Launching and landing on different elevations

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.900
Let's do a slightly
more complicated

00:00:02.900 --> 00:00:05.710
two-dimensional projectile
motion problem now.

00:00:05.710 --> 00:00:11.830
So in this situation, I am going
to launch the projectile off

00:00:11.830 --> 00:00:12.470
of a platform.

00:00:15.340 --> 00:00:17.660
And then it is going to
land on another platform.

00:00:20.750 --> 00:00:27.190
And I'm going to fire the
projectile at an angle.

00:00:27.190 --> 00:00:28.970
Let me draw this a
little bit better.

00:00:28.970 --> 00:00:30.460
So I'm going to
fire the projectile

00:00:30.460 --> 00:00:35.020
at an angle of-- let me
use a not so clean number.

00:00:35.020 --> 00:00:39.920
Let's say it is a
53-degree angle.

00:00:39.920 --> 00:00:43.720
And it's coming
out of the cannon.

00:00:43.720 --> 00:00:45.960
And let me make it clear.

00:00:45.960 --> 00:00:49.100
So it's coming out--
let me do it this way,

00:00:49.100 --> 00:00:50.880
just to make it 100% clear.

00:00:53.720 --> 00:00:56.660
So this angle right
over here is 53 degrees.

00:00:59.210 --> 00:01:01.650
And we are going
to have it come out

00:01:01.650 --> 00:01:07.110
of the muzzle of the cannon
with a velocity of 90

00:01:07.110 --> 00:01:07.860
meters per second.

00:01:10.970 --> 00:01:13.430
And just to give ourselves
a sense of the heights,

00:01:13.430 --> 00:01:16.630
or how high it's being
launched from-- so

00:01:16.630 --> 00:01:19.420
from the muzzle of the
cannon down to here.

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So this height right
over here, let's say

00:01:23.550 --> 00:01:26.780
that that is 25 meters.

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And let's say that this height
right over here is 9 meters.

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And so we're essentially
launching this

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from a height of 25 meters.

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I know in the last video,
even though I drew the cannon

00:01:41.660 --> 00:01:43.050
like this, we
assumed that it was

00:01:43.050 --> 00:01:45.260
being launched from
an altitude of 0

00:01:45.260 --> 00:01:47.930
and then landing back
at an altitude of 0.

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Here we're assuming
we're launching it

00:01:50.380 --> 00:01:52.300
from an altitude of 25
meters, because that's

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when it's leaving the muzzle.

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And it's going to
start decelerating

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at least in the
vertical direction

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as soon as it leaves the muzzle.

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And then we're
assuming it's not going

00:02:01.470 --> 00:02:03.020
to land back at
the same altitude.

00:02:03.020 --> 00:02:05.490
It's going to land at
a different altitude.

00:02:05.490 --> 00:02:07.839
So how do we think
about this problem?

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So the first thing
you'll always want to do

00:02:09.630 --> 00:02:13.370
is divide your velocity
vector into its horizontal

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and vertical components.

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You use the vertical
component to figure out

00:02:16.800 --> 00:02:18.467
how long it's going
to stay in the air.

00:02:18.467 --> 00:02:20.800
And then you use the horizontal
component to figure out,

00:02:20.800 --> 00:02:23.529
given how long it's in the
air, how far did it travel.

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And once again,
we're going to assume

00:02:25.070 --> 00:02:28.120
that air resistance
is negligible.

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So just based on what we
did in the last video--

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and I'll go through all of the
steps in this one, as well.

00:02:35.740 --> 00:02:43.010
So if we draw our vector, the
length here is going to be 90.

00:02:43.010 --> 00:02:44.430
This is our velocity vector.

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The angle over here between
the x-axis and our vector

00:02:47.320 --> 00:02:49.430
is 53 degrees.

00:02:49.430 --> 00:02:51.970
And let me draw the
horizontal component.

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The horizontal component
would look like this.

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And the vertical component--
I'll do it in orange.

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The vertical component would
look like-- that's not orange.

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The vertical component
will look like this.

00:03:06.350 --> 00:03:08.930
And so the vertical
component of the vector--

00:03:08.930 --> 00:03:11.180
what would be the length of
this side right over here?

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Well, this is the opposite side.

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We know from basic
trigonometry sine

00:03:17.630 --> 00:03:20.780
of an angle is opposite
over the hypotenuse.

00:03:20.780 --> 00:03:27.490
So we know that the
sine of 53 degrees

00:03:27.490 --> 00:03:31.860
is equal to this
opposite side, is

00:03:31.860 --> 00:03:33.810
equal to the vertical velocity.

00:03:41.350 --> 00:03:43.150
And I could write
it's the magnitude

00:03:43.150 --> 00:03:44.556
of the vertical velocity.

00:03:44.556 --> 00:03:46.180
I write that subscript
y, because we're

00:03:46.180 --> 00:03:47.360
in the y direction.

00:03:47.360 --> 00:03:50.230
That's the vertical
direction, over the length

00:03:50.230 --> 00:03:53.020
of the hypotenuse,
over the magnitude

00:03:53.020 --> 00:03:54.810
of our original vector.

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Or we can get that this
side right over here.

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If we multiply both
sides by 90, we

00:04:01.370 --> 00:04:03.670
get that the
magnitude of that side

00:04:03.670 --> 00:04:10.250
is going to be equal to 90
times the sine of 53 degrees.

00:04:14.460 --> 00:04:18.190
Now, if we want to do
the horizontal component,

00:04:18.190 --> 00:04:20.700
the horizontal side
is adjacent to this.

00:04:20.700 --> 00:04:22.590
Cosine, soh cah toa.

00:04:22.590 --> 00:04:24.615
Cosine is adjacent
over hypotenuse.

00:04:27.400 --> 00:04:33.180
So the horizontal
component of our velocity,

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I'll say in the x direction,
over the hypotenuse, over 90,

00:04:39.580 --> 00:04:46.060
is equal to the
cosine of 53 degrees.

00:04:46.060 --> 00:04:48.150
Cosine is adjacent
over hypotenuse.

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Adjacent, that's
this length, over 90.

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Multiply both sides by 90.

00:04:53.460 --> 00:04:57.440
You get that the
horizontal component

00:04:57.440 --> 00:05:02.705
is equal to 90 times
cosine of 53 degrees.

00:05:07.970 --> 00:05:10.550
Now, how do we
figure out how long

00:05:10.550 --> 00:05:11.720
this thing stays in the air?

00:05:11.720 --> 00:05:13.770
Well, we'll use the
vertical component for that.

00:05:16.680 --> 00:05:19.250
And especially since we're
dealing with different levels,

00:05:19.250 --> 00:05:22.460
we can't use that more basic
reasoning, that hey, whatever

00:05:22.460 --> 00:05:24.080
velocity we start
off at, it's going

00:05:24.080 --> 00:05:26.170
to be the same velocity but
in the opposite direction.

00:05:26.170 --> 00:05:27.390
Or the same magnitude
and velocity

00:05:27.390 --> 00:05:28.570
but the opposite direction.

00:05:28.570 --> 00:05:30.860
Because we're not going
to the same elevation.

00:05:30.860 --> 00:05:33.630
But what we could do is
we can use the formula

00:05:33.630 --> 00:05:37.070
that we derived in
the previous video.

00:05:37.070 --> 00:05:39.360
That the displacement--
let me just copy and paste

00:05:39.360 --> 00:05:41.120
this a little bit
lower so we can use it.

00:05:45.550 --> 00:05:47.500
So I'll stick it
right over here.

00:05:47.500 --> 00:05:49.240
So we could use this.

00:05:49.240 --> 00:05:52.286
We know that the displacement is
equal to the initial velocity--

00:05:52.286 --> 00:05:54.160
and we're dealing with
the vertical direction

00:05:54.160 --> 00:05:56.680
right here-- times the change
in time plus the acceleration,

00:05:56.680 --> 00:05:59.480
times the change in time
squared, divided by 2.

00:05:59.480 --> 00:06:02.354
So how do we use this to figure
out how long were in the air?

00:06:02.354 --> 00:06:03.520
So what is the displacement?

00:06:03.520 --> 00:06:07.370
If we're starting
at 25 meters high,

00:06:07.370 --> 00:06:09.840
and we're going
to 9 meters high.

00:06:09.840 --> 00:06:13.740
So over the course, while
this thing is traveling,

00:06:13.740 --> 00:06:17.660
it will be displaced
downwards 16 meters.

00:06:21.320 --> 00:06:23.360
Or another way to
think about it is

00:06:23.360 --> 00:06:28.160
our displacement in
the vertical direction

00:06:28.160 --> 00:06:31.130
is going to be equal
to negative 16 meters.

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Let me write that a
little bit bigger.

00:06:32.840 --> 00:06:34.760
Negative 16 meters, right?

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Because 25 minus 9 is 16.

00:06:38.200 --> 00:06:42.060
And so we can put
that into the formula

00:06:42.060 --> 00:06:44.110
that we derived in
the previous video.

00:06:44.110 --> 00:06:47.920
We get negative 16-- I won't
write the units here just so

00:06:47.920 --> 00:06:52.680
that we don't take up
too much real estate,

00:06:52.680 --> 00:06:54.270
so that at least
it looks simple--

00:06:54.270 --> 00:06:56.404
is equal to the
initial velocity.

00:06:56.404 --> 00:06:58.570
We're dealing with just the
vertical dimension here.

00:06:58.570 --> 00:07:00.236
So we're just dealing
with the vertical.

00:07:02.244 --> 00:07:04.410
And remember, it's negative
because our displacement

00:07:04.410 --> 00:07:05.560
is going to be downwards.

00:07:05.560 --> 00:07:08.170
We're losing altitude.

00:07:08.170 --> 00:07:10.610
So our vertical velocity--
we already figured that out.

00:07:10.610 --> 00:07:13.670
It is 90 times the
sine of 53 degrees.

00:07:18.440 --> 00:07:21.109
Actually, let me do
it in that same color.

00:07:21.109 --> 00:07:22.900
The first time we do
a new type of problem,

00:07:22.900 --> 00:07:26.450
it's good to know-- so 90
times the sine of 53 degrees,

00:07:26.450 --> 00:07:33.230
times our change in time,
is equal to the acceleration

00:07:33.230 --> 00:07:36.980
of-- due to the force of gravity
for an object in free fall

00:07:36.980 --> 00:07:40.390
is going to be negative 9.8
meters per second squared.

00:07:40.390 --> 00:07:42.710
But we're dividing that by 2.

00:07:42.710 --> 00:07:50.020
So we're going to have minus 4.9
meters per second squared times

00:07:50.020 --> 00:07:50.956
delta t squared.

00:07:50.956 --> 00:07:52.330
Times our change
in time squared.

00:07:56.670 --> 00:07:58.660
So how do we solve
something like this?

00:07:58.660 --> 00:08:00.530
You can't just factor
out a t and solve it.

00:08:00.530 --> 00:08:03.200
But you might recognize
this is a quadratic equation

00:08:03.200 --> 00:08:03.880
right over here.

00:08:03.880 --> 00:08:05.588
And the way you solve
quadratic equations

00:08:05.588 --> 00:08:09.230
is you get everything onto
one side of this equation.

00:08:09.230 --> 00:08:11.320
And then you either
factor it out.

00:08:11.320 --> 00:08:12.870
But more likely
in this situation,

00:08:12.870 --> 00:08:14.810
we will use the
quadratic formula,

00:08:14.810 --> 00:08:16.630
which we've proved
in other videos

00:08:16.630 --> 00:08:20.250
and hopefully given you the
intuition for it-- to actually

00:08:20.250 --> 00:08:22.869
solve for the times where
your elevation, where

00:08:22.869 --> 00:08:24.660
your displacement in
the vertical direction

00:08:24.660 --> 00:08:26.430
is negative 16 meters.

00:08:26.430 --> 00:08:28.250
And you'll get two
solutions here.

00:08:28.250 --> 00:08:30.940
And one of the solutions will
be a negative change in time.

00:08:30.940 --> 00:08:33.580
So it'll be like at
sometime in the past,

00:08:33.580 --> 00:08:35.500
you were also at
negative 16 meters.

00:08:35.500 --> 00:08:37.280
That's nonsensical
for this problem.

00:08:37.280 --> 00:08:40.440
So we'll want to take
the positive value here.

00:08:40.440 --> 00:08:46.250
So let's put all of this on
one side of the equation.

00:08:46.250 --> 00:08:47.845
So let's add 16 to both sides.

00:08:50.790 --> 00:08:54.230
On the left-hand side,
you just get a 0.

00:08:54.230 --> 00:08:55.871
0 is equal to--
and I'll write it

00:08:55.871 --> 00:08:58.370
in kind of the traditional way
that we're used to seeing it.

00:08:58.370 --> 00:09:00.078
I'll write the highest
degree term first.

00:09:00.078 --> 00:09:03.435
So negative 4.9 times
delta t squared.

00:09:06.570 --> 00:09:25.320
And then we have plus 90 sine
of 53 degrees times delta t,

00:09:25.320 --> 00:09:26.335
and then plus 16.

00:09:28.850 --> 00:09:32.170
I'm going to do that in yellow.

00:09:32.170 --> 00:09:35.406
All of this is equal to 0.

00:09:35.406 --> 00:09:37.530
And this, once again, is
just a quadratic equation.

00:09:37.530 --> 00:09:39.560
We can find its roots.

00:09:39.560 --> 00:09:41.400
And the roots will be
in terms of delta t.

00:09:41.400 --> 00:09:45.030
We can solve for delta t
using the quadratic formula.

00:09:45.030 --> 00:09:50.540
So we get delta t-- and if
this is very unfamiliar to you,

00:09:50.540 --> 00:09:53.570
review the videos on Khan
Academy in the algebra playlist

00:09:53.570 --> 00:09:54.630
on the quadratic formula.

00:09:54.630 --> 00:09:55.550
And if you don't know
where it came from,

00:09:55.550 --> 00:09:57.210
we also prove it for you.

00:09:57.210 --> 00:10:00.950
So it's equal to negative B.
B is this term right here,

00:10:00.950 --> 00:10:03.000
the coefficient on the delta t.

00:10:03.000 --> 00:10:07.844
So it's going to be negative
90 sine of 53 degrees.

00:10:07.844 --> 00:10:09.510
I'll write the quadratic
formula up here

00:10:09.510 --> 00:10:12.870
for those of you who
don't fully remember it.

00:10:12.870 --> 00:10:16.750
So if I'm trying to
solve Ax squared plus Bx

00:10:16.750 --> 00:10:20.050
plus C is equal to 0,
the roots over here

00:10:20.050 --> 00:10:24.600
are going to be negative B plus
or minus the square root of B

00:10:24.600 --> 00:10:31.940
squared minus 4AC, all
of that over 2 times A.

00:10:31.940 --> 00:10:33.700
These are going to
be the x values that

00:10:33.700 --> 00:10:36.114
satisfy this equation up here.

00:10:36.114 --> 00:10:37.530
So that's all I'm
doing over here.

00:10:37.530 --> 00:10:38.660
This is the B value.

00:10:38.660 --> 00:10:43.159
Negative B plus or
minus-- and it's

00:10:43.159 --> 00:10:45.450
going to turn out that we
only care about the plus one,

00:10:45.450 --> 00:10:47.590
because that's going to
give us the positive value.

00:10:47.590 --> 00:10:48.923
But I'll just write it out here.

00:10:48.923 --> 00:10:53.810
Plus or minus the square
root of B squared.

00:10:53.810 --> 00:10:56.060
So it's this quantity squared.

00:10:56.060 --> 00:11:03.610
So it is 90 sine of 53
degrees squared, minus 4--

00:11:03.610 --> 00:11:05.770
we're going to run
[? out that ?] little space.

00:11:05.770 --> 00:11:16.120
So minus 4 times A,
which is negative 4.9.

00:11:16.120 --> 00:11:21.130
Well, let me just write negative
4.9 times C. C over here

00:11:21.130 --> 00:11:23.065
is 16, times 16.

00:11:23.065 --> 00:11:25.350
Let me put this radical
all the way over here.

00:11:25.350 --> 00:11:26.720
All of that over 2A.

00:11:30.360 --> 00:11:32.600
So A is negative 4.9.

00:11:32.600 --> 00:11:37.800
2 times A is negative 9.8.

00:11:37.800 --> 00:11:40.110
So now we can get
the calculator out

00:11:40.110 --> 00:11:41.604
to figure out our
change in time.

00:11:41.604 --> 00:11:44.020
And I'm just going to focus
on the positive version of it.

00:11:44.020 --> 00:11:45.810
I'll leave it up to you to
find the negative version

00:11:45.810 --> 00:11:48.101
and see that it'll give you
a negative value for change

00:11:48.101 --> 00:11:48.690
in time.

00:11:48.690 --> 00:11:50.370
And that's
nonsensical, so we only

00:11:50.370 --> 00:11:52.080
care about the
positive change in time

00:11:52.080 --> 00:11:58.290
where we get to a displacement
of negative 16 meters.

00:11:58.290 --> 00:12:00.470
Let's get the calculator out.

00:12:00.470 --> 00:12:03.770
So we get-- want to
do this carefully.

00:12:03.770 --> 00:12:09.809
We have negative 90
sine of 53 degrees plus.

00:12:09.809 --> 00:12:12.100
I'm doing the plus version
here because that'll give us

00:12:12.100 --> 00:12:17.320
the positive value--
plus the square root of.

00:12:17.320 --> 00:12:18.680
And I'll do this in parentheses.

00:12:18.680 --> 00:12:23.730
90 sine of 53 degrees squared.

00:12:23.730 --> 00:12:25.400
That's that part right there.

00:12:25.400 --> 00:12:26.830
These two negatives cancel out.

00:12:26.830 --> 00:12:29.840
So I could say this is
plus 4 times positive 4.9.

00:12:29.840 --> 00:12:37.450
So plus 4 times 4.9 times 16.

00:12:37.450 --> 00:12:40.630
And then that closes
off our entire radical.

00:12:40.630 --> 00:12:43.710
And so this will give me
the numerator up here.

00:12:43.710 --> 00:12:44.900
That gives me the numerator.

00:12:44.900 --> 00:12:53.580
And then I want to divide that
by-- did I do the negative 90?

00:12:53.580 --> 00:12:55.710
Oh, and I just realized
that I made a mistake.

00:12:55.710 --> 00:12:59.240
I said that the positive
version would give you

00:12:59.240 --> 00:13:00.040
the positive time.

00:13:00.040 --> 00:13:01.373
But now we realize that's wrong.

00:13:01.373 --> 00:13:03.960
Because when I took the positive
version, when I put a plus up

00:13:03.960 --> 00:13:07.960
here, I get a positive
2.14 for the numerator.

00:13:07.960 --> 00:13:10.810
But then we divide
it by negative 9.8.

00:13:10.810 --> 00:13:12.310
We're going to get
a negative value.

00:13:12.310 --> 00:13:14.560
So that's not going to be
the time that we care about.

00:13:14.560 --> 00:13:16.950
We care about the time where
this is a negative value.

00:13:16.950 --> 00:13:19.340
So let me re-enter that.

00:13:19.340 --> 00:13:20.860
So let me do the negative value.

00:13:20.860 --> 00:13:24.580
So let me move
back a little bit.

00:13:24.580 --> 00:13:31.979
And then let me replace
this with a minus sign.

00:13:31.979 --> 00:13:33.770
So I'm going to look
at the negative value,

00:13:33.770 --> 00:13:35.660
because I want
the positive time.

00:13:35.660 --> 00:13:38.559
And so now my numerator
here is a negative value.

00:13:38.559 --> 00:13:40.350
And so this is actually
what we care about.

00:13:40.350 --> 00:13:42.850
We care about the
numerator's a negative value.

00:13:42.850 --> 00:13:49.760
You divide by
negative 9.8, and you

00:13:49.760 --> 00:13:55.530
get 14.8-- I'll just
round-- 14.89 seconds.

00:13:55.530 --> 00:13:58.500
So delta t, the
positive version,

00:13:58.500 --> 00:14:01.920
is equal to 14.89 seconds.

00:14:01.920 --> 00:14:05.832
And so my initial comment about
using the positive version

00:14:05.832 --> 00:14:08.290
was wrong because we have this
denominator that's negative.

00:14:08.290 --> 00:14:10.640
So you want the
numerator to be negative.

00:14:10.640 --> 00:14:12.370
And only when the
numerator is negative

00:14:12.370 --> 00:14:14.520
will the whole
expression be positive.

00:14:14.520 --> 00:14:18.052
And so we got this positive
time of 14.89 seconds.

00:14:18.052 --> 00:14:19.260
I'm going to leave you there.

00:14:19.260 --> 00:14:21.094
And the next part of
the video-- actually, I

00:14:21.094 --> 00:14:23.509
might as well just solve it
instead of making a new video.

00:14:23.509 --> 00:14:24.960
Although this is running long.

00:14:24.960 --> 00:14:31.800
So the amount of time that we're
in the air is 14.89 seconds.

00:14:31.800 --> 00:14:36.192
So if I were to ask you the
horizontal displacement,

00:14:36.192 --> 00:14:37.650
it's going to be
the amount of time

00:14:37.650 --> 00:14:42.130
we're in the air times your
constant horizontal velocity.

00:14:42.130 --> 00:14:45.090
And we already figured out our
constant horizontal velocity.

00:14:45.090 --> 00:14:49.700
So if you want to figure
out how far along the x-axis

00:14:49.700 --> 00:14:54.210
we get displaced, we
just take this time

00:14:54.210 --> 00:14:57.730
times-- that just means
our previous answer--

00:14:57.730 --> 00:15:04.050
times this value right here,
times 90 cosine of 53 degrees.

00:15:04.050 --> 00:15:06.760
And that gives 806 meters.

00:15:06.760 --> 00:15:11.323
So this displacement right
over here is 806 meters.


Different way to determine time in air

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.000
In the last video, where we
launched something at an angle,

00:00:03.000 --> 00:00:05.910
we figured out the horizontal
and the vertical components

00:00:05.910 --> 00:00:07.425
of that launched velocity.

00:00:07.425 --> 00:00:09.460
And we used a vertical
component to figure out

00:00:09.460 --> 00:00:11.630
how long that thing
is in the air.

00:00:11.630 --> 00:00:13.950
And then we used that
time to figure out

00:00:13.950 --> 00:00:17.444
how far it will travel, given
a constant horizontal velocity.

00:00:17.444 --> 00:00:18.860
What I want to do
in this video is

00:00:18.860 --> 00:00:20.890
to redo part of that problem.

00:00:20.890 --> 00:00:23.300
And I want to redo
the part to figure out

00:00:23.300 --> 00:00:26.750
how long this object
stays in the air.

00:00:26.750 --> 00:00:29.490
And I'm going to do it in a
slightly more complicated way

00:00:29.490 --> 00:00:31.380
than the way I did
in the last video,

00:00:31.380 --> 00:00:33.590
but it's kind of a
more powerful way.

00:00:33.590 --> 00:00:35.924
And we'll see that it's more
powerful in future videos.

00:00:35.924 --> 00:00:38.340
So the last video, to figure
out how long it's in the air,

00:00:38.340 --> 00:00:38.923
we said, look.

00:00:38.923 --> 00:00:41.300
We're starting and stopping
at the same elevation.

00:00:41.300 --> 00:00:43.945
Whatever our vertical
velocity is here upward,

00:00:43.945 --> 00:00:46.070
we're going to have the
same magnitude of velocity,

00:00:46.070 --> 00:00:47.440
but it's going to be downward.

00:00:47.440 --> 00:00:51.294
We used that insight to figure
out the change in velocity,

00:00:51.294 --> 00:00:52.960
and then we used that
change in velocity

00:00:52.960 --> 00:00:56.570
and the constant acceleration
due to the force of gravity

00:00:56.570 --> 00:00:59.264
on an object in free fall, to
figure out the time in the air.

00:00:59.264 --> 00:01:00.930
And it was pretty
simple mathematically.

00:01:00.930 --> 00:01:06.292
What we're going to do now
is, to derive a formula that's

00:01:06.292 --> 00:01:07.750
useful in physics,
although, I like

00:01:07.750 --> 00:01:09.126
to always derive
it from scratch.

00:01:09.126 --> 00:01:11.208
And that's how we're going
to derive it over here.

00:01:11.208 --> 00:01:13.167
And you've see me derive
this formula before.

00:01:13.167 --> 00:01:15.250
But just to start off, we
need to remind ourselves

00:01:15.250 --> 00:01:17.875
that the vertical velocity that
we calculated in the last video

00:01:17.875 --> 00:01:19.690
was five meters per second.

00:01:19.690 --> 00:01:21.490
And given that
vertical velocity,

00:01:21.490 --> 00:01:24.550
let's figure out how long
this thing stays in the air.

00:01:24.550 --> 00:01:28.850
So our vertical velocity, our
velocity in the y direction,

00:01:28.850 --> 00:01:31.010
is five meters per second.

00:01:31.010 --> 00:01:33.150
And also as a reminder,
our convention,

00:01:33.150 --> 00:01:35.650
when we're dealing with
vectors in one dimension,

00:01:35.650 --> 00:01:37.700
and this is essentially
in one dimension

00:01:37.700 --> 00:01:40.450
now, since we know it's
in the y direction,

00:01:40.450 --> 00:01:43.442
is positive means up,
negative means down,

00:01:43.442 --> 00:01:45.400
when we're dealing with
the vertical direction.

00:01:45.400 --> 00:01:47.441
When we're dealing with
the horizontal direction,

00:01:47.441 --> 00:01:51.110
positive means to the right,
negative means to the left.

00:01:51.110 --> 00:01:54.136
So just to make sure we
have those conventions down.

00:01:54.136 --> 00:01:55.510
So how can we
figure out how long

00:01:55.510 --> 00:01:57.130
this thing stays in the air?

00:01:57.130 --> 00:02:00.190
So we can remind ourselves
that the displacement,

00:02:00.190 --> 00:02:02.010
and everything we're
talking about now,

00:02:02.010 --> 00:02:03.480
is, well, this
will actually apply

00:02:03.480 --> 00:02:05.490
whether in any one dimension.

00:02:05.490 --> 00:02:15.160
But the displacement is
equal to the average velocity

00:02:15.160 --> 00:02:17.545
times our change in time.

00:02:17.545 --> 00:02:20.110
Times the amount of
time that elapsed.

00:02:20.110 --> 00:02:22.180
And what is our
average velocity?

00:02:22.180 --> 00:02:26.910
Our average velocity is-- if we
assume constant acceleration--

00:02:26.910 --> 00:02:33.384
it is our final velocity
plus our initial velocity.

00:02:33.384 --> 00:02:34.800
Actually let me
write it this way,

00:02:34.800 --> 00:02:36.430
it'll make it a
little bit cleaner.

00:02:36.430 --> 00:02:42.650
It is our initial velocity
plus our final velocity divided

00:02:42.650 --> 00:02:43.150
by 2.

00:02:43.150 --> 00:02:45.870
So literally the arithmetic
mean of our initial

00:02:45.870 --> 00:02:47.240
and final velocities.

00:02:47.240 --> 00:02:52.040
And we're going to multiply
that times change in time.

00:02:52.040 --> 00:02:54.750
Now, what is our final
velocity going to be?

00:02:54.750 --> 00:02:57.410
Our final velocity--
I'll write it over here.

00:02:57.410 --> 00:03:09.176
Our final velocity is the same
thing as your initial velocity,

00:03:09.176 --> 00:03:13.520
plus acceleration,
your acceleration

00:03:13.520 --> 00:03:17.360
times how much
time has passed by.

00:03:17.360 --> 00:03:23.250
This right here is
your final velocity.

00:03:23.250 --> 00:03:27.460
So let's substitute that back
into this, right over here,

00:03:27.460 --> 00:03:29.030
and what do we get?

00:03:29.030 --> 00:03:33.330
We get our initial velocity,
so our average velocity

00:03:33.330 --> 00:03:36.080
is now our initial velocity
plus our final velocity.

00:03:36.080 --> 00:03:38.720
The final velocity
is this over here.

00:03:38.720 --> 00:03:42.940
So it's our initial velocity
plus the acceleration

00:03:42.940 --> 00:03:47.300
times the amount of
time that passes by.

00:03:47.300 --> 00:03:49.130
All of that divided
by-- let me do

00:03:49.130 --> 00:03:51.540
that same color-- all
of that divided by 2.

00:03:54.070 --> 00:03:55.710
And then all of
this is multiplied

00:03:55.710 --> 00:03:58.270
by the amount of
time that passes by.

00:03:58.270 --> 00:04:02.030
Once again, this is
our total displacement.

00:04:02.030 --> 00:04:06.870
So this part right over here--
let me do it in magenta--

00:04:06.870 --> 00:04:08.210
we have to vi's.

00:04:08.210 --> 00:04:09.050
All right.

00:04:09.050 --> 00:04:11.930
So that simplifies to two vi's.

00:04:11.930 --> 00:04:13.960
2 times our initial velocities.

00:04:13.960 --> 00:04:17.000
And so this expression right
here simplifies to two times vi

00:04:17.000 --> 00:04:19.194
divided by 2 is our
initial velocity.

00:04:21.740 --> 00:04:25.440
And then we have-- well,
I'll just do it all here.

00:04:25.440 --> 00:04:27.920
And then I'll multiply it times
the time elapse after that.

00:04:27.920 --> 00:04:33.260
Plus the acceleration
times time,

00:04:33.260 --> 00:04:35.000
the acceleration times
the amount of time

00:04:35.000 --> 00:04:37.840
that's passed by, divided by 2.

00:04:37.840 --> 00:04:40.150
And all of this is multiplied
by the amount of time

00:04:40.150 --> 00:04:41.830
that passes by.

00:04:41.830 --> 00:04:44.390
And this is our
total displacement.

00:04:44.390 --> 00:04:46.520
And we're almost done.

00:04:46.520 --> 00:04:48.740
And if you have to take
a physics exam under time

00:04:48.740 --> 00:04:50.820
pressure, it's not
ridiculous to not

00:04:50.820 --> 00:04:52.310
have to rederive it every time.

00:04:52.310 --> 00:04:54.020
But I want to emphasize it.

00:04:54.020 --> 00:04:57.390
You should realize that these
formulas come from almost just

00:04:57.390 --> 00:04:59.871
common, basic sense
principles, and you

00:04:59.871 --> 00:05:01.120
should be able to derive them.

00:05:01.120 --> 00:05:03.860
But it's good to memorize them
if you have a time pressure

00:05:03.860 --> 00:05:04.360
exam.

00:05:04.360 --> 00:05:06.930
So now if we multiply
this delta t,

00:05:06.930 --> 00:05:08.760
you get the total
displacement is

00:05:08.760 --> 00:05:16.240
equal to your initial velocity
times your change in time,

00:05:16.240 --> 00:05:22.720
plus-- I'll do the same
color-- plus your acceleration

00:05:22.720 --> 00:05:25.740
over 2 times-- you have a
delta t times a delta t.

00:05:25.740 --> 00:05:27.490
Remember, delta just
means change in time.

00:05:27.490 --> 00:05:29.450
This triangle just
means change in time.

00:05:29.450 --> 00:05:31.400
So change in time
times change in time

00:05:31.400 --> 00:05:32.890
is change in time squared.

00:05:36.089 --> 00:05:38.630
And this expression right here,
this is in some physics books

00:05:38.630 --> 00:05:41.490
or in some classes, they'll
just give you this formula.

00:05:41.490 --> 00:05:43.640
But I wanted to just
show you that it's

00:05:43.640 --> 00:05:46.030
derived from very,
very simple principles.

00:05:46.030 --> 00:05:49.200
From just, displacement is
average velocity times time.

00:05:49.200 --> 00:05:50.770
What is average velocity?

00:05:50.770 --> 00:05:52.090
What is final velocity?

00:05:52.090 --> 00:05:55.050
It's just initial velocity
plus acceleration times time.

00:05:55.050 --> 00:05:57.550
And you'll get this
expression right over here.

00:05:57.550 --> 00:05:59.937
So I did this to figure
out how much time we're

00:05:59.937 --> 00:06:01.020
going to spend in the air.

00:06:01.020 --> 00:06:03.640
So how do we do that for
this particular problem?

00:06:03.640 --> 00:06:05.767
Well, we know our
initial velocity,

00:06:05.767 --> 00:06:07.600
that's the launch of
the vertical component,

00:06:07.600 --> 00:06:09.260
because we're dealing with just
the vertical component here.

00:06:09.260 --> 00:06:11.020
We know our initial velocity.

00:06:11.020 --> 00:06:13.600
We know the acceleration
due to gravity.

00:06:13.600 --> 00:06:15.950
And in this situation,
we want to figure out

00:06:15.950 --> 00:06:17.640
when the displacement
is going to be--

00:06:17.640 --> 00:06:19.350
and this is going to be
a little bit unintuitive,

00:06:19.350 --> 00:06:20.320
but hopefully it
makes sense-- when

00:06:20.320 --> 00:06:21.737
the displacement
is going to be 0.

00:06:21.737 --> 00:06:24.320
Because there's going to be two
times when the displacement is

00:06:24.320 --> 00:06:25.270
going to be 0.

00:06:25.270 --> 00:06:28.330
There's going to be the time
right when we launch it,

00:06:28.330 --> 00:06:31.090
right at time 0 the
displacement is going to be 0.

00:06:31.090 --> 00:06:34.470
Then this thing's going to be
launched and then come back

00:06:34.470 --> 00:06:35.420
to the ground.

00:06:35.420 --> 00:06:37.170
And then when it comes
back to the ground,

00:06:37.170 --> 00:06:39.160
the displacement is
going to be at 0 again.

00:06:39.160 --> 00:06:42.080
So if we set the displacement
to be equal to 0,

00:06:42.080 --> 00:06:45.080
we should get two answers
to that, and one of them

00:06:45.080 --> 00:06:47.280
should have a non-zero
time component.

00:06:47.280 --> 00:06:50.060
One of them will probably
be 0, and the other one

00:06:50.060 --> 00:06:52.820
will have a non-zero
time, and that's

00:06:52.820 --> 00:06:54.230
the one that we should go for.

00:06:54.230 --> 00:06:56.410
Because that's the
second time, that's

00:06:56.410 --> 00:06:58.390
the time when we've
hit the ground.

00:06:58.390 --> 00:07:00.010
So let's apply that.

00:07:00.010 --> 00:07:01.510
So we have the
displacement is going

00:07:01.510 --> 00:07:05.540
to be 0, so 0 is equal to--
our initial velocity is

00:07:05.540 --> 00:07:07.460
5 meters per second upwards.

00:07:07.460 --> 00:07:10.100
I won't write the units
just for the sake of space.

00:07:10.100 --> 00:07:16.670
5 meters per second upwards
times change in time, plus,

00:07:16.670 --> 00:07:18.310
what's our acceleration?

00:07:18.310 --> 00:07:21.630
Well, our convention is if
somethings going downwards,

00:07:21.630 --> 00:07:24.830
if the one dimensional vector is
going downwards, it's negative.

00:07:24.830 --> 00:07:29.100
So the acceleration is negative
9.8 meters per second squared.

00:07:29.100 --> 00:07:30.970
You divide that by 2.

00:07:30.970 --> 00:07:34.030
So we get-- actually just
let me write it all out.

00:07:34.030 --> 00:07:35.540
Actually let me
write it like this.

00:07:35.540 --> 00:07:38.400
It's negative 9.8 meters
per second squared.

00:07:38.400 --> 00:07:41.010
I'm not writing the units
here just for simplicity,

00:07:41.010 --> 00:07:43.680
just so that we don't
take up too much space.

00:07:43.680 --> 00:07:47.570
And then all of this
times delta t squared.

00:07:47.570 --> 00:07:49.860
And lucky for us, this
one is pretty simple,

00:07:49.860 --> 00:07:52.264
we don't have to resort to
factoring, or we will factor,

00:07:52.264 --> 00:07:54.430
but we don't have to resort
to the quadratic formula

00:07:54.430 --> 00:07:57.510
or anything like that to solve
this quadratic equation, right

00:07:57.510 --> 00:07:58.010
over here.

00:07:58.010 --> 00:07:59.270
We can factor out a delta t.

00:07:59.270 --> 00:08:07.650
So we get 0 is equal to delta
t times 5 minus 9.8 over 4.2

00:08:07.650 --> 00:08:10.390
is 4.9.

00:08:10.390 --> 00:08:12.890
And if you factor out a delta
t out of delta t squared,

00:08:12.890 --> 00:08:16.090
you're left with a delta t.

00:08:16.090 --> 00:08:18.880
So there's two
solutions here, There's

00:08:18.880 --> 00:08:20.630
two delta t's that
will give us 0.

00:08:20.630 --> 00:08:22.570
Either this delta
t is equal to 0,

00:08:22.570 --> 00:08:24.780
so either delta
t is equal to 0--

00:08:24.780 --> 00:08:26.140
that's just our starting point.

00:08:26.140 --> 00:08:28.570
So that's not the time
that we're interested in,

00:08:28.570 --> 00:08:30.760
where no time has passed by.

00:08:30.760 --> 00:08:33.551
Or all of this business
could be equal to 0.

00:08:33.551 --> 00:08:35.550
If we have two things
when we take their product

00:08:35.550 --> 00:08:38.419
that equal 0, that means
either one or both of them

00:08:38.419 --> 00:08:39.600
are equal to 0.

00:08:39.600 --> 00:08:48.350
So we have 5 minus 4.9
delta t is equal to 0.

00:08:48.350 --> 00:08:52.360
We could add 4.9 delta t to
both sides of this equation.

00:08:52.360 --> 00:08:56.720
And we will get 5 is
equal to 4.9 delta t.

00:09:00.730 --> 00:09:08.720
And then we divide both sides
by 4.9 to get the elapsed time.

00:09:08.720 --> 00:09:12.900
So delta t is equal
to, 5 divided by 4.9

00:09:12.900 --> 00:09:19.790
is-- we're at the home
stretch-- 5 divided by 4.9

00:09:19.790 --> 00:09:22.530
is equal to 1.02 seconds.

00:09:22.530 --> 00:09:26.050
So it's equal to 1.02 seconds.

00:09:26.050 --> 00:09:28.177
And so once again, we
got the same exact answer

00:09:28.177 --> 00:09:30.510
we got in the last one, we
just used a different method.

00:09:30.510 --> 00:09:32.960
And that's one of the really
fun things about physics

00:09:32.960 --> 00:09:35.050
is as long as you're
doing logically

00:09:35.050 --> 00:09:36.890
consistent correct
things, there's

00:09:36.890 --> 00:09:39.150
many ways to often
get that answer,

00:09:39.150 --> 00:09:41.070
but it should be
the same answer.

00:09:41.070 --> 00:09:43.390
If it's not, you
probably did something

00:09:43.390 --> 00:09:47.120
wrong in one of those
two ways of doing it.


Projectile at an angle

WEBVTT
Kind: captions
Language: en

00:00:00.767 --> 00:00:02.045
- [Voiceover] So I've got a rocket here.

00:00:02.045 --> 00:00:05.377
And this rocket is going
to launch a projectile,

00:00:05.377 --> 00:00:07.424
maybe it's a rock of some kind,

00:00:07.424 --> 00:00:10.641
with the velocity of
ten meters per second.

00:00:10.641 --> 00:00:13.340
And the direction of that velocity

00:00:13.340 --> 00:00:15.239
is going to be be 30 degrees,

00:00:15.239 --> 00:00:17.856
30 degrees upwards from the horizontal.

00:00:17.856 --> 00:00:20.347
Or the angle between the
direction of the launch

00:00:20.347 --> 00:00:22.777
and horizontal is 30 degrees.

00:00:22.777 --> 00:00:24.221
And what we want to
figure out in this video

00:00:24.221 --> 00:00:26.798
is how far does the rock travel?

00:00:26.798 --> 00:00:29.678
We want to figure out how,

00:00:29.678 --> 00:00:31.595
how far does it travel?

00:00:32.859 --> 00:00:34.109
Does it travel?

00:00:35.819 --> 00:00:37.529
And to simplify this problem,

00:00:37.529 --> 00:00:38.362
what we're gonna do is

00:00:38.362 --> 00:00:40.484
we're gonna break down
this velocity vector

00:00:40.484 --> 00:00:43.309
into its vertical and
horizontal components.

00:00:43.309 --> 00:00:44.888
We're going to use a vertical component,

00:00:44.888 --> 00:00:46.513
so let me just draw it visually.

00:00:46.513 --> 00:00:47.976
So this velocity vector can be

00:00:47.976 --> 00:00:50.458
broken down into its vertical

00:00:50.458 --> 00:00:52.958
and its horizontal components.

00:00:53.953 --> 00:00:56.120
And its horizontal components.

00:00:56.120 --> 00:00:58.233
So we're gonna get some
vertical component,

00:00:58.233 --> 00:01:01.356
some amount of velocity
in the upwards direction,

00:01:01.356 --> 00:01:02.261
and we can figure,

00:01:02.261 --> 00:01:04.223
we can use that to figure out

00:01:04.223 --> 00:01:06.813
how long will this rock stay in the air.

00:01:06.813 --> 00:01:07.776
Because it doesn't matter

00:01:07.776 --> 00:01:09.580
what its horizontal component is.

00:01:09.580 --> 00:01:11.855
Its vertical component is gonna determine

00:01:11.855 --> 00:01:14.142
how quickly it decelerates due to gravity

00:01:14.142 --> 00:01:15.884
and then re-accelerated, and essentially

00:01:15.884 --> 00:01:18.049
how long it's going to be the air.

00:01:18.049 --> 00:01:20.626
And once we figure out
how long it's in the air,

00:01:20.626 --> 00:01:22.971
we can multiply it by,

00:01:22.971 --> 00:01:24.320
we can multiply it by the

00:01:24.320 --> 00:01:26.923
horizontal component of the velocity,

00:01:26.923 --> 00:01:29.338
and that will tell us how far it travels.

00:01:29.338 --> 00:01:30.883
And, once again, the assumption

00:01:30.883 --> 00:01:31.962
that were making this videos

00:01:31.962 --> 00:01:34.679
is that air resistance is negligible.

00:01:34.679 --> 00:01:37.837
Obviously, if there was
significant air resistance,

00:01:37.837 --> 00:01:40.634
this horizontal velocity
would not stay constant

00:01:40.634 --> 00:01:42.144
while it's traveling through the air.

00:01:42.144 --> 00:01:43.452
But we're going to assume that it does,

00:01:43.452 --> 00:01:44.925
that this does not change,

00:01:44.925 --> 00:01:45.758
that it is negligible.

00:01:45.758 --> 00:01:47.562
We can assume that were
doing this experiment

00:01:47.562 --> 00:01:49.278
on the moon if we wanted to have a,

00:01:49.278 --> 00:01:51.245
if we wanted to view it in purer terms.

00:01:51.245 --> 00:01:53.114
But let's solve the problem.

00:01:53.114 --> 00:01:54.113
So the first that we want to do

00:01:54.113 --> 00:01:57.247
is we wanna break down
this velocity vector.

00:01:57.247 --> 00:01:59.105
We want to break down this velocity vector

00:01:59.105 --> 00:02:02.031
that has a magnitude of
ten meters per second.

00:02:02.031 --> 00:02:05.815
And has an angle of 30
degrees with the horizontal.

00:02:05.815 --> 00:02:08.149
We want to break it down it
with x- and y-components,

00:02:08.149 --> 00:02:10.750
or its horizontal and vertical components.

00:02:10.750 --> 00:02:13.307
so that's its horizontal,

00:02:13.307 --> 00:02:14.742
let me draw a little bit better,

00:02:14.742 --> 00:02:16.808
that's its horizontal component,

00:02:16.808 --> 00:02:19.502
and that its vertical
component looks like this.

00:02:19.502 --> 00:02:21.662
This is its vertical component.

00:02:21.662 --> 00:02:24.074
So let's do the vertical component first.

00:02:24.074 --> 00:02:26.082
So how do we figure out
the vertical component

00:02:26.082 --> 00:02:29.483
given that we know the
hypotenuse of this right triangle

00:02:29.483 --> 00:02:31.469
and we know this angle right over here.

00:02:31.469 --> 00:02:34.325
And the angle, and the side,
this vertical component,

00:02:34.325 --> 00:02:36.147
or the length of that vertical component,

00:02:36.147 --> 00:02:37.703
or the magnitude of it,

00:02:37.703 --> 00:02:40.245
is opposite the angle.

00:02:40.245 --> 00:02:41.743
So we want to figure out the opposite.

00:02:41.743 --> 00:02:44.471
We have to hypotenuse, so once again

00:02:44.471 --> 00:02:46.221
we write down so-cah,

00:02:47.530 --> 00:02:49.191
so-ca-toh-ah.

00:02:49.191 --> 00:02:52.338
Sin is opposite over hypotenuse.

00:02:52.338 --> 00:02:54.521
So we know that the sin,

00:02:54.521 --> 00:02:56.354
the sin of 30 degrees,

00:02:57.424 --> 00:02:59.513
the sin of 30 degrees,

00:02:59.513 --> 00:03:02.045
is going to be equal to the magnitude

00:03:02.045 --> 00:03:04.181
of our vertical component.

00:03:04.181 --> 00:03:07.750
So this is the magnitude of velocity,

00:03:07.750 --> 00:03:10.193
I'll say the velocity in the y direction.

00:03:10.193 --> 00:03:11.480
That's the vertical direction,

00:03:11.480 --> 00:03:13.546
y is the upwards direction.

00:03:13.546 --> 00:03:17.505
Is equal to the magnitude of our velocity

00:03:17.505 --> 00:03:20.710
of the velocity in the y direction.

00:03:20.710 --> 00:03:24.984
Divided by the magnitude
of the hypotenuse,

00:03:24.984 --> 00:03:26.935
or the magnitude of our original vector.

00:03:26.935 --> 00:03:29.970
Divided by ten meters per second.

00:03:29.970 --> 00:03:31.803
Ten meters per second.

00:03:32.931 --> 00:03:34.710
And then, to solve for this quantity

00:03:34.710 --> 00:03:35.924
right over here,

00:03:35.924 --> 00:03:38.165
we multiply both sides by 10.

00:03:38.165 --> 00:03:41.076
And you get 10, sin of 30.

00:03:41.076 --> 00:03:42.909
10, sin of 30 degrees.

00:03:44.840 --> 00:03:47.314
10 sin of 30 degrees is going to be

00:03:47.314 --> 00:03:50.221
equal to the magnitude of our,

00:03:50.221 --> 00:03:53.425
the magnitude of our vertical component.

00:03:53.425 --> 00:03:56.397
And so what is the sin of 30 degrees?

00:03:56.397 --> 00:03:58.344
And this, you might have memorized this

00:03:58.344 --> 00:04:00.096
from your basic trigonometry class.

00:04:00.096 --> 00:04:01.993
You can get the calculator
out if you want,

00:04:01.993 --> 00:04:04.037
but sin of 30 degrees is
pretty straightforward.

00:04:04.037 --> 00:04:06.858
It is 1/2. So sin of 30 degrees,

00:04:06.858 --> 00:04:09.262
use a calculator if you
don't remember that, or

00:04:09.262 --> 00:04:10.214
you remember it now so

00:04:10.214 --> 00:04:12.261
sin of 30 degrees is 1/2.

00:04:12.261 --> 00:04:14.186
And so 10 times 1/2

00:04:14.186 --> 00:04:16.508
is going to be five.

00:04:16.508 --> 00:04:17.785
So, and I forgot the units there,

00:04:17.785 --> 00:04:19.729
so it's five meters per second.

00:04:19.729 --> 00:04:21.893
Is equal to the magnitude,

00:04:21.893 --> 00:04:25.876
is equal to the magnitude
of our vertical component.

00:04:25.876 --> 00:04:27.396
Let me get that in the right color.

00:04:27.396 --> 00:04:31.417
It's equal to the magnitude
of our vertical component.

00:04:31.417 --> 00:04:32.392
So what does that do?

00:04:32.392 --> 00:04:33.225
What we're,

00:04:33.225 --> 00:04:35.973
this projectile, because
vertical component

00:04:35.973 --> 00:04:37.529
is five meters per second,

00:04:37.529 --> 00:04:40.153
it will stay in the air
the same amount of time

00:04:40.153 --> 00:04:42.351
as anything that has a vertical

00:04:42.351 --> 00:04:44.014
component of five meters per second.

00:04:44.014 --> 00:04:47.459
If you threw a rock or
projectile straight up

00:04:47.459 --> 00:04:50.397
at a velocity five meters per second,

00:04:50.397 --> 00:04:52.124
that rocket projectile will stay

00:04:52.124 --> 00:04:54.462
up in the air as long as this one here

00:04:54.462 --> 00:04:56.946
because they have the
same vertical component.

00:04:56.946 --> 00:05:00.070
So let's think about how
long it will stay in the air.

00:05:00.070 --> 00:05:01.567
Since were dealing with a situation

00:05:01.567 --> 00:05:03.448
where we're starting in the ground

00:05:03.448 --> 00:05:04.946
and we're also finishing

00:05:04.946 --> 00:05:06.652
at the same elevation,

00:05:06.652 --> 00:05:09.067
and were assuming the air
resistance is negligible,

00:05:09.067 --> 00:05:10.983
we can do a little bit
of a simplification here.

00:05:10.983 --> 00:05:12.144
Although I'll do another version where

00:05:12.144 --> 00:05:13.440
we're doing the more complicated,

00:05:13.440 --> 00:05:17.137
but I guess the way that
applies to more situations.

00:05:17.137 --> 00:05:18.220
We could say,

00:05:18.220 --> 00:05:19.394
we could say "well what is our

00:05:19.394 --> 00:05:21.469
"change in velocity here?"

00:05:21.469 --> 00:05:23.431
So if we think about just
the vertical velocity,

00:05:23.431 --> 00:05:25.114
our initial velocity,

00:05:25.114 --> 00:05:26.368
let me write it this way.

00:05:26.368 --> 00:05:29.050
Our initial velocity, and we're talking,

00:05:29.050 --> 00:05:30.471
let me label all of this.

00:05:30.471 --> 00:05:32.062
So we're talking only in the vertical.

00:05:32.062 --> 00:05:33.385
Let me do all the vertical stuff

00:05:33.385 --> 00:05:34.660
that we wrote in blue.

00:05:34.660 --> 00:05:37.655
So vertical, were dealing
with the vertical here.

00:05:37.655 --> 00:05:40.755
So our initial velocity,
in the vertical direction,

00:05:40.755 --> 00:05:42.833
our initial velocity in
the vertical direction

00:05:42.833 --> 00:05:45.655
is going to be five meters per second.

00:05:45.655 --> 00:05:47.803
Is going to be five meters per second.

00:05:47.803 --> 00:05:50.625
And we're going to use
a convention, that up,

00:05:50.625 --> 00:05:55.259
that up is positive and
that down is negative.

00:05:55.259 --> 00:05:57.697
And now what is going to
be our final velocity?

00:05:57.697 --> 00:05:59.055
We're going to be going up

00:05:59.055 --> 00:06:01.017
and would be decelerated by gravity,

00:06:01.017 --> 00:06:02.887
We're gonna be stationary at some point.

00:06:02.887 --> 00:06:05.661
And then were to start
accelerating back down.

00:06:05.661 --> 00:06:08.599
And, if we assume that air
resistance is negligible,

00:06:08.599 --> 00:06:11.478
when we get back to ground level,

00:06:11.478 --> 00:06:13.776
we will have the same
magnitude of velocity

00:06:13.776 --> 00:06:15.983
but will be going in
the opposite direction.

00:06:15.983 --> 00:06:17.237
So our final velocity, remember,

00:06:17.237 --> 00:06:18.096
we're just talking about the

00:06:18.096 --> 00:06:19.911
vertical component right now.

00:06:19.911 --> 00:06:21.461
We haven't even thought
about the horizontal.

00:06:21.461 --> 00:06:22.585
We're just trying to figure out

00:06:22.585 --> 00:06:25.603
how long does this thing stay in the air?

00:06:25.603 --> 00:06:29.121
So its final velocity is
going to be negative five.

00:06:29.121 --> 00:06:30.886
Negative five

00:06:30.886 --> 00:06:32.069
meters per second.

00:06:32.069 --> 00:06:33.219
And this is initial velocity,

00:06:33.219 --> 00:06:35.704
the final velocity is going
to be looking like that.

00:06:35.704 --> 00:06:38.305
Same magnitude, just in
the opposite direction.

00:06:38.305 --> 00:06:39.825
So what's our change in velocity

00:06:39.825 --> 00:06:41.381
in the vertical direction?

00:06:41.381 --> 00:06:45.189
Change in velocity, in
the vertical direction,

00:06:45.189 --> 00:06:47.397
or in the y-direction, is going to be

00:06:47.397 --> 00:06:51.994
our final velocity, negative
five meters per second,

00:06:51.994 --> 00:06:53.550
minus our initial velocity,

00:06:53.550 --> 00:06:55.895
minus five meters per second,

00:06:55.895 --> 00:06:58.710
which is equal to negative 10

00:06:58.710 --> 00:07:00.646
meters per second.

00:07:00.646 --> 00:07:02.692
So how do we use this information

00:07:02.692 --> 00:07:05.051
to figure out how long it's in the air?

00:07:05.051 --> 00:07:06.340
Well we know!

00:07:06.340 --> 00:07:10.055
We know that our vertical, our change

00:07:10.055 --> 00:07:13.840
our change in our, in
our vertical velocity,

00:07:13.840 --> 00:07:15.918
is going to be the same thing

00:07:15.918 --> 00:07:17.518
or it's equal to

00:07:17.518 --> 00:07:19.135
our acceleration

00:07:19.135 --> 00:07:20.842
in the vertical direction

00:07:20.842 --> 00:07:22.911
times the change in time.

00:07:22.911 --> 00:07:25.767
Times the amount of time that passes by.

00:07:25.767 --> 00:07:28.663
What's our acceleration
in the vertical direction?

00:07:28.663 --> 00:07:30.602
What's the acceleration due to gravity,

00:07:30.602 --> 00:07:32.204
or acceleration that gravity,

00:07:32.204 --> 00:07:35.446
that the force of gravity
has an object in freefall?

00:07:35.446 --> 00:07:37.281
and so this, right here, is going to be

00:07:37.281 --> 00:07:41.321
negative 9.8 meters per second squared.

00:07:41.321 --> 00:07:42.609
So this quantity over here is

00:07:42.609 --> 00:07:44.084
negative 10 meters per second,

00:07:44.084 --> 00:07:45.048
we figured that out,

00:07:45.048 --> 00:07:47.079
that's gonna be the change in velocity.

00:07:47.079 --> 00:07:49.889
Negative 10 meters per second

00:07:49.889 --> 00:07:51.607
is going to be equal to

00:07:51.607 --> 00:07:53.267
negative 9.8,

00:07:53.267 --> 00:07:56.286
negative 9.8 meters per second squared

00:07:56.286 --> 00:07:58.922
times our change in time.

00:07:58.922 --> 00:08:00.584
So to figure out the total amount of time

00:08:00.584 --> 00:08:01.479
that we are the air,

00:08:01.479 --> 00:08:03.701
we just divide both sides by

00:08:03.701 --> 00:08:06.932
negative 9.8 meters per second squared.

00:08:06.932 --> 00:08:08.219
So we get,

00:08:08.219 --> 00:08:09.052
lets just do that,

00:08:09.052 --> 00:08:11.297
I wanna do that in the same color.

00:08:11.297 --> 00:08:12.850
So I do it in,

00:08:12.850 --> 00:08:14.174
that's not, well, that close enough.

00:08:14.174 --> 00:08:16.269
So we get negative 9.8

00:08:16.269 --> 00:08:18.731
meters per second squared.

00:08:18.731 --> 00:08:22.085
Negative 9.8 meters per second squared.

00:08:22.085 --> 00:08:23.247
That cancels out,

00:08:23.247 --> 00:08:25.325
and I get my change in time.

00:08:25.325 --> 00:08:26.625
And I'll just get the calculator.

00:08:26.625 --> 00:08:28.123
I have a negative divided by a negative

00:08:28.123 --> 00:08:29.365
so that's a positive, which is good,

00:08:29.365 --> 00:08:31.246
because we want to go in positive time.

00:08:31.246 --> 00:08:32.930
We assume that the elapsed time

00:08:32.930 --> 00:08:34.462
is a positive one.

00:08:34.462 --> 00:08:35.420
And so what we get?

00:08:35.420 --> 00:08:37.933
If I get my calculator out,

00:08:37.933 --> 00:08:39.471
I get my calculator out.

00:08:39.471 --> 00:08:41.342
I have, this is the same thing as

00:08:41.342 --> 00:08:43.789
positive 10 divided by 9.8.

00:08:43.789 --> 00:08:45.372
10, divided by 9.8.

00:08:47.148 --> 00:08:48.315
Gives me 1.02.

00:08:49.923 --> 00:08:51.756
I'll just round to two digits

00:08:51.756 --> 00:08:53.289
right over there.

00:08:53.289 --> 00:08:55.706
So that gives me 1.02 seconds

00:08:58.119 --> 00:08:59.767
So our change in time,

00:08:59.767 --> 00:09:03.012
so this right over here is 1.02.

00:09:03.012 --> 00:09:05.538
So our change in time,

00:09:05.538 --> 00:09:09.033
delta t, I'm using lowercase now but

00:09:09.033 --> 00:09:10.810
I can make this all lower case.

00:09:10.810 --> 00:09:12.143
Is equal to 1.02

00:09:13.617 --> 00:09:14.700
1.02 seconds.

00:09:16.571 --> 00:09:18.297
Now how do we use this information

00:09:18.297 --> 00:09:21.990
to figure out how far this thing travels?

00:09:21.990 --> 00:09:23.802
Well if we assume that it retains

00:09:23.802 --> 00:09:27.064
its horizontal component of
its velocity the whole time,

00:09:27.064 --> 00:09:29.189
we just assume we can this multiply that

00:09:29.189 --> 00:09:30.825
times our change in time

00:09:30.825 --> 00:09:32.799
and we'll get the total displacement

00:09:32.799 --> 00:09:35.665
in the horizontal direction.

00:09:35.665 --> 00:09:36.826
So to do that, we need to figure out

00:09:36.826 --> 00:09:38.104
this horizontal component,

00:09:38.104 --> 00:09:38.937
which we didn't do yet.

00:09:38.937 --> 00:09:41.933
So this is the component of our velocity

00:09:41.933 --> 00:09:44.646
in the x direction, or
the horizontal direction.

00:09:44.646 --> 00:09:48.181
Once again, we break out a
little bit of trigonometry.

00:09:48.181 --> 00:09:51.037
This side is adjacent to the angle,

00:09:51.037 --> 00:09:54.049
so the adjacent over hypotenuse
is the cosine of the angle.

00:09:54.049 --> 00:09:57.009
Cosine of an angle is
adjacent over hypotenuse.

00:09:57.009 --> 00:09:58.426
So we get cosine.

00:10:00.132 --> 00:10:02.594
Cosine of 30 degrees,

00:10:02.594 --> 00:10:04.114
I just want to make sure
I color-code it right,

00:10:04.114 --> 00:10:07.215
cosine of 30 degrees is equal to

00:10:07.215 --> 00:10:08.982
the adjacent side.

00:10:08.982 --> 00:10:10.723
Is equal to the adjacent side,

00:10:10.723 --> 00:10:14.362
which is the magnitude of
our horizontal component,

00:10:14.362 --> 00:10:17.722
is equal to the adjacent
side over the hypotenuse.

00:10:17.722 --> 00:10:19.889
Over 10 meters per second.

00:10:21.194 --> 00:10:23.817
multiply both sides by
10 meters per second,

00:10:23.817 --> 00:10:27.776
you get the magnitude
of our adjacent side,

00:10:27.776 --> 00:10:29.553
color transitioning is difficult,

00:10:29.553 --> 00:10:32.617
the magnitude of our adjacent side

00:10:32.617 --> 00:10:35.079
is equal to 10 meters per second.

00:10:35.079 --> 00:10:37.206
Is equal to 10 meters per second.

00:10:37.206 --> 00:10:38.751
Times the cosine,

00:10:38.751 --> 00:10:41.334
times the cosine of 30 degrees.

00:10:42.711 --> 00:10:44.187
And you might not remember

00:10:44.187 --> 00:10:45.568
the cosine of 30 degrees,

00:10:45.568 --> 00:10:47.252
you can use a calculator for this.

00:10:47.252 --> 00:10:49.063
Or you can just, if you do remember it,

00:10:49.063 --> 00:10:50.428
you know that it's the square root

00:10:50.428 --> 00:10:52.530
of three over two.

00:10:52.530 --> 00:10:54.921
Square root of three over two.

00:10:54.921 --> 00:10:56.848
So to figure out the actual component,

00:10:56.848 --> 00:10:58.787
I'll stop to get a
calculator out if I want,

00:10:58.787 --> 00:10:59.732
well I don't have to use it,

00:10:59.732 --> 00:11:00.885
do it just yet,

00:11:00.885 --> 00:11:01.974
because I have 10 times

00:11:01.974 --> 00:11:03.797
the square root of three over two.

00:11:03.797 --> 00:11:05.933
Which is going to be 10
divided by two is five.

00:11:05.933 --> 00:11:07.998
So it's going to be five
times the square root of

00:11:07.998 --> 00:11:10.687
three meters per second.

00:11:10.687 --> 00:11:12.539
So if I wanna figure out

00:11:12.539 --> 00:11:14.906
the entire horizontal displacement,

00:11:14.906 --> 00:11:16.508
so let's think about it this way,

00:11:16.508 --> 00:11:19.051
the horizontal displacement,

00:11:19.051 --> 00:11:19.884
that's what we get for it,

00:11:19.884 --> 00:11:20.879
we're trying to figure out,

00:11:20.879 --> 00:11:23.555
the horizontal displacement,

00:11:23.555 --> 00:11:25.183
a S for displacement,

00:11:25.183 --> 00:11:28.481
is going to be equal
to the average velocity

00:11:28.481 --> 00:11:32.244
in the x direction, or
the horizontal direction.

00:11:32.244 --> 00:11:33.787
And that's just going to be this five

00:11:33.787 --> 00:11:35.174
square root of three meters per second

00:11:35.174 --> 00:11:36.840
because it doesn't change.

00:11:36.840 --> 00:11:38.245
So it's gonna be five,

00:11:38.245 --> 00:11:40.513
I don't want to do that same color,

00:11:40.513 --> 00:11:45.355
is going to be the five square
roots of 3 meters per second

00:11:45.355 --> 00:11:46.899
times the change in time,

00:11:46.899 --> 00:11:49.244
times how long it is in the air.

00:11:49.244 --> 00:11:52.402
And we figure that out! Its 1.02 seconds.

00:11:52.402 --> 00:11:53.985
Times 1.02 seconds.

00:11:55.297 --> 00:11:57.205
The seconds cancel out with seconds,

00:11:57.205 --> 00:11:58.503
and we'll get that answers in meters,

00:11:58.503 --> 00:12:01.909
and now we get our calculator
out to figure it out.

00:12:01.909 --> 00:12:06.292
so we have five time the
square root of three,

00:12:06.292 --> 00:12:07.209
times 1.02.

00:12:09.812 --> 00:12:11.812
It gives us 8.83 meters,

00:12:13.059 --> 00:12:13.892
just to round it.

00:12:13.892 --> 00:12:16.281
So this is going to be equal to,

00:12:16.281 --> 00:12:18.698
this is going to be equal to,

00:12:19.636 --> 00:12:21.603
this is going to be oh, sorry.

00:12:21.603 --> 00:12:24.478
this is going to be equal to 8.8,

00:12:24.478 --> 00:12:25.708
is that the number I got?

00:12:25.708 --> 00:12:26.541
8.83,

00:12:27.710 --> 00:12:28.710
8.83 meters.

00:12:29.883 --> 00:12:30.815
And we're done.

00:12:30.815 --> 00:12:32.731
And the next video, I'm gonna try to,

00:12:32.731 --> 00:12:36.572
I'll show you another way
of solving for this delta t.

00:12:36.572 --> 00:12:37.405
To show you, really, that there's

00:12:37.405 --> 00:12:39.101
multiple ways to solve this.

00:12:39.101 --> 00:12:40.269
It's a little bit more complicated

00:12:40.269 --> 00:12:42.284
but it's also a little bit more powerful

00:12:42.284 --> 00:12:45.860
if we don't start and end
at the same elevation.


Visualizing vectors in 2 dimensions

WEBVTT
Kind: captions
Language: en

00:00:00.183 --> 00:00:01.350
- [Voiceover] All the problems

00:00:01.350 --> 00:00:02.703
we've been dealing with so far

00:00:02.703 --> 00:00:06.363
have essentially been
happening in one dimension.

00:00:06.363 --> 00:00:08.476
You could go forward or back.

00:00:08.476 --> 00:00:10.380
So you could go forward or back.

00:00:10.380 --> 00:00:11.750
Or right or left.

00:00:11.750 --> 00:00:14.769
Or you could go up or down.

00:00:14.769 --> 00:00:16.859
What I wanna start to
talk about in this video

00:00:16.859 --> 00:00:19.134
is what happens when we
extend that to two dimensions

00:00:19.134 --> 00:00:21.294
or we can even just extend
what we're doing in this video

00:00:21.294 --> 00:00:24.521
to three or four, really an
arbitrary number of dimensions.

00:00:24.521 --> 00:00:26.379
Although if you're dealing
with classical mechanics

00:00:26.379 --> 00:00:29.746
you normally don't have to go
more than three dimensions.

00:00:29.746 --> 00:00:32.114
And if you're gonna deal
with more than one dimension,

00:00:32.114 --> 00:00:33.182
especially in two dimensions,

00:00:33.182 --> 00:00:35.945
we're also gonna be dealing
with two-dimensional vectors.

00:00:35.945 --> 00:00:37.826
And I just wanna make
sure, through this video,

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that we understand at least the basics

00:00:39.335 --> 00:00:40.821
of two-dimensional vectors.

00:00:40.821 --> 00:00:42.006
Remember, a vector is something

00:00:42.006 --> 00:00:44.978
that has both magnitude and direction.

00:00:44.978 --> 00:00:46.208
So the first thing I wanna do

00:00:46.208 --> 00:00:47.741
is just give you a visual understanding

00:00:47.741 --> 00:00:50.899
of how vectors in two
dimensions would add.

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So let's say I have a vector right here.

00:00:54.498 --> 00:00:55.915
That is vector A.

00:00:57.494 --> 00:00:59.815
So, once again, its magnitude is specified

00:00:59.815 --> 00:01:01.928
by the length of this arrow.

00:01:01.928 --> 00:01:03.623
And its direction is specified

00:01:03.623 --> 00:01:05.388
by the direction of the arrow.

00:01:05.388 --> 00:01:06.851
So it's going in that direction.

00:01:06.851 --> 00:01:08.569
Now let's say I have another vector.

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Let's call it vector B.

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Let's call it vector B.

00:01:12.865 --> 00:01:14.444
It looks like this.

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Now what I wanna do in this video

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is think about what happens when I add

00:01:17.973 --> 00:01:19.924
vector A to vector B.

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So there's a couple things to think about

00:01:21.549 --> 00:01:23.175
when you visually depict vectors.

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The important thing is,
for example, for vector A,

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that you get the length right

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and you get the direction right.

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Where you actually draw it doesn't matter.

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So this could be vector A.

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This could also be vector A.

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Notice, it has the same length

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and it has the same direction.

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This is also vector A.

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I could draw vector A up
here. It does not matter.

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I could draw vector A up there.

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I could draw vector B.

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I could draw vector B over here.

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It's still vector B.

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It still has the same
magnitude and direction.

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Notice, we're not saying

00:01:52.399 --> 00:01:54.082
that its tail has to
start at the same place

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that vector A's tail starts at.

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I could draw vector B over here.

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So I can always have the same vector

00:02:00.502 --> 00:02:02.035
but I can shift it around.

00:02:02.035 --> 00:02:03.010
So I can move it up there.

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As long as it has the same
magnitude, the same length,

00:02:05.425 --> 00:02:08.165
and the same direction.

00:02:08.165 --> 00:02:09.697
And the whole reason I'm doing that

00:02:09.697 --> 00:02:11.996
is because the way to
visually add vectors...

00:02:11.996 --> 00:02:14.246
If I wanted to add vector A

00:02:17.007 --> 00:02:18.340
plus vector B...

00:02:20.720 --> 00:02:22.980
And I'll show you how to
do it more analytically

00:02:22.980 --> 00:02:24.977
in a future video.

00:02:24.977 --> 00:02:28.088
I can literally draw vector A.

00:02:28.088 --> 00:02:29.435
I draw vector A.

00:02:29.435 --> 00:02:31.362
So that's vector A, right over there.

00:02:31.362 --> 00:02:33.173
And then I can draw vector B,

00:02:33.173 --> 00:02:37.980
but I put the tail of vector
B to the head of vector A.

00:02:37.980 --> 00:02:39.559
So I shift vector B over

00:02:39.559 --> 00:02:42.276
so its tail is right at
the head of vector A.

00:02:42.276 --> 00:02:46.037
And then vector B would
look something like this.

00:02:46.037 --> 00:02:47.755
It would look something like this.

00:02:47.755 --> 00:02:51.636
And then if you go from the tail of A

00:02:51.636 --> 00:02:54.044
all the way to the head of B,

00:02:54.044 --> 00:02:55.986
all the way to the head of B,

00:02:55.986 --> 00:02:58.236
and you call that vector C,

00:02:59.668 --> 00:03:01.918
that is the sum of A and B.

00:03:05.500 --> 00:03:07.883
And it should make sense,
if you think about it.

00:03:07.883 --> 00:03:10.191
Let's say these were displacement vectors.

00:03:10.191 --> 00:03:12.118
So A shows that you're being displaced

00:03:12.118 --> 00:03:14.022
this much in this direction.

00:03:14.022 --> 00:03:15.601
B shows that you're being displaced

00:03:15.601 --> 00:03:18.062
this much in this direction.

00:03:18.062 --> 00:03:20.268
So the length of B in that direction.

00:03:20.268 --> 00:03:22.637
And if I were to say you
have a displacement of A,

00:03:22.637 --> 00:03:24.680
and then you have a displacement of B,

00:03:24.680 --> 00:03:26.631
what is your total displacement?

00:03:26.631 --> 00:03:28.488
So you would have had to be,

00:03:28.488 --> 00:03:30.439
I guess, shifted this
far in this direction,

00:03:30.439 --> 00:03:33.225
and then you would be shifted
this far in this direction.

00:03:33.225 --> 00:03:35.315
So the net amount that you've been shifted

00:03:35.315 --> 00:03:37.498
is this far in that direction.

00:03:37.498 --> 00:03:40.261
So that's why this would
be the sum of those.

00:03:40.261 --> 00:03:42.351
Now we can use that same idea

00:03:42.351 --> 00:03:45.207
to break down any vector in two dimensions

00:03:45.207 --> 00:03:48.063
into, we could say, into its components.

00:03:48.063 --> 00:03:49.595
And I'll give you a better sense

00:03:49.595 --> 00:03:51.917
of what that means in a second.

00:03:51.917 --> 00:03:53.750
So if I have vector A.

00:03:54.812 --> 00:03:56.680
Let me pick a new letter.

00:03:56.680 --> 00:03:59.513
Let's call this vector "vector X."

00:04:00.974 --> 00:04:03.253
Let's call this "vector X."

00:04:03.253 --> 00:04:07.336
I can say that vector X
is going to be the sum of

00:04:09.172 --> 00:04:11.587
this vector right here in green

00:04:11.587 --> 00:04:14.420
and this vector right here in red.

00:04:16.533 --> 00:04:19.088
Notice, X starts at the
tail of the green vector

00:04:19.088 --> 00:04:22.478
and goes all the way to the
head of the magenta vector.

00:04:22.478 --> 00:04:24.126
And the magenta vector starts

00:04:24.126 --> 00:04:25.682
at the head of the green vector

00:04:25.682 --> 00:04:27.377
and then finishes, I guess,

00:04:27.377 --> 00:04:30.999
well where it finishes is
where vector X finishes.

00:04:30.999 --> 00:04:32.857
And the reason why I do this...

00:04:32.857 --> 00:04:35.016
And, you know, hopefully from this

00:04:35.016 --> 00:04:36.200
comparable explanation right here,

00:04:36.200 --> 00:04:38.998
says, okay, look, the green
vector plus the magenta vector

00:04:38.998 --> 00:04:40.987
gives us this X vector.

00:04:40.987 --> 00:04:41.933
That should make sense.

00:04:41.933 --> 00:04:44.571
I put the head of the green vector

00:04:44.571 --> 00:04:48.054
to the tail of this magenta
vector right over here.

00:04:48.054 --> 00:04:50.027
But the whole reason why I did this is,

00:04:50.027 --> 00:04:53.905
if I can express X as a
sum of these two vectors,

00:04:53.905 --> 00:04:56.924
it then breaks down X into
its vertical component

00:04:56.924 --> 00:04:59.780
and its horizontal component.

00:04:59.780 --> 00:05:01.447
So I could call this

00:05:04.546 --> 00:05:06.526
the horizontal component,

00:05:06.526 --> 00:05:08.802
or I should say the vertical component.

00:05:08.802 --> 00:05:10.311
X vertical.

00:05:10.311 --> 00:05:13.981
And then I could call this over here

00:05:13.981 --> 00:05:15.443
the X horizontal.

00:05:15.443 --> 00:05:16.999
Or another way I could draw it,

00:05:16.999 --> 00:05:18.578
I could shift this X vertical over.

00:05:18.578 --> 00:05:20.250
Remember, it doesn't
matter where I draw it,

00:05:20.250 --> 00:05:22.340
as long as it has the same
magnitude and direction.

00:05:22.340 --> 00:05:24.012
And I could draw it like this.

00:05:24.012 --> 00:05:24.929
X vertical.

00:05:25.846 --> 00:05:26.891
And so what you see is

00:05:26.891 --> 00:05:30.391
is that you could express this vector X...

00:05:31.238 --> 00:05:32.975
Let me do it in the same colors.

00:05:32.975 --> 00:05:34.392
You can express this vector X

00:05:34.392 --> 00:05:38.409
as the sum of its horizontal
and its vertical components.

00:05:38.409 --> 00:05:42.576
As the sum of its horizontal
and its vertical components.

00:05:43.865 --> 00:05:45.677
Now we're gonna see over and over again

00:05:45.677 --> 00:05:47.813
that this is super powerful

00:05:47.813 --> 00:05:49.025
because what it can do is

00:05:49.025 --> 00:05:50.687
it can turn a two-dimensional problem

00:05:50.687 --> 00:05:53.120
into two separate
one-dimensional problems,

00:05:53.120 --> 00:05:54.908
one acting in a horizontal direction,

00:05:54.908 --> 00:05:57.369
one acting in a vertical direction.

00:05:57.369 --> 00:05:59.459
Now let's do it a little
bit more mathematical.

00:05:59.459 --> 00:06:01.085
I've just been telling you
about length and all of that.

00:06:01.085 --> 00:06:02.524
But let's actually break down...

00:06:02.524 --> 00:06:04.498
Let me just show you what this means,

00:06:04.498 --> 00:06:08.051
to break down the components of a vector.

00:06:08.051 --> 00:06:12.218
So let's say that I have a
vector that looks like this.

00:06:13.474 --> 00:06:15.865
Let me do my best to...

00:06:15.865 --> 00:06:19.696
Let's say I have a vector
that looks like this.

00:06:19.696 --> 00:06:21.322
It's length is five.

00:06:21.322 --> 00:06:23.739
So let me call this vector A.

00:06:25.748 --> 00:06:28.915
So vector A's length is equal to five.

00:06:31.403 --> 00:06:33.110
And let's say that its direction...

00:06:33.110 --> 00:06:35.664
We're gonna give its
direction by the angle

00:06:35.664 --> 00:06:38.334
between the direction its pointing in

00:06:38.334 --> 00:06:40.354
and the positive X axis.

00:06:40.354 --> 00:06:42.932
So maybe I'll draw an axis over here.

00:06:42.932 --> 00:06:44.395
So let's say that this right over here

00:06:44.395 --> 00:06:47.065
is the positive Y axis going
in the vertical direction.

00:06:47.065 --> 00:06:49.898
This right over here
is the positive X axis

00:06:49.898 --> 00:06:51.895
going in the horizontal direction.

00:06:51.895 --> 00:06:54.635
And to specify this vector's direction

00:06:54.635 --> 00:06:57.119
I will give this angle right over here.

00:06:57.119 --> 00:06:58.884
And I'm gonna give a very peculiar angle,

00:06:58.884 --> 00:07:00.974
but I picked this for a specific reason,

00:07:00.974 --> 00:07:03.040
just so things work out neatly in the end.

00:07:03.040 --> 00:07:05.792
And I'm gonna give it in degrees.

00:07:05.792 --> 00:07:07.542
It's 36.8699 degrees.

00:07:11.912 --> 00:07:13.897
So I'm picking that particular number

00:07:13.897 --> 00:07:15.174
for a particular reason.

00:07:15.174 --> 00:07:17.287
Now what I wanna do is I wanna figure out

00:07:17.287 --> 00:07:21.745
this vector's horizontal
and vertical component.

00:07:21.745 --> 00:07:22.883
So I wanna break it down

00:07:22.883 --> 00:07:25.321
into something that's
going straight up or down

00:07:25.321 --> 00:07:27.782
and something that's going
straight right or left.

00:07:27.782 --> 00:07:29.059
So how do I do this?

00:07:29.059 --> 00:07:31.080
Well, one, I could just
draw them, visually,

00:07:31.080 --> 00:07:32.055
see what they look like.

00:07:32.055 --> 00:07:36.420
So its vertical component
would look like this.

00:07:36.420 --> 00:07:37.837
It would start...

00:07:38.825 --> 00:07:41.096
Its vertical component
would look like this.

00:07:41.096 --> 00:07:44.579
And its horizontal component
would look like this.

00:07:44.579 --> 00:07:46.718
Its horizontal component
would look like this.

00:07:46.718 --> 00:07:48.079
The horizontal component,
the way I drew it,

00:07:48.079 --> 00:07:50.884
it would start where vector A starts

00:07:50.884 --> 00:07:54.971
and go as far in the X
direction as vector A's tip,

00:07:54.971 --> 00:07:56.341
but only in the X direction,

00:07:56.341 --> 00:07:59.614
and then you need to, to get
back to the head of vector A,

00:07:59.614 --> 00:08:01.601
you need to have its vertical component.

00:08:01.601 --> 00:08:03.134
And we can sometimes call this,

00:08:03.134 --> 00:08:05.944
we could call the vertical
component over here A sub Y,

00:08:05.944 --> 00:08:08.080
just so that it's moving
in the Y direction.

00:08:08.080 --> 00:08:12.724
And we can call this
horizontal component A sub X.

00:08:12.724 --> 00:08:14.790
Now what I wanna do is I wanna figure out

00:08:14.790 --> 00:08:18.505
the magnitude of A sub Y and A sub X.

00:08:18.505 --> 00:08:20.108
So how do we do that?

00:08:20.108 --> 00:08:21.338
Well, the way we drew this,

00:08:21.338 --> 00:08:23.591
I've essentially set up
a right triangle for us.

00:08:23.591 --> 00:08:25.982
This is a right triangle.

00:08:25.982 --> 00:08:27.724
We know the length of this triangle,

00:08:27.724 --> 00:08:31.137
or the length of this side, or
the length of the hypotenuse.

00:08:31.137 --> 00:08:33.993
That's going to be the
magnitude of vector A.

00:08:33.993 --> 00:08:37.708
And so the magnitude of
vector A is equal to five.

00:08:37.708 --> 00:08:39.218
We already knew that up here.

00:08:39.218 --> 00:08:41.191
So how do we figure out the sides?

00:08:41.191 --> 00:08:44.118
Well, we could use a little
bit of basic trigonometry.

00:08:44.118 --> 00:08:46.765
If we know the angle, and
we know the hypotenuse,

00:08:46.765 --> 00:08:50.294
how do we figure out the
opposite side to the angle?

00:08:50.294 --> 00:08:51.896
So this right here,

00:08:51.896 --> 00:08:55.588
this right here is the
opposite side to the angle.

00:08:55.588 --> 00:08:57.771
And if we forgot some of
our basic trigonometry

00:08:57.771 --> 00:09:00.720
we can relearn it right now.

00:09:00.720 --> 00:09:01.720
Soh-cah-toa.

00:09:03.347 --> 00:09:05.285
Sine is opposite over hypotenuse.

00:09:05.285 --> 00:09:07.293
Cosine is adjacent over hypotenuse.

00:09:07.293 --> 00:09:10.172
Tangent is opposite over adjacent.

00:09:10.172 --> 00:09:14.073
So we have the angle,
we want the opposite,

00:09:14.073 --> 00:09:16.335
and we have the hypotenuse.

00:09:16.335 --> 00:09:17.585
So we could say

00:09:18.723 --> 00:09:20.973
that the sine of our angle,

00:09:22.202 --> 00:09:23.119
the sine of

00:09:24.762 --> 00:09:26.012
36.899 degrees,

00:09:29.104 --> 00:09:32.688
is going to be equal to the
opposite over the hypotenuse.

00:09:32.688 --> 00:09:34.360
The opposite side of the angle

00:09:34.360 --> 00:09:37.332
is the magnitude of our Y component.

00:09:37.332 --> 00:09:41.499
...is going to be equal to the
magnitude of our Y component,

00:09:42.699 --> 00:09:44.623
the magnitude of our Y component,

00:09:44.623 --> 00:09:46.843
over the magnitude of the hypotenuse,

00:09:46.843 --> 00:09:48.399
over this length over here,

00:09:48.399 --> 00:09:51.116
which we know is going
to be equal to five.

00:09:51.116 --> 00:09:53.346
Or if you multiply both sides by five,

00:09:53.346 --> 00:09:54.763
you get five sine

00:09:55.924 --> 00:09:57.424
of 36.899 degrees,

00:09:59.676 --> 00:10:03.843
is equal to the magnitude
of the vertical component

00:10:06.579 --> 00:10:08.390
of our vector A.

00:10:08.390 --> 00:10:10.062
Now before I take out the calculator

00:10:10.062 --> 00:10:11.270
and figure out what this is,

00:10:11.270 --> 00:10:14.892
let me do the same thing for
the horizontal component.

00:10:14.892 --> 00:10:18.979
Over here we know this side
is adjacent to the angle.

00:10:18.979 --> 00:10:20.627
And we know the hypotenuse.

00:10:20.627 --> 00:10:24.018
And so cosine deals with
adjacent and hypotenuse.

00:10:24.018 --> 00:10:27.685
So we know that the
cosine of 36.899 degrees

00:10:29.777 --> 00:10:30.944
is equal to...

00:10:31.936 --> 00:10:33.631
Cosine is adjacent over hypotenuse.

00:10:33.631 --> 00:10:37.714
So it's equal to the
magnitude of our X component

00:10:38.600 --> 00:10:39.808
over the hypotenuse.

00:10:39.808 --> 00:10:41.410
The hypotenuse here has...

00:10:41.410 --> 00:10:43.686
Or the magnitude of the
hypotenuse, I should say,

00:10:43.686 --> 00:10:45.381
which has a length of five.

00:10:45.381 --> 00:10:47.656
Once again, we multiply
both sides by five,

00:10:47.656 --> 00:10:51.823
and we get five times the
cosine of 36.899 degrees

00:10:53.136 --> 00:10:56.886
is equal to the magnitude
of our X component.

00:10:58.500 --> 00:11:00.288
So let's figure out what these are.

00:11:00.288 --> 00:11:02.099
Let me get the calculator out.

00:11:02.099 --> 00:11:04.682
Let me get my trusty TI-85 out.

00:11:05.582 --> 00:11:08.020
I wanna make sure it's in degree mode.

00:11:08.020 --> 00:11:09.553
So let me check.

00:11:09.553 --> 00:11:11.782
Yep, we're in degree
mode right over there.

00:11:11.782 --> 00:11:14.197
Don't wanna... Make sure
we're not in radian mode.

00:11:14.197 --> 00:11:15.834
Now let's exit that.

00:11:15.834 --> 00:11:20.001
And we have the vertical
component is equal to five times

00:11:20.969 --> 00:11:23.219
the sine of 36.899 degrees,

00:11:27.679 --> 00:11:31.744
which is, if we round
it, right at about three.

00:11:31.744 --> 00:11:33.625
So this is equal to...

00:11:33.625 --> 00:11:36.179
So the magnitude of our vertical component

00:11:36.179 --> 00:11:37.679
is equal to three.

00:11:39.964 --> 00:11:41.218
And then let's do the same thing

00:11:41.218 --> 00:11:43.635
for our horizontal component.

00:11:45.865 --> 00:11:47.948
So now we have five times

00:11:49.734 --> 00:11:52.151
the cosine of 36.899 degrees,

00:11:54.489 --> 00:11:57.780
is, if once again we round it to, I guess,

00:11:57.780 --> 00:12:01.375
our hundredths place,
we get it to being four.

00:12:01.375 --> 00:12:03.625
So we get it to being four.

00:12:04.718 --> 00:12:07.737
So we see here is a
situation where we have...

00:12:07.737 --> 00:12:11.243
This is a classic three-four-five
Pythagorean triangle.

00:12:11.243 --> 00:12:15.410
The magnitude of our
horizontal component is four.

00:12:17.130 --> 00:12:21.270
The magnitude of our vertical
component, right over here,

00:12:21.270 --> 00:12:22.601
is equal to three.

00:12:22.601 --> 00:12:23.855
And once again, you might say,

00:12:23.855 --> 00:12:27.129
Sal, why are we going
through all of this trouble?

00:12:27.129 --> 00:12:27.962
And we'll see in the next video

00:12:27.962 --> 00:12:30.133
that if we say something has a velocity,

00:12:30.133 --> 00:12:32.641
in this direction, of
five meters per second,

00:12:32.641 --> 00:12:36.426
we could break that down into
two component velocities.

00:12:36.426 --> 00:12:39.096
We could say that that's
going in the upwards direction

00:12:39.096 --> 00:12:40.512
at three meters per second,

00:12:40.512 --> 00:12:43.601
and it's also going to the right
in the horizontal direction

00:12:43.601 --> 00:12:45.342
at four meters per second.

00:12:45.342 --> 00:12:47.107
And it allows us to break up the problem

00:12:47.107 --> 00:12:48.523
into two simpler problems,

00:12:48.523 --> 00:12:50.334
into two one-dimensional problems,

00:12:50.334 --> 00:12:53.313
instead of a bigger two-dimensional one.


Impact velocity from given height

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:04.030
What I want to do in this video
is answer an age-old question,

00:00:04.030 --> 00:00:06.630
or at least an interesting
question to me.

00:00:06.630 --> 00:00:10.280
And the question is, let's
say I have a ledge here--

00:00:10.280 --> 00:00:12.510
I have a ledge or
cliff, or maybe

00:00:12.510 --> 00:00:14.630
this is a building of some kind.

00:00:14.630 --> 00:00:17.230
And let's say it has height, h.

00:00:17.230 --> 00:00:21.970
So let's say it has a height
of h, right over here.

00:00:21.970 --> 00:00:25.480
And what I'm curious about is
if I were to either-- Let's

00:00:25.480 --> 00:00:29.690
say that this is me over
here, so this is me.

00:00:29.690 --> 00:00:32.560
If I were to either
jump, myself--

00:00:32.560 --> 00:00:35.310
that's not recommended
for very large h's.

00:00:35.310 --> 00:00:38.530
Or If I were to throw something,
maybe a rock off of this

00:00:38.530 --> 00:00:42.530
ledge, how fast would
either myself or that rock

00:00:42.530 --> 00:00:48.110
be going right before
it hits the ground?

00:00:48.110 --> 00:00:49.980
And like all of the
other videos we're

00:00:49.980 --> 00:00:52.110
doing on projectile
motion right now,

00:00:52.110 --> 00:00:54.490
we're going to ignore
air resistance.

00:00:54.490 --> 00:00:57.390
And for small h's and
for small velocities,

00:00:57.390 --> 00:00:59.200
that's actually reasonable.

00:00:59.200 --> 00:01:02.250
Or if the object
is very aerodynamic

00:01:02.250 --> 00:01:04.200
and is kind of dense,
then the air resistance

00:01:04.200 --> 00:01:05.220
will matter less.

00:01:05.220 --> 00:01:08.110
If it's me kind of belly
flopping from a high altitude,

00:01:08.110 --> 00:01:10.840
then the air resistance
will start to matter a lot.

00:01:10.840 --> 00:01:15.670
But for the sake of simplicity,
we're going to assume no air.

00:01:15.670 --> 00:01:17.650
Or we're not going
to take into effect

00:01:17.650 --> 00:01:19.382
the effects of air resistance.

00:01:19.382 --> 00:01:20.840
Or we could assume
that we're doing

00:01:20.840 --> 00:01:24.380
this on an Earth-like planet
that has no atmosphere.

00:01:24.380 --> 00:01:25.480
However you want to do it.

00:01:25.480 --> 00:01:27.100
So let's just think
about the problem.

00:01:27.100 --> 00:01:28.730
And just so you know,
some of you might say,

00:01:28.730 --> 00:01:29.670
that's not realistic.

00:01:29.670 --> 00:01:32.410
But this actually would be
realistic for a small h.

00:01:32.410 --> 00:01:35.920
If you were to jump off of the
roof of a one-story building,

00:01:35.920 --> 00:01:37.922
air resistance will not
be a major component

00:01:37.922 --> 00:01:39.130
in determining your speed if.

00:01:39.130 --> 00:01:41.620
It was to be a much larger
building, then all of a sudden

00:01:41.620 --> 00:01:42.120
it matters.

00:01:42.120 --> 00:01:43.800
And I don't recommend you
do any of these things.

00:01:43.800 --> 00:01:45.300
Those are all very
dangerous things.

00:01:45.300 --> 00:01:46.700
Much better to do
it with a rock.

00:01:46.700 --> 00:01:49.490
So that's actually the example
we're going to be considering.

00:01:49.490 --> 00:01:52.190
So let's just think
about this a little bit.

00:01:52.190 --> 00:01:56.474
We want to figure
out-- So at the top,

00:01:56.474 --> 00:01:57.890
right when the
thing gets dropped,

00:01:57.890 --> 00:01:59.540
right when the
rock gets dropped,

00:01:59.540 --> 00:02:05.021
you have an initial
velocity of 0.

00:02:05.021 --> 00:02:07.270
And once again, we're going
to use the convention here

00:02:07.270 --> 00:02:11.120
that positive velocity
means upwards,

00:02:11.120 --> 00:02:16.230
or a positive vector means up,
a negative vector means down.

00:02:16.230 --> 00:02:18.650
So we're going to have an
initial velocity over here

00:02:18.650 --> 00:02:19.630
of 0.

00:02:19.630 --> 00:02:21.720
And then at the
bottom we're going

00:02:21.720 --> 00:02:30.650
to have some final
velocity here that

00:02:30.650 --> 00:02:33.400
is going to be a
negative number.

00:02:33.400 --> 00:02:36.680
So it's going to have some
negative value over here.

00:02:36.680 --> 00:02:39.290
So this is going to be negative.

00:02:39.290 --> 00:02:43.340
This is going to be a negative
number right over there.

00:02:43.340 --> 00:02:46.070
And we know that the
acceleration of gravity

00:02:46.070 --> 00:02:49.400
for an object on free
fall, an object in free

00:02:49.400 --> 00:02:51.580
fall near the
surface of the earth.

00:02:51.580 --> 00:02:54.500
We know it, and we're going
to assume that it's constant.

00:02:54.500 --> 00:02:56.740
So our constant
acceleration is going

00:02:56.740 --> 00:03:01.440
to be negative 9.8 meters
per second squared.

00:03:01.440 --> 00:03:03.380
So what we're going
to do is given an h,

00:03:03.380 --> 00:03:05.480
and given that their
initial velocity is 0

00:03:05.480 --> 00:03:07.810
and that our acceleration
is negative 9.8 meters per

00:03:07.810 --> 00:03:11.870
squared, we want to figure
out what our final velocity is

00:03:11.870 --> 00:03:14.120
going to be right before
we hit the ground.

00:03:14.120 --> 00:03:15.640
We're going to
assume that this h is

00:03:15.640 --> 00:03:17.120
given in meters,
right over here.

00:03:17.120 --> 00:03:18.495
And we'll get an
answer in meters

00:03:18.495 --> 00:03:20.390
per second for that
final velocity.

00:03:20.390 --> 00:03:22.190
So let's see how we
can figure it out.

00:03:22.190 --> 00:03:24.703
So we know some basic things.

00:03:24.703 --> 00:03:26.786
And the whole point of
these is to really show you

00:03:26.786 --> 00:03:29.680
that you can always derive
these more interesting questions

00:03:29.680 --> 00:03:32.410
from very basic
things that we know.

00:03:32.410 --> 00:03:41.450
So we know that displacement is
equal to average velocity times

00:03:41.450 --> 00:03:42.270
change in time.

00:03:45.030 --> 00:03:48.510
And we know that
average velocity--

00:03:48.510 --> 00:03:50.690
if we assume
acceleration is constant,

00:03:50.690 --> 00:03:53.090
which we are doing--
average velocity

00:03:53.090 --> 00:04:01.770
is the final velocity plus
the initial velocity over 2.

00:04:01.770 --> 00:04:06.940
And then our change in time,
the amount of elapsed time

00:04:06.940 --> 00:04:10.940
that goes by-- this is
our change in velocity.

00:04:10.940 --> 00:04:12.750
So elapsed time
is the same thing.

00:04:12.750 --> 00:04:16.870
I write it over here--
is our change in velocity

00:04:16.870 --> 00:04:20.160
divided by our acceleration.

00:04:20.160 --> 00:04:22.120
And just to make sure
you understand this,

00:04:22.120 --> 00:04:24.230
it just comes
straight from the idea

00:04:24.230 --> 00:04:26.300
that acceleration-- or
let me write it this way--

00:04:26.300 --> 00:04:32.280
that change in velocity is
just acceleration times time.

00:04:32.280 --> 00:04:36.310
Or I should say, acceleration
times change in time.

00:04:36.310 --> 00:04:39.780
So if you divide both sides of
this equation by acceleration,

00:04:39.780 --> 00:04:42.107
you get this right over here.

00:04:42.107 --> 00:04:43.940
So that is what our
displacement-- Remember,

00:04:43.940 --> 00:04:45.830
I want an expression
for displacement

00:04:45.830 --> 00:04:48.040
in terms of the things
we know and the one

00:04:48.040 --> 00:04:50.180
thing that we want to find out.

00:04:50.180 --> 00:04:55.240
Well, for this example
right over here,

00:04:55.240 --> 00:04:56.490
we know a couple of things.

00:04:56.490 --> 00:04:58.680
Well actually, let me
take it step by step.

00:04:58.680 --> 00:05:00.585
We know that our
initial velocity is 0.

00:05:03.130 --> 00:05:06.290
So this first expression
for the example we're doing,

00:05:06.290 --> 00:05:10.760
the average velocity is going
to be our final velocity divided

00:05:10.760 --> 00:05:13.270
by 2, since our
initial velocity is 0.

00:05:13.270 --> 00:05:19.480
Our change in velocity
is the same thing

00:05:19.480 --> 00:05:22.420
as final velocity
minus initial velocity.

00:05:25.070 --> 00:05:28.489
And once again, we know that
the initial velocity is 0 here.

00:05:28.489 --> 00:05:30.280
So our change in velocity
is the same thing

00:05:30.280 --> 00:05:32.040
as our final velocity.

00:05:32.040 --> 00:05:33.589
So once again,
this will be times.

00:05:33.589 --> 00:05:35.380
Instead of writing
change in velocity here,

00:05:35.380 --> 00:05:37.624
we could just write
our final velocity

00:05:37.624 --> 00:05:38.790
because we're starting at 0.

00:05:38.790 --> 00:05:39.930
Initial velocity is 0.

00:05:39.930 --> 00:05:46.421
So times our final velocity
divided by our acceleration.

00:05:46.421 --> 00:05:48.670
Final velocity is the same
thing as change in velocity

00:05:48.670 --> 00:05:51.230
because initial velocity was 0.

00:05:51.230 --> 00:05:56.080
And all of this is going
to be our displacement.

00:05:56.080 --> 00:05:59.460
And now it looks like
we have everything

00:05:59.460 --> 00:06:01.040
written in things we know.

00:06:01.040 --> 00:06:04.270
So if we multiply both
sides of this expression

00:06:04.270 --> 00:06:11.620
or both sides of this equation
by 2 times our acceleration

00:06:11.620 --> 00:06:12.840
on that side.

00:06:12.840 --> 00:06:14.300
And we multiply
the left-hand side

00:06:14.300 --> 00:06:18.300
by-- I'll do the same colors--
2 times our acceleration.

00:06:21.720 --> 00:06:27.140
On the left hand side, we
get 2 times our acceleration

00:06:27.140 --> 00:06:35.070
times our displacement
is going to be equal to,

00:06:35.070 --> 00:06:38.084
on the right hand side, the
2 cancels out with the 2,

00:06:38.084 --> 00:06:40.250
the acceleration cancels
out with the acceleration--

00:06:40.250 --> 00:06:42.410
it will be equal
to the velocity,

00:06:42.410 --> 00:06:43.920
our final velocity squared.

00:06:49.280 --> 00:06:52.480
Final velocity times
final velocity.

00:06:52.480 --> 00:06:56.380
And so we can just solve
for final velocity here.

00:06:56.380 --> 00:07:01.090
So we know our acceleration
is negative 9.8 meters

00:07:01.090 --> 00:07:02.840
per second squared.

00:07:02.840 --> 00:07:04.750
So let me write this over here.

00:07:04.750 --> 00:07:07.450
So this is negative 9.8.

00:07:07.450 --> 00:07:10.870
So we have 2 times
negative 9.8--

00:07:10.870 --> 00:07:12.290
let me just multiply that out.

00:07:12.290 --> 00:07:19.080
So that's negative 19.6
meters per second squared.

00:07:19.080 --> 00:07:21.640
And then what's our
displacement going to be?

00:07:21.640 --> 00:07:23.470
What's the displacement
over the course

00:07:23.470 --> 00:07:27.534
of dropping this rock off of
this ledge or off of this roof?

00:07:27.534 --> 00:07:29.950
So you might be tempted to say
that our displacement is h.

00:07:29.950 --> 00:07:32.010
But remember, these
are vector quantities,

00:07:32.010 --> 00:07:34.540
so you want to make sure
you get the direction right.

00:07:34.540 --> 00:07:39.080
From where the rock started to
where it ends, what's it doing?

00:07:39.080 --> 00:07:41.470
It's going to go
a distance of h,

00:07:41.470 --> 00:07:44.210
but it's going to go a
distance of h downwards.

00:07:44.210 --> 00:07:48.170
And our convention
is down is negative.

00:07:48.170 --> 00:07:52.150
So in this example,
our displacement

00:07:52.150 --> 00:07:55.730
from when it leaves your hand
to when it hits the ground,

00:07:55.730 --> 00:07:58.800
the displacement is going
to be equal to negative h.

00:07:58.800 --> 00:08:00.600
It's going to travel
a distance of h,

00:08:00.600 --> 00:08:02.996
but it's going to travel
that distance downwards.

00:08:02.996 --> 00:08:05.370
And that's why this vector
notion is very important here.

00:08:05.370 --> 00:08:07.610
Our convention is
very important here.

00:08:07.610 --> 00:08:14.690
So our displacement over here is
going to be negative h meters.

00:08:17.730 --> 00:08:19.270
So this is the
variable, and this

00:08:19.270 --> 00:08:20.640
is the shorthand for meters.

00:08:20.640 --> 00:08:22.630
So when you multiply
these two things out,

00:08:22.630 --> 00:08:24.930
lucky for us these
negatives cancel out,

00:08:24.930 --> 00:08:37.330
and you get 19.6h meters squared
per second squared is equal

00:08:37.330 --> 00:08:39.535
to our final velocity squared.

00:08:44.219 --> 00:08:45.760
And notice, when
you square something

00:08:45.760 --> 00:08:47.130
you lose the sign information.

00:08:47.130 --> 00:08:49.359
If our final velocity was
positive, you square it,

00:08:49.359 --> 00:08:50.650
you still get a positive value.

00:08:50.650 --> 00:08:52.191
If it was negative
and you square it,

00:08:52.191 --> 00:08:53.521
you still get a positive value.

00:08:53.521 --> 00:08:55.020
But remember, in
this example, we're

00:08:55.020 --> 00:08:56.270
going to be moving downward.

00:08:56.270 --> 00:08:58.350
So we want the negative
version of this.

00:08:58.350 --> 00:09:01.000
So to really figure
out our final velocity,

00:09:01.000 --> 00:09:03.640
we take, essentially,
the negative square root

00:09:03.640 --> 00:09:07.480
of both sides of this equation.

00:09:07.480 --> 00:09:10.380
So if we were to take the square
root of both sides of this,

00:09:10.380 --> 00:09:12.000
you take the square
root of that side,

00:09:12.000 --> 00:09:14.330
you take the square root of
that side, you will get--

00:09:14.330 --> 00:09:21.870
and I'll flip them around-- your
final velocity, we could say,

00:09:21.870 --> 00:09:25.444
is equal to the
square root of 19.6h.

00:09:25.444 --> 00:09:27.860
And you can even take the
square root of the meter squared

00:09:27.860 --> 00:09:30.590
per second squared, treat
them almost like variables,

00:09:30.590 --> 00:09:31.700
even though they're units.

00:09:31.700 --> 00:09:33.241
And then outside of
the radical sign,

00:09:33.241 --> 00:09:35.616
you will get a
meters per second.

00:09:35.616 --> 00:09:37.240
And the thing I want
to be careful here

00:09:37.240 --> 00:09:39.031
is if we just take the
principal root here,

00:09:39.031 --> 00:09:41.600
the principal root here is
the positive square root.

00:09:41.600 --> 00:09:44.180
But we know that our velocity
is going to be downwards here,

00:09:44.180 --> 00:09:45.930
because that is our convention.

00:09:45.930 --> 00:09:50.627
So we want to make sure we
get the negative square root.

00:09:50.627 --> 00:09:52.210
So let's try it out
with some numbers.

00:09:52.210 --> 00:09:54.230
We've essentially
solved what we set out

00:09:54.230 --> 00:09:56.460
to solve at the
beginning of this video,

00:09:56.460 --> 00:09:59.997
how fast would we be falling,
as a function of the height.

00:09:59.997 --> 00:10:01.580
Well, let's try it
out with something.

00:10:01.580 --> 00:10:03.910
So let's say that the
height is-- I don't know,

00:10:03.910 --> 00:10:09.550
let's say the height
is 5 meters, which

00:10:09.550 --> 00:10:12.400
would be probably
jumping off of a

00:10:12.400 --> 00:10:14.910
or throwing a rock off
of a one-story, maybe

00:10:14.910 --> 00:10:18.270
a commercial one-story building.

00:10:18.270 --> 00:10:20.730
That's about 5 meters,
would be about 15 feet.

00:10:20.730 --> 00:10:23.170
So yeah, about the roof
of a commercial building,

00:10:23.170 --> 00:10:24.850
give or take.

00:10:24.850 --> 00:10:27.090
So let's turn it on.

00:10:27.090 --> 00:10:28.470
And so what do we get?

00:10:28.470 --> 00:10:38.420
If we put 5 meters in here, we
get 19.6 times 5 gives us 98.

00:10:38.420 --> 00:10:41.070
So almost 100.

00:10:41.070 --> 00:10:43.610
And then, we want to take
the square root of that,

00:10:43.610 --> 00:10:45.140
so it's going to be almost 10.

00:10:45.140 --> 00:10:51.020
So the square root of
98 gives us roughly 9.9.

00:10:51.020 --> 00:10:53.330
And we want the negative
square root of that.

00:10:53.330 --> 00:10:56.410
in that situation, when
the height is 5 meters-- So

00:10:56.410 --> 00:10:59.492
if you jump off of a
one-story commercial building,

00:10:59.492 --> 00:11:01.950
right at the bottom, or if you
throw a rock off that, right

00:11:01.950 --> 00:11:03.908
at the bottom, right
before it hits the ground,

00:11:03.908 --> 00:11:11.160
it will have a velocity of
negative 9.9 meters per second.

00:11:11.160 --> 00:11:15.590
So negative 9.9
meters per second.

00:11:15.590 --> 00:11:17.650
I'll leave it up to
you, as an exercise,

00:11:17.650 --> 00:11:20.020
to figure out how fast
this is either kilometers

00:11:20.020 --> 00:11:23.490
per hour or miles per hour
because it's pretty fast.

00:11:23.490 --> 00:11:24.990
It's not something
you's want to do.

00:11:24.990 --> 00:11:26.550
And this is just off of
a one-story building.

00:11:26.550 --> 00:11:28.170
But you can usually
figure this out.

00:11:28.170 --> 00:11:30.250
You could use this for,
really, any height as

00:11:30.250 --> 00:11:33.370
long as we're reasonably close
to the surface of the earth

00:11:33.370 --> 00:11:35.820
and you ignore the
effects of air resistance.

00:11:35.820 --> 00:11:38.280
At really high heights,
especially if the object is not

00:11:38.280 --> 00:11:40.380
that aerodynamic,
then air resistance

00:11:40.380 --> 00:11:42.930
will start to matter a lot.