12:19.060 --> 00:12:21.100
the vertical component.

00:12:21.100 --> 00:12:25.620
All of that over 5 square
roots of 2 squared.

00:12:25.620 --> 00:12:27.510
And now what does this give us?

00:12:27.510 --> 00:12:30.050
So now we literally have
a quadratic equation

00:12:30.050 --> 00:12:31.530
with only one variable.

00:12:31.530 --> 00:12:33.050
So we can solve for this.

00:12:33.050 --> 00:12:34.120
But let me rewrite it.

00:12:34.120 --> 00:12:35.060
Let me simplify it.

00:12:35.060 --> 00:12:36.920
So we have our
vertical component

00:12:36.920 --> 00:12:40.249
is equal to-- now we have
a 5 square roots of 2

00:12:40.249 --> 00:12:42.040
in the numerator and
one in the denominator

00:12:42.040 --> 00:12:43.270
and they cancel out.

00:12:43.270 --> 00:12:48.040
So we get the square root of 3
times our vertical component,

00:12:48.040 --> 00:12:50.010
the magnitude of our
vertical component.

00:12:50.010 --> 00:12:52.540
It's actually also specifying,
well, the magnitude,

00:12:52.540 --> 00:12:57.180
we can say, for now, although
I'm not using that notation.

00:12:57.180 --> 00:13:01.235
So then we have minus 4.9
times this quantity squared.

00:13:01.235 --> 00:13:03.360
So that's going to be, the
square root of 3 squared

00:13:03.360 --> 00:13:10.340
is 3 times the vertical
component squared.

00:13:10.340 --> 00:13:14.561
And then over 5 squared,
which is 25, times 2.

00:13:14.561 --> 00:13:16.060
That's the square
root of 2 squares.

00:13:16.060 --> 00:13:19.230
So 25 times 2 is 50.

00:13:19.230 --> 00:13:22.490
And so we get, if we were to
simplify this a little bit

00:13:22.490 --> 00:13:22.990
more.

00:13:22.990 --> 00:13:25.710
Let's subtract this
from both sides.

00:13:25.710 --> 00:13:28.140
We get-- I'll do it
all in one color-- 0

00:13:28.140 --> 00:13:35.320
is equal to the square
root of 3 minus 1

00:13:35.320 --> 00:13:39.790
times our vertical component.

00:13:39.790 --> 00:13:41.880
Because if we subtract
this from both sides,

00:13:41.880 --> 00:13:44.520
that's square root of 3
times our vertical component

00:13:44.520 --> 00:13:46.327
minus 1 times our
vertical component.

00:13:46.327 --> 00:13:48.160
So that's the square
root of 3 minus 1 times

00:13:48.160 --> 00:13:49.570
our vertical component.

00:13:49.570 --> 00:13:51.290
And then we have all
of this business.

00:13:51.290 --> 00:14:01.800
Minus 4.9 times 3 over 50 times
our vertical component squared.

00:14:01.800 --> 00:14:03.540
And lucky for us, we
can just factor out

00:14:03.540 --> 00:14:08.080
one of these s sub y's over
here, one of these vectors.

00:14:08.080 --> 00:14:11.040
And so we can get-- and I'll
just do that in place so that I

00:14:11.040 --> 00:14:13.540
don't-- well, let me just-- I
don't want to skip many steps.

00:14:13.540 --> 00:14:20.080
So this is equal to the square
root of 3 minus 1, minus 4.9.

00:14:20.080 --> 00:14:22.410
Let me do it in that color.

00:14:22.410 --> 00:14:25.370
So let me do it like this.

00:14:25.370 --> 00:14:32.360
It's equal to the square root
of 3 minus 1 minus 4.9 times

00:14:32.360 --> 00:14:38.910
3 over 50 times one of these,
times our vertical component.

00:14:38.910 --> 00:14:42.110
And then we factored one of
those vertical components out.

00:14:42.110 --> 00:14:44.160
So we factored one of them out.

00:14:44.160 --> 00:14:47.150
So the vertical component
of our displacement

00:14:47.150 --> 00:14:50.350
could either be-- so we have
the product of two things that

00:14:50.350 --> 00:14:51.404
equal 0.

00:14:51.404 --> 00:14:52.820
So our vertical
displacement could

00:14:52.820 --> 00:14:56.230
be 0, which is true, because
at some point in the path

00:14:56.230 --> 00:14:58.450
we literally had 0
vertical displacement.

00:14:58.450 --> 00:14:59.950
That was literally
where we started.

00:14:59.950 --> 00:15:01.280
But that's not the
answer we're looking for.

00:15:01.280 --> 00:15:03.480
We're looking for this
vertical displacement.

00:15:03.480 --> 00:15:05.050
So either this is going to be 0.

00:15:05.050 --> 00:15:06.841
But that's just kind
of the obvious answer.

00:15:06.841 --> 00:15:10.330
Or all of this business
is going to be equal to 0.

00:15:10.330 --> 00:15:13.520
But this is pretty easy
to solve for 0 over here.

00:15:13.520 --> 00:15:20.412
So we get square root
of 3 minus 1 minus 4.9.

00:15:20.412 --> 00:15:22.120
Let me just calculate
all of these things

00:15:22.120 --> 00:15:24.490
just so that I don't have
to keep writing them.

00:15:24.490 --> 00:15:35.570
So we get the square root of
3 minus 1 is equal to 0.73205.

00:15:35.570 --> 00:15:37.920
So I'll just write 0.732 here.

00:15:37.920 --> 00:15:41.990
So this is equal to 0.732.

00:15:41.990 --> 00:15:43.810
That's that part
right over there.

00:15:43.810 --> 00:15:47.370
And then 4.94.

00:15:47.370 --> 00:15:49.629
I know this problem
is getting long.

00:15:49.629 --> 00:15:51.920
Just pause it and take a
break if you're getting tired.

00:15:51.920 --> 00:15:59.170
4.9 times 3 divided by
50 is equal to 0.294.

00:15:59.170 --> 00:16:03.580
So minus 0.294.

00:16:03.580 --> 00:16:06.710
I could put a 0 out front just
to make it clear where we are.

00:16:06.710 --> 00:16:10.820
Times this, times our
vertical component.

00:16:10.820 --> 00:16:12.650
This could also be equal to 0.

00:16:12.650 --> 00:16:14.480
Either this is 0 or that is 0.

00:16:14.480 --> 00:16:16.490
When this is 0, it gives
us the obvious answer.

00:16:16.490 --> 00:16:17.970
We're more interested in this.

00:16:17.970 --> 00:16:22.610
To solve for this, we can
add this to both sides.

00:16:22.610 --> 00:16:30.280
And we get 0.732 is
equal to negative 0.294

00:16:30.280 --> 00:16:32.430
times the vertical component.

00:16:32.430 --> 00:16:34.380
We are in the home stretch.

00:16:34.380 --> 00:16:42.261
We divide both sides by
this, by negative 0.29.

00:16:42.261 --> 00:16:43.760
Oh sorry, this will
now be positive.

00:16:43.760 --> 00:16:47.520
I almost made a careless mistake
after 16 minutes of video.

00:16:47.520 --> 00:16:54.590
So now we divide
both sides by 0.294

00:16:54.590 --> 00:16:58.130
and we get our
vertical displacement.

00:16:58.130 --> 00:17:00.480
So this, I think, a drum
roll might be in order.

00:17:00.480 --> 00:17:04.530
So we have 0.732 and
all of this business,

00:17:04.530 --> 00:17:10.000
but I'll just round it there,
divided by 0.294 gives us

00:17:10.000 --> 00:17:15.280
a vertical displacement of,
if we round, 2.50 meters.

00:17:15.280 --> 00:17:18.160
Or 2.49 meters, I should say.

00:17:18.160 --> 00:17:23.800
So this is equal to 2.49 meters.

00:17:23.800 --> 00:17:25.380
This is exciting.

00:17:25.380 --> 00:17:29.070
This is equal to 2.49 meters.

00:17:29.070 --> 00:17:31.280
And now we can figure out
the horizontal displacement

00:17:31.280 --> 00:17:33.260
pretty easily,
because we know that

00:17:33.260 --> 00:17:35.870
the horizontal displacement
is square root of 3 times

00:17:35.870 --> 00:17:37.639
the vertical displacement.

00:17:37.639 --> 00:17:38.680
So let's figure that out.

00:17:38.680 --> 00:17:40.370
That's the vertical
displacement.

00:17:40.370 --> 00:17:43.510
Let's multiply that times
the square root of 3.

00:17:43.510 --> 00:17:46.550
And we get 4.31 meters.

00:17:46.550 --> 00:17:49.580
So we get the
horizontal displacement.

00:17:49.580 --> 00:17:54.580
The horizontal displacement
is equal to 4.31 meters.

00:17:54.580 --> 00:17:59.080
So this is equal to 4.31 meters.

00:17:59.080 --> 00:18:01.280
So we actually now know
the total displacement

00:18:01.280 --> 00:18:03.834
in the vertical direction and
in the horizontal direction.

00:18:03.834 --> 00:18:05.000
And I'll leave it up to you.

00:18:05.000 --> 00:18:07.930
If you wanted to figure out
exactly how far along the hill

00:18:07.930 --> 00:18:10.010
we traveled, you
can just use both

00:18:10.010 --> 00:18:12.130
of these values in the
Pythagorean theorem

00:18:12.130 --> 00:18:13.960
to essentially figure
out the hypotenuse

00:18:13.960 --> 00:18:16.061
of this right triangle.


Total final velocity for projectile

WEBVTT
Kind: captions
Language: en

00:00:00.311 --> 00:00:02.560
In the last video, I told
you that we would figure out

00:00:02.560 --> 00:00:05.580
the final velocity of
when this thing lands.

00:00:05.580 --> 00:00:06.400
So let's do that.

00:00:06.400 --> 00:00:08.570
I forgot to do it
in the last video.

00:00:08.570 --> 00:00:10.460
So let's figure out
the final velocity--

00:00:10.460 --> 00:00:12.320
the vertical and the
horizontal components

00:00:12.320 --> 00:00:13.730
of that final velocity.

00:00:13.730 --> 00:00:16.950
And then we can reconstruct
the total final velocity.

00:00:16.950 --> 00:00:18.710
So the horizontal
component is easy,

00:00:18.710 --> 00:00:20.918
because we already know that
the horizontal component

00:00:20.918 --> 00:00:23.975
of its velocity is this value
right over here, which we--

00:00:23.975 --> 00:00:26.060
this 30 cosine of 80 degrees.

00:00:26.060 --> 00:00:28.270
And that's not going to
change at any point in time.

00:00:28.270 --> 00:00:32.119
So this is going to be
the horizontal component

00:00:32.119 --> 00:00:35.381
of the projectile's
velocity when it lands.

00:00:35.381 --> 00:00:36.880
But what we need
to do is figure out

00:00:36.880 --> 00:00:40.310
the vertical component
of its velocity.

00:00:40.310 --> 00:00:42.950
Well, one thing we did
figure out in the last video,

00:00:42.950 --> 00:00:47.260
we figured out what the time
in the air is going to be.

00:00:47.260 --> 00:00:50.970
And we know a way of figuring
out our final velocity

00:00:50.970 --> 00:00:55.580
from an initial velocity
given our time in the air.

00:00:55.580 --> 00:00:58.127
We know that a
change in velocity --

00:00:58.127 --> 00:00:59.960
and we're only dealing
in the vertical now--

00:00:59.960 --> 00:01:01.501
we're only dealing
with the vertical,

00:01:01.501 --> 00:01:04.730
because the horizontal velocity
is not going to change.

00:01:04.730 --> 00:01:07.290
We've assumed that air
resistance is negligible.

00:01:07.290 --> 00:01:09.690
So we're only dealing with
the vertical component

00:01:09.690 --> 00:01:10.450
right over here.

00:01:13.210 --> 00:01:15.680
We know that the
change in velocity--

00:01:15.680 --> 00:01:17.340
or, we could say
the horizontal--

00:01:17.340 --> 00:01:21.770
the vertical component of
the change in velocity,

00:01:21.770 --> 00:01:28.680
is equal to the vertical
component of the acceleration

00:01:28.680 --> 00:01:30.430
times time.

00:01:30.430 --> 00:01:31.930
Now, we know what
the change in time

00:01:31.930 --> 00:01:35.810
is, we know it is-- I'll just
write down times our time.

00:01:35.810 --> 00:01:40.250
And what is our
change in velocity?

00:01:40.250 --> 00:01:45.490
Well, our change in velocity
is our final vertical velocity

00:01:45.490 --> 00:01:47.894
minus our initial
vertical velocity.

00:01:47.894 --> 00:01:49.810
And we know what our
initial vertical velocity

00:01:49.810 --> 00:01:51.160
is, we solved for it.

00:01:51.160 --> 00:01:53.860
Our initial vertical
velocity, we figured out,

00:01:53.860 --> 00:01:56.800
was 29.54 meters per second.

00:01:56.800 --> 00:02:02.220
That's 30 sine of 80 degrees,
29.54 meters per second.

00:02:02.220 --> 00:02:08.190
So this is going to be minus
29.54 meters per second,

00:02:08.190 --> 00:02:11.540
is equal to-- our acceleration
in the vertical direction

00:02:11.540 --> 00:02:14.400
is negative, because
it's accelerating us

00:02:14.400 --> 00:02:18.830
downwards, negative 9.8
meters per second squared.

00:02:18.830 --> 00:02:21.860
And our time in the
air is 5.67 seconds.

00:02:21.860 --> 00:02:25.460
Times 5.67 seconds.

00:02:25.460 --> 00:02:27.840
And so we can solve for
the vertical component

00:02:27.840 --> 00:02:29.610
of our final velocity.

00:02:29.610 --> 00:02:31.830
So once again, this is
the vertical component.

00:02:31.830 --> 00:02:33.360
This isn't the total one.

00:02:33.360 --> 00:02:35.060
So, the vertical component.

00:02:35.060 --> 00:02:36.884
Let me-- well I wrote
vertical up here.

00:02:36.884 --> 00:02:38.300
So there's the
vertical component.

00:02:38.300 --> 00:02:39.730
So let's solve for this.

00:02:39.730 --> 00:02:42.190
So if you add 29.54
to both sides,

00:02:42.190 --> 00:02:46.590
you get the vertical component
of your final velocity.

00:02:46.590 --> 00:02:48.230
Well, this is a
vertical component,

00:02:48.230 --> 00:02:50.390
I didn't mark it
up here properly--

00:02:50.390 --> 00:02:57.670
is equal to 29.54 meters
per second plus 9.8 plus --

00:02:57.670 --> 00:03:00.790
or I should say minus--
meters per second.

00:03:00.790 --> 00:03:08.120
Minus 9.8 meters per second
squared, times 5.67 seconds.

00:03:08.120 --> 00:03:10.550
The seconds cancel out
with one of these seconds.

00:03:10.550 --> 00:03:12.630
So everything is
meters per second.

00:03:12.630 --> 00:03:17.820
And so, get the
calculator out again,

00:03:17.820 --> 00:03:27.580
we have 29.54 minus
9.8 times 5.67.

00:03:27.580 --> 00:03:35.550
So we get our change-- our final
velocity is negative 26.03.

00:03:35.550 --> 00:03:40.564
So this is negative
26.03 meters per second.

00:03:40.564 --> 00:03:41.980
And you might say
wait, wait Sal ,

00:03:41.980 --> 00:03:45.290
what is this negative 26.03
meters per second mean?

00:03:45.290 --> 00:03:48.010
Remember, when we're dealing
in the vertical dimension,

00:03:48.010 --> 00:03:51.050
positive means up,
negative mean down.

00:03:51.050 --> 00:03:55.820
So it means we're going 26.03
meters per second downwards.

00:03:55.820 --> 00:04:00.860
Downwards, right when we land.

00:04:00.860 --> 00:04:04.880
So what is our
total velocity when

00:04:04.880 --> 00:04:07.190
we fall back to that landing?

00:04:07.190 --> 00:04:10.440
So the vertical
component of our velocity

00:04:10.440 --> 00:04:18.459
is negative 29.06 times .03
in the downward direction.

00:04:18.459 --> 00:04:21.450
And the horizontal component
of our velocity, we know,

00:04:21.450 --> 00:04:24.010
hadn't changed the entire time.

00:04:24.010 --> 00:04:28.410
That, we figured out, was
30 cosine of 80 degrees.

00:04:28.410 --> 00:04:31.720
So that over here, is
30 cosine of 80 degrees.

00:04:31.720 --> 00:04:33.760
I'll get the calculator
out to calculate it.

00:04:33.760 --> 00:04:40.290
30 cosine of 80 degrees,
which is equal to 5.21.

00:04:40.290 --> 00:04:44.080
So this is 5.21
meters per second.

00:04:44.080 --> 00:04:46.390
These are both in
meters per second.

00:04:46.390 --> 00:04:48.570
So what is the total velocity?

00:04:48.570 --> 00:04:50.530
Well, I can do
the head to tails.

00:04:50.530 --> 00:04:55.730
So I can shift this guy
over so that its tail is

00:04:55.730 --> 00:04:58.530
at the head of the blue vector.

00:04:58.530 --> 00:04:59.780
So it would look like that.

00:04:59.780 --> 00:05:01.500
The length of
this-- the magnitude

00:05:01.500 --> 00:05:06.397
of our vertical
component, is 29.03.

00:05:06.397 --> 00:05:08.480
And then we could just use
the Pythagorean theorem

00:05:08.480 --> 00:05:12.949
to figure out the magnitude of
the total velocity upon impact.

00:05:12.949 --> 00:05:14.490
So the length of
that-- we could just

00:05:14.490 --> 00:05:16.470
use the Pythagorean theorem.

00:05:16.470 --> 00:05:18.710
So the magnitude of our
total velocity, that's

00:05:18.710 --> 00:05:19.980
this length right over here.

00:05:23.730 --> 00:05:25.550
The magnitude of
our total velocity,

00:05:25.550 --> 00:05:27.800
our total final velocity
I guess we can say,

00:05:27.800 --> 00:05:31.540
is going to be equal to-- well
that's-- let me write it this

00:05:31.540 --> 00:05:32.412
way.

00:05:32.412 --> 00:05:33.870
The magnitude of
our total velocity

00:05:33.870 --> 00:05:36.328
is going to be equal to square
root-- this is just straight

00:05:36.328 --> 00:05:40.640
from the Pythagorean
theorem-- of 5.21

00:05:40.640 --> 00:05:48.070
squared plus 29.03 squared.

00:05:48.070 --> 00:05:53.010
And we get it as
being the second--

00:05:53.010 --> 00:06:04.350
the square root of 5.21
squared plus 29.03 squared

00:06:04.350 --> 00:06:08.870
gives us 29.49
meters per second.

00:06:08.870 --> 00:06:12.180
This is equal to 29.49
meters per second.

00:06:12.180 --> 00:06:14.440
That is the magnitude
of our final velocity,

00:06:14.440 --> 00:06:17.120
but we also need to
figure out its direction.

00:06:17.120 --> 00:06:19.160
And so we need to
figure out this angle.

00:06:19.160 --> 00:06:22.320
And now we're talking about
an angle below the horizontal.

00:06:22.320 --> 00:06:24.734
Or, if you wanted to view
it in kind of pure terms,

00:06:24.734 --> 00:06:26.234
it would be a
negative angle-- or we

00:06:26.234 --> 00:06:28.560
could say an angle
below the horizontal.

00:06:28.560 --> 00:06:30.430
So what is this angle
right over here?

00:06:30.430 --> 00:06:32.160
So if we view it as
a positive angle just

00:06:32.160 --> 00:06:34.240
in the traditional
trigonometric way,

00:06:34.240 --> 00:06:37.602
we could say that the-- we could
use any of the trig functions,

00:06:37.602 --> 00:06:38.685
we could even use tangent.

00:06:38.685 --> 00:06:39.590
Let's use tangent.

00:06:39.590 --> 00:06:42.980
We could say that the
tangent of the angle, is

00:06:42.980 --> 00:06:45.410
equal to the opposite
over the adjacent--

00:06:45.410 --> 00:06:50.690
is equal to 29.03 over 5.21.

00:06:50.690 --> 00:06:54.140
Or that theta is equal
to the inverse tangent,

00:06:54.140 --> 00:07:02.340
or the arctangent
of 29.03 over 5.21.

00:07:02.340 --> 00:07:09.570
And that gives us-- we take
the inverse tangent of 29.03

00:07:09.570 --> 00:07:17.380
divided by 5.21, and
we get 79.8 degrees.

00:07:17.380 --> 00:07:21.740
But it's going to be
79.8 degrees south,

00:07:21.740 --> 00:07:23.285
or, I guess, below
the horizontal.

00:07:26.040 --> 00:07:30.270
Or you could view this as an
angle of negative 79.8 degrees

00:07:30.270 --> 00:07:34.380
above the horizontal,
either one of those work.

00:07:34.380 --> 00:07:37.721
What's neat about this, is we
figured out our final velocity

00:07:37.721 --> 00:07:38.220
vector.

00:07:38.220 --> 00:07:40.990
The entire vector, we know
what that entire vector is.

00:07:43.650 --> 00:07:49.790
It is 29.49 meters
per second at 79.8

00:07:49.790 --> 00:07:52.210
degrees below the horizontal.


Total displacement for projectile

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:01.740
- [Instructor] Let's do another example

00:00:01.740 --> 00:00:03.060
where we are projecting something

00:00:03.060 --> 00:00:04.220
and it lands at a different level

00:00:04.220 --> 00:00:06.140
and we'll also figure out
some other interesting things.

00:00:06.140 --> 00:00:08.760
We'll figure out what the
actual velocity vector

00:00:08.760 --> 00:00:10.200
is when it's landing, so both

00:00:10.200 --> 00:00:12.410
its magnitude and its direction.

00:00:12.410 --> 00:00:13.950
So let's do a situation.

00:00:13.950 --> 00:00:17.290
So let's say we are launching
something from ground level,

00:00:17.290 --> 00:00:21.180
and we're gonna launch it at a
pretty steep angle over here.

00:00:21.180 --> 00:00:23.493
So let's say we launch it with an angle,

00:00:24.380 --> 00:00:27.520
let's say we launch it with an angle of,

00:00:27.520 --> 00:00:30.300
let me just, let's say it's 80 degrees.

00:00:30.300 --> 00:00:34.493
We launch it at an 80
degree angle, at, let me.

00:00:35.540 --> 00:00:38.060
We launch it at an 80 degree angle,

00:00:38.060 --> 00:00:41.840
80 degree angle, and
it is going to be going

00:00:41.840 --> 00:00:43.610
at 30 meters per second,

00:00:43.610 --> 00:00:46.300
30 meters per second, so that's
the length of this vector,

00:00:46.300 --> 00:00:47.960
that's the magnitude of that vector,

00:00:47.960 --> 00:00:50.750
and let's say we want to
make it land on this landing,

00:00:50.750 --> 00:00:52.680
and this landing right over here,

00:00:52.680 --> 00:00:55.510
this landing right over here has a height,

00:00:55.510 --> 00:00:59.800
this landing over here
has a height of 10 meters.

00:00:59.800 --> 00:01:03.140
This height over here is 10 meters,

00:01:03.140 --> 00:01:04.180
so what I wanna do first

00:01:04.180 --> 00:01:07.800
of all is I wanna figure out
how far along the landing,

00:01:07.800 --> 00:01:11.610
how far along the landing
do I actually land,

00:01:11.610 --> 00:01:13.310
how far along the landing.

00:01:13.310 --> 00:01:16.410
So, and maybe I'll add some
other information right here.

00:01:16.410 --> 00:01:19.090
So say from this launching
point to the beginning

00:01:19.090 --> 00:01:20.690
of the landing, let's say

00:01:20.690 --> 00:01:23.820
that this right over here is two meters,

00:01:23.820 --> 00:01:24.800
two meters, so we just want

00:01:24.800 --> 00:01:28.010
to know how far along
the landing do we land.

00:01:28.010 --> 00:01:30.400
So like we did before, we want to

00:01:30.400 --> 00:01:32.070
break this vector into its horizontal

00:01:32.070 --> 00:01:33.610
and vertical components,

00:01:33.610 --> 00:01:35.380
its vertical and vertical components,

00:01:35.380 --> 00:01:37.150
and I'm gonna go a little
bit faster in this video,

00:01:37.150 --> 00:01:38.270
'cause hopefully we're getting the knack

00:01:38.270 --> 00:01:39.920
for this type of thing.

00:01:39.920 --> 00:01:43.810
So our velocity, our
velocity in the vertical

00:01:43.810 --> 00:01:44.990
or our vertical component

00:01:44.990 --> 00:01:48.580
of our velocity is going to
be equal to the magnitude

00:01:48.580 --> 00:01:51.240
of our total velocity,
30 meters per second,

00:01:51.240 --> 00:01:54.830
times, and it's going to
be the sine of 80 degrees,

00:01:54.830 --> 00:01:57.210
'cause sine is opposite over hypotenuse,

00:01:57.210 --> 00:02:01.600
times the sine of 80 degrees,

00:02:01.600 --> 00:02:03.820
and our, and we'll just get
it out of the way right now,

00:02:03.820 --> 00:02:05.200
the horizontal component

00:02:05.200 --> 00:02:08.060
of our velocity is going
to be 30 meters per second,

00:02:08.060 --> 00:02:09.350
and I'm not writing the units here,

00:02:09.350 --> 00:02:13.440
just to save space, times
the cosine of 80 degrees.

00:02:13.440 --> 00:02:15.900
Once again, cosine is adjacent,

00:02:15.900 --> 00:02:19.300
adjacent, adjacent over hypotenuse,

00:02:19.300 --> 00:02:20.730
and if you feel like I'm skipping steps,

00:02:20.730 --> 00:02:23.970
in the last few videos, I go
into this in much more detail.

00:02:23.970 --> 00:02:27.750
So, how much time do we spend in the air?

00:02:27.750 --> 00:02:30.660
How much time do we spend in the air?

00:02:30.660 --> 00:02:33.220
So, once again, in the last few videos,

00:02:33.220 --> 00:02:35.800
we saw that we can look at our,

00:02:35.800 --> 00:02:38.043
we can look at displacement,

00:02:39.120 --> 00:02:40.610
if we want to figure out time in the air,

00:02:40.610 --> 00:02:45.120
we know that displacement is
equal to the initial velocity,

00:02:45.120 --> 00:02:47.580
initial velocity times time,

00:02:47.580 --> 00:02:51.630
plus acceleration, let me
write times change in times,

00:02:51.630 --> 00:02:52.810
since that's more technically,

00:02:52.810 --> 00:02:56.000
that's technically more correct,

00:02:56.000 --> 00:02:58.700
plus acceleration times change

00:02:58.700 --> 00:03:03.540
in time squared, over, over two.

00:03:03.540 --> 00:03:06.440
Now, in our situation, we know
what our initial velocity is,

00:03:06.440 --> 00:03:08.980
we are talking about the vertical
direction right over here,

00:03:08.980 --> 00:03:10.790
so our initial velocity
is going to be this.

00:03:10.790 --> 00:03:12.670
We are trying to figure
out how time in the air,

00:03:12.670 --> 00:03:14.810
and the vertical
component determines that,

00:03:14.810 --> 00:03:17.360
because at some point, when
it hits back to the ground,

00:03:17.360 --> 00:03:18.850
it's not gonna be traveling anymore,

00:03:18.850 --> 00:03:20.980
so that's what's determining
its time in the air.

00:03:20.980 --> 00:03:22.990
So we know that, we know the acceleration.

00:03:22.990 --> 00:03:24.250
Remember, our convention,

00:03:24.250 --> 00:03:26.470
when we are dealing with
the vertical dimension

00:03:26.470 --> 00:03:29.570
is up is positive, down is negative,

00:03:29.570 --> 00:03:33.760
so this is negative 9.8
meters per second squared,

00:03:33.760 --> 00:03:36.680
and then we can, and then
what is the total displacement

00:03:36.680 --> 00:03:37.780
that we are gonna have?

00:03:37.780 --> 00:03:39.530
Well, we are starting at ground level,

00:03:39.530 --> 00:03:41.540
and we are just talking about
the vertical, remember that.

00:03:41.540 --> 00:03:44.060
So our total displacement
is going to be 10 meters,

00:03:44.060 --> 00:03:46.860
so this value right here
is going to be 10 meters,

00:03:46.860 --> 00:03:49.810
so it simplifies to 10 meters,

00:03:49.810 --> 00:03:51.540
I won't write the units here,

00:03:51.540 --> 00:03:55.760
is equals 10 is equal to,
what's 30 sine of 80 degrees?

00:03:55.760 --> 00:04:00.760
So we have 30 times the sine
of 80 degrees, gives us 29.54.

00:04:03.570 --> 00:04:05.840
29.54.

00:04:05.840 --> 00:04:10.840
So this is 29.54 times our change in time,

00:04:11.750 --> 00:04:13.670
times our change in time,

00:04:13.670 --> 00:04:16.320
and so this is negative
9.8 divided by two,

00:04:16.320 --> 00:04:19.290
so it's negative, I'll
do that same green color,

00:04:19.290 --> 00:04:22.310
four point, that's not
the same green color,

00:04:22.310 --> 00:04:25.920
negative 4.9 meters per second squared.

00:04:25.920 --> 00:04:27.440
I'm not writing the units here,

00:04:27.440 --> 00:04:30.950
times delta t squared,
times delta t squared,

00:04:30.950 --> 00:04:33.150
and then we can subtract
10 from both sides

00:04:33.150 --> 00:04:36.760
and write this in a traditional
quadratic equation form,

00:04:36.760 --> 00:04:39.580
so we get negative, and I'm
gonna flip the sides too,

00:04:39.580 --> 00:04:44.580
negative 4.9 times delta t squared,

00:04:45.020 --> 00:04:50.020
and then we have, plus
29.54 times delta t,

00:04:51.090 --> 00:04:55.230
minus 10, minus 10 is equal to zero,

00:04:55.230 --> 00:04:58.250
and then we can use the quadratic
formula to find the roots

00:04:58.250 --> 00:05:00.300
of this, so the delta t's

00:05:00.300 --> 00:05:02.670
that satisfy this quadratic equation are

00:05:02.670 --> 00:05:04.710
going to be negative b,

00:05:04.710 --> 00:05:08.400
so negative 29.54, plus

00:05:08.400 --> 00:05:13.240
or minus the square root of 29.54 squared,

00:05:13.240 --> 00:05:18.050
b squared, minus four, minus four,

00:05:18.050 --> 00:05:19.953
times a, which is negative 4.9,

00:05:21.070 --> 00:05:22.660
for the negative times a
negative is a positive.

00:05:22.660 --> 00:05:27.140
So it's plus four times
positive 4.9, times,

00:05:27.140 --> 00:05:29.170
or I shouldn't have, I
shouldn't have jumped so fast

00:05:29.170 --> 00:05:30.020
to get rid of the negative.

00:05:30.020 --> 00:05:33.810
So it's gonna be minus four times a,

00:05:33.810 --> 00:05:36.810
which is negative 4.9, times c,

00:05:36.810 --> 00:05:39.740
which is negative 10, so just a times c,

00:05:39.740 --> 00:05:42.590
negative 4.9 times negative 10, let me

00:05:42.590 --> 00:05:46.520
just write negative 4.9 times
negative 10, these two guys,

00:05:46.520 --> 00:05:49.010
their signs are going to cancel out.

00:05:49.010 --> 00:05:54.010
All of that, all of that
over, all of that over two a.

00:05:54.230 --> 00:05:58.390
All of that over negative 4.9
times two, so negative 9.8.

00:05:58.390 --> 00:05:59.930
And like we saw in the last video,

00:05:59.930 --> 00:06:01.610
we want a positive value for this.

00:06:01.610 --> 00:06:03.240
A negative time is nonsensical,

00:06:03.240 --> 00:06:05.070
that's kind of going into the past,

00:06:05.070 --> 00:06:07.140
so we want a positive value.

00:06:07.140 --> 00:06:09.250
And since we have a
negative in the denominator,

00:06:09.250 --> 00:06:11.570
we want to have a negative value up here,

00:06:11.570 --> 00:06:13.520
and if we already have
a negative value here,

00:06:13.520 --> 00:06:14.980
if we have subtract
from that native value,

00:06:14.980 --> 00:06:16.770
we're definitely gonna have
a negative value up here,

00:06:16.770 --> 00:06:18.310
and then you divide by a negative value,

00:06:18.310 --> 00:06:19.210
we'll get a positive value,

00:06:19.210 --> 00:06:22.550
so we can really focus on
the subtracting the radical,

00:06:22.550 --> 00:06:23.510
and you could try it out.

00:06:23.510 --> 00:06:24.990
If you tried the positive version,

00:06:24.990 --> 00:06:28.130
you're gonna get a negative
value for this entire thing.

00:06:28.130 --> 00:06:30.720
Do that on, you can try
that out after this video,

00:06:30.720 --> 00:06:32.700
just to verify that you
get a nonsensical answer.

00:06:32.700 --> 00:06:34.660
So let's use a negative right over here,

00:06:34.660 --> 00:06:39.660
so we have negative 29.54
minus the square root of,

00:06:44.150 --> 00:06:49.150
the square root of 29.54 squared,

00:06:49.220 --> 00:06:51.180
and then we have minus four,

00:06:51.180 --> 00:06:54.780
minus four, times negative 4.9
times negative 10, these two,

00:06:54.780 --> 00:06:55.613
when you take the product,

00:06:55.613 --> 00:06:58.810
it's positive 49, so times,

00:06:58.810 --> 00:07:03.810
times 49, and I should add
a parentheses over there,

00:07:04.040 --> 00:07:05.980
so let me insert a parentheses.

00:07:05.980 --> 00:07:10.980
You can insert parentheses,
okay, so times 49.

00:07:11.790 --> 00:07:13.620
So this right here will
give me the numerator

00:07:13.620 --> 00:07:16.680
if I evaluate this, got me negative value,

00:07:16.680 --> 00:07:19.470
and I divide that by negative 9.8, so

00:07:19.470 --> 00:07:24.470
divided by negative 9.8
gives me 5.67, 5.67 seconds.

00:07:28.530 --> 00:07:31.440
This is equal to 5.67 seconds,

00:07:31.440 --> 00:07:33.070
and you could keep the units in there

00:07:33.070 --> 00:07:35.340
and make sure that all of the
dimensional analysis works,

00:07:35.340 --> 00:07:37.770
and I think you'll find that it does.

00:07:37.770 --> 00:07:39.920
So our total, our total vertical,

00:07:39.920 --> 00:07:41.030
the total time in the air

00:07:41.030 --> 00:07:46.030
is 5.6, the total time in
the air is 5.67 seconds.

00:07:47.700 --> 00:07:49.120
Now, what I want to do,

00:07:49.120 --> 00:07:51.910
the whole point of this is to figure out,

00:07:51.910 --> 00:07:55.570
is to figure out how far
along this landing we land.

00:07:55.570 --> 00:07:57.000
Well, our horizontal component

00:07:57.000 --> 00:07:59.320
of our velocity is right over here.

00:07:59.320 --> 00:08:02.500
We know that, we know
that our displacement

00:08:02.500 --> 00:08:04.240
in the horizontal direction,

00:08:04.240 --> 00:08:06.610
our displacement in the
horizontal direction will be

00:08:06.610 --> 00:08:10.030
our velocity in the horizontal direction,

00:08:10.030 --> 00:08:11.180
and it's a constant velocity,

00:08:11.180 --> 00:08:12.850
so it's the same thing
as our average velocity

00:08:12.850 --> 00:08:15.890
in the horizontal direction,
times the change in time.

00:08:15.890 --> 00:08:17.360
Well, the change in time is,

00:08:17.360 --> 00:08:20.270
so I'll just that times,
our change in time.

00:08:20.270 --> 00:08:24.140
So this is going to be equal
to 30 cosine of 80 degrees,

00:08:24.140 --> 00:08:28.320
30 cosine of 80 degrees,

00:08:28.320 --> 00:08:33.320
times 5.67 seconds, times 5.67 seconds,

00:08:33.520 --> 00:08:35.030
and this, I won't write the units.

00:08:35.030 --> 00:08:36.710
This is meters per second times seconds,

00:08:36.710 --> 00:08:38.780
it'll give us an answer in meters.

00:08:38.780 --> 00:08:40.810
So once again, we have,

00:08:40.810 --> 00:08:45.310
this is our time, this is
the 5.67, times 30 cosine

00:08:46.830 --> 00:08:51.830
of 80 degrees, gives us 29.53 meters.

00:08:52.680 --> 00:08:55.620
So our total horizontal traveling,

00:08:55.620 --> 00:08:57.550
displacement I guess, we could say,

00:08:57.550 --> 00:09:01.550
is 29, I already forgot
the number, 29.53 meters.

00:09:01.550 --> 00:09:05.950
29.53 meters, and I'll
write this as a vector,

00:09:05.950 --> 00:09:08.000
that is our horizontal displacement,

00:09:08.000 --> 00:09:11.800
it's equal to 29.53 meters.

00:09:11.800 --> 00:09:15.140
Now, we've done a lot of
deconstructing vectors,

00:09:15.140 --> 00:09:16.720
what I thought would be
interesting in this video

00:09:16.720 --> 00:09:18.330
is to construct a vector.

00:09:18.330 --> 00:09:20.150
So we know our horizontal displacement.

00:09:20.150 --> 00:09:21.980
We also know our vertical displacement,

00:09:21.980 --> 00:09:25.800
it's positive 10 meters, so
what's our total displacement?

00:09:25.800 --> 00:09:27.230
Let me write this down.

00:09:27.230 --> 00:09:29.250
So we have a horizontal displacement.

00:09:29.250 --> 00:09:34.250
We have a horizontal
displacement of 29.53 meters,

00:09:34.910 --> 00:09:38.210
and we have a vertical
displacement of plus 10 meters.

00:09:38.210 --> 00:09:42.340
We have a vertical displacement
of positive 10 meters,

00:09:42.340 --> 00:09:45.360
and so what is our total displacement?

00:09:45.360 --> 00:09:48.540
What is our total
displacement going to be?

00:09:48.540 --> 00:09:51.000
Well, we could use the
Pythagorean theorem now here.

00:09:51.000 --> 00:09:52.470
The square of the magnitude

00:09:52.470 --> 00:09:54.910
of our total displacement is
going to be equal to the sum

00:09:54.910 --> 00:09:57.250
of these two squares, or another way,

00:09:57.250 --> 00:09:59.250
this is just the Pythagorean theorem,

00:09:59.250 --> 00:10:02.150
so we could have, if we call
this length over here c,

00:10:02.150 --> 00:10:03.294
or let me just write it,

00:10:03.294 --> 00:10:06.810
this is the magnitude of our
displacement right over here,

00:10:06.810 --> 00:10:10.100
the magnitude of our total
displacement squared is

00:10:10.100 --> 00:10:14.230
going to be equal to 10
squared plus 29.53 squared.

00:10:14.230 --> 00:10:19.230
So it's going to be 10
squared plus 29.53 squared.

00:10:19.640 --> 00:10:23.330
I'm gonna do that same
color, plus 29.53 squared.

00:10:26.040 --> 00:10:27.220
If we want to solve for this,

00:10:27.220 --> 00:10:30.120
we just take the square
root of both sides,

00:10:30.120 --> 00:10:31.750
so I'll just do that in place.

00:10:31.750 --> 00:10:34.810
So if we just take the
square root of both sides,

00:10:34.810 --> 00:10:38.090
we will get the magnitude
of our total displacement,

00:10:38.090 --> 00:10:40.810
and I can get the calculator
out once again to do that.

00:10:40.810 --> 00:10:42.990
So the magnitude of our total displacement

00:10:42.990 --> 00:10:46.000
is the square root of, 10 squared

00:10:46.000 --> 00:10:50.480
is just 100, plus 29, I can even use,

00:10:50.480 --> 00:10:51.950
well, I could use all of this information,

00:10:51.950 --> 00:10:53.560
so that I don't even lose some precision.

00:10:53.560 --> 00:10:55.050
I could say second answer,

00:10:55.050 --> 00:10:56.670
that literally means the previous answer,

00:10:56.670 --> 00:10:59.913
which is that 29.53 squared,

00:11:00.870 --> 00:11:05.700
gives us a total
displacement of 31.18 meters.

00:11:05.700 --> 00:11:09.330
So this is equal to 31.18 meters,

00:11:09.330 --> 00:11:11.428
and of course, it's a vector,

00:11:11.428 --> 00:11:13.410
so we've only given you the magnitude,

00:11:13.410 --> 00:11:15.070
we also need the direction.

00:11:15.070 --> 00:11:16.930
So one way to specify direction

00:11:16.930 --> 00:11:19.580
is to give you the angle
with the horizontal,

00:11:19.580 --> 00:11:22.410
and let's call that angle theta,

00:11:22.410 --> 00:11:25.550
and once again, we can use our trig,

00:11:25.550 --> 00:11:27.400
our trig functions over here.

00:11:27.400 --> 00:11:30.460
We could use pretty much
any of the trig functions,

00:11:30.460 --> 00:11:32.330
but we know the opposite side

00:11:32.330 --> 00:11:35.470
is 10, we know the hypotenuse here

00:11:35.470 --> 00:11:38.380
is 31.18, so why not use sine,

00:11:38.380 --> 00:11:40.590
sine is opposite over hypotenuse.

00:11:40.590 --> 00:11:42.740
So we know that the sine

00:11:42.740 --> 00:11:47.740
of theta is going to be
equal to 10 over 31.18.

00:11:49.330 --> 00:11:54.330
10 over 31.18, or if you
want to solve for theta,

00:11:54.950 --> 00:11:56.790
you take the arcsine of both sides,

00:11:56.790 --> 00:11:58.500
or the inverse sine of both sides,

00:11:58.500 --> 00:12:01.660
you get theta is equal to the inverse sine

00:12:01.660 --> 00:12:06.627
or I could write arcsine over
here, of 10, of 10 over 31.8.

00:12:10.727 --> 00:12:13.010
10 over 31.18, I should say.

00:12:13.010 --> 00:12:15.810
So let's once again get the
calculator out to figure out

00:12:15.810 --> 00:12:18.380
that value, so I'm gonna
take the inverse sine.

00:12:18.380 --> 00:12:20.123
Once again, so this is the
same thing as the arcsine.

00:12:20.123 --> 00:12:22.180
This just says, give me the angle

00:12:22.180 --> 00:12:23.770
that when I take its sine,

00:12:23.770 --> 00:12:25.440
I get this value right over here,

00:12:25.440 --> 00:12:28.930
so the inverse sine of 10

00:12:28.930 --> 00:12:31.890
divided by our previous answer, 31.18.

00:12:31.890 --> 00:12:36.060
I'll just say that our
previous answer is equal to,

00:12:36.060 --> 00:12:40.453
so this says, give me the angle
whose sine is 10 over 31.18.

00:12:41.350 --> 00:12:45.100
So I have 18.7 degrees,

00:12:45.100 --> 00:12:49.450
or 18.71 degrees, so this
is equal to 18.71 degrees,

00:12:51.290 --> 00:12:55.390
18.71 degrees above the horizontal.

00:12:55.390 --> 00:12:56.800
So here we've constructed a vector,

00:12:56.800 --> 00:12:58.390
we took its vertical component

00:12:58.390 --> 00:12:59.810
and its horizontal component

00:12:59.810 --> 00:13:02.120
and we were able to figure
out the total vector,

00:13:02.120 --> 00:13:05.590
so this projectile in this situation is,

00:13:05.590 --> 00:13:06.600
its total displacement,

00:13:06.600 --> 00:13:07.910
so just to make it clear,

00:13:07.910 --> 00:13:10.460
its path will look something like this,

00:13:10.460 --> 00:13:12.613
its path will look something like this.

00:13:13.630 --> 00:13:15.470
Its path is gonna look
something like this,

00:13:15.470 --> 00:13:18.250
and we've just calculated
its total displacement,

00:13:18.250 --> 00:13:21.410
its total displacement is 31.18 meters,

00:13:21.410 --> 00:13:24.760
18.71 degrees above the horizontal.

00:13:24.760 --> 00:13:25.770
And the other thing I've realized

00:13:25.770 --> 00:13:26.710
is when I started this problem,

00:13:26.710 --> 00:13:28.850
I asked you, I think I was asking you

00:13:28.850 --> 00:13:30.910
how far along the platform,

00:13:30.910 --> 00:13:34.190
and we've figured out its
total horizontal displacement,

00:13:34.190 --> 00:13:36.980
so if you wanna know how
far along the platform,

00:13:36.980 --> 00:13:38.960
the platform starts two
meters to the right,

00:13:38.960 --> 00:13:41.950
so it's really 27.53 meters along

00:13:41.950 --> 00:13:44.673
the platform is where it lands.