Deriving max projectile displacement given time

WEBVTT
Kind: captions
Language: en

00:00:00.462 --> 00:00:02.420
Just want to follow up
on the last video, where

00:00:02.420 --> 00:00:05.030
we threw balls in the air, and
saw how long they stayed up

00:00:05.030 --> 00:00:05.530
in the air.

00:00:05.530 --> 00:00:08.109
And we used that to figure
out how fast we initially

00:00:08.109 --> 00:00:10.499
threw the ball and how
high they went in the air.

00:00:10.499 --> 00:00:12.790
And in the last video, we
did it with specific numbers.

00:00:12.790 --> 00:00:13.800
In this video, I
just want to see

00:00:13.800 --> 00:00:16.129
if we can derive some
interesting formulas so that we

00:00:16.129 --> 00:00:19.040
can do the computations
really fast in our brains,

00:00:19.040 --> 00:00:21.840
while we're playing this game
out on some type of a field,

00:00:21.840 --> 00:00:24.880
and we don't necessarily
have any paper around.

00:00:24.880 --> 00:00:28.880
So let's say that the ball
is in the air for delta t.

00:00:28.880 --> 00:00:31.120
Delta t is equal
to time in the air.

00:00:34.560 --> 00:00:36.340
Then we know that
the time up is going

00:00:36.340 --> 00:00:38.800
to be half that, which is the
same thing as the time down.

00:00:38.800 --> 00:00:42.880
The time up is going to
be equal to delta t-- I'm

00:00:42.880 --> 00:00:44.570
going to do that
in the same color--

00:00:44.570 --> 00:00:49.380
is going to be equal to the
time in the air divided by 2.

00:00:49.380 --> 00:00:52.060
So what was our
initial velocity?

00:00:52.060 --> 00:00:54.370
Well, all we have to
do is remind ourselves

00:00:54.370 --> 00:00:57.780
that the change
in velocity, which

00:00:57.780 --> 00:01:00.020
is the same thing as
the final velocity

00:01:00.020 --> 00:01:02.026
minus the initial velocity.

00:01:02.026 --> 00:01:03.400
So the final
velocity-- remember,

00:01:03.400 --> 00:01:05.691
we're just talking about half
of the path of this ball.

00:01:05.691 --> 00:01:08.175
So the time that
it gets released,

00:01:08.175 --> 00:01:10.610
and it's going at kind of
its maximum upward velocity,

00:01:10.610 --> 00:01:13.470
and it goes slower, and slower,
slower, all the way until it's

00:01:13.470 --> 00:01:15.220
stationary for just
a moment, and then

00:01:15.220 --> 00:01:17.080
it starts going down again.

00:01:17.080 --> 00:01:19.810
Now remember, the acceleration
is constant downwards

00:01:19.810 --> 00:01:21.524
this entire time.

00:01:21.524 --> 00:01:23.190
So what is the final
velocity if we just

00:01:23.190 --> 00:01:24.680
consider half of the time?

00:01:24.680 --> 00:01:25.870
Well, it's the time.

00:01:25.870 --> 00:01:26.830
It's 0.

00:01:26.830 --> 00:01:32.580
So it's going to be 0
minus our initial velocity,

00:01:32.580 --> 00:01:34.040
when it was taking off.

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That's our change in velocity.

00:01:36.130 --> 00:01:40.500
This is our change
in velocity, is

00:01:40.500 --> 00:01:46.560
going to be equal to the
acceleration of gravity,

00:01:46.560 --> 00:01:49.930
negative 9.8 meters
per second squared--

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or the acceleration
due to gravity

00:01:51.820 --> 00:01:54.800
when an object is in free fall,
to be technically correct--

00:01:54.800 --> 00:01:59.100
times the time that
we are going up.

00:01:59.100 --> 00:02:03.194
So times delta t up,
which is the same thing.

00:02:03.194 --> 00:02:04.110
I won't even write it.

00:02:04.110 --> 00:02:07.510
Delta t up is the same
thing as our total time

00:02:07.510 --> 00:02:12.590
in the air divided by 2.

00:02:12.590 --> 00:02:17.670
And so we get negative
the initial velocity is

00:02:17.670 --> 00:02:20.740
equal to-- this thing,
when you divide it by 2,

00:02:20.740 --> 00:02:25.100
is going to be 4.9 meters
per second squared--

00:02:25.100 --> 00:02:30.047
we still have our negative
out front-- times our delta t.

00:02:30.047 --> 00:02:31.880
Remember, this is our
total time in the air,

00:02:31.880 --> 00:02:32.770
not just the time up.

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This is our total
time in the air.

00:02:34.590 --> 00:02:37.580
And then we multiply both
sides times a negative.

00:02:37.580 --> 00:02:40.290
We get our initial
velocity is just

00:02:40.290 --> 00:02:47.330
going to be equal to 4.9
meters per second squared times

00:02:47.330 --> 00:02:52.710
the total time that
we are in the air.

00:02:52.710 --> 00:02:55.030
Or you could say
it's going to be

00:02:55.030 --> 00:02:58.254
9.8 meters per second squared
times half of the time

00:02:58.254 --> 00:02:59.170
that we're in the air.

00:02:59.170 --> 00:03:01.880
Either of those would get
you the same calculation.

00:03:01.880 --> 00:03:04.620
So let's figure out our total
distance, or the distance

00:03:04.620 --> 00:03:07.150
that we travel in the time up.

00:03:07.150 --> 00:03:09.530
So that'll give us
our peak distance.

00:03:09.530 --> 00:03:12.850
Remember that distance-- or
I should say displacement,

00:03:12.850 --> 00:03:14.650
in this situation--
displacement is

00:03:14.650 --> 00:03:21.380
equal to average velocity
times change in time.

00:03:21.380 --> 00:03:24.670
The change in time that we
care about is the time up.

00:03:24.670 --> 00:03:26.980
So that is our delta t over 2.

00:03:26.980 --> 00:03:30.620
Our total time divided by 2.

00:03:30.620 --> 00:03:33.720
This is our time up.

00:03:37.120 --> 00:03:39.316
And what's our average velocity?

00:03:39.316 --> 00:03:40.690
Well, the average
velocity, if we

00:03:40.690 --> 00:03:44.470
assume constant acceleration,
is your initial velocity

00:03:44.470 --> 00:03:47.570
plus your final velocity over 2.

00:03:47.570 --> 00:03:50.490
It's really just the
mean of the two things.

00:03:50.490 --> 00:03:52.900
Well, we know what our
initial velocity is.

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Our initial velocity is
this thing over here.

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So this is this thing over here.

00:03:57.250 --> 00:03:58.840
Our final velocity--
remember, we're

00:03:58.840 --> 00:04:00.390
just talking about
the first half

00:04:00.390 --> 00:04:02.140
of the time the
ball is in the air--

00:04:02.140 --> 00:04:04.730
so it's final velocity is 0.

00:04:04.730 --> 00:04:08.290
We're talking when it gets
to this peak point, right

00:04:08.290 --> 00:04:11.980
over here-- that's from two
videos ago-- that peak point

00:04:11.980 --> 00:04:13.040
right over there.

00:04:13.040 --> 00:04:15.470
So our average
velocity is just going

00:04:15.470 --> 00:04:20.290
to be this stuff divided by 2.

00:04:20.290 --> 00:04:25.870
So it's going to be 4.9 meters
per second squared times delta

00:04:25.870 --> 00:04:32.380
t over 2.

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So this right here, this
is our average velocity.

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Velocity average.

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So let's stick that
back over here.

00:04:39.810 --> 00:04:43.510
So our maximum
displacement is going

00:04:43.510 --> 00:04:45.890
to be our average
velocity-- so that

00:04:45.890 --> 00:04:51.160
is 4.9 meters per second
squared-- times delta

00:04:51.160 --> 00:04:55.980
t, all of that over 2.

00:04:55.980 --> 00:05:00.060
And then we multiply it
again times the time up.

00:05:00.060 --> 00:05:03.270
So times delta t over 2 again.

00:05:03.270 --> 00:05:04.980
This is the same thing.

00:05:04.980 --> 00:05:06.444
These are the same thing.

00:05:06.444 --> 00:05:07.610
And then we can simplify it.

00:05:07.610 --> 00:05:12.790
Our maximum displacement is
equal to 4.9 meters per second

00:05:12.790 --> 00:05:22.340
squared times delta t
squared, all of that over 4.

00:05:22.340 --> 00:05:27.150
And then we can just
divide 4.9 divided by 4.

00:05:27.150 --> 00:05:37.275
4.9 divided by 4 is-- let me
just get the calculator out.

00:05:37.275 --> 00:05:39.275
I don't want to do that
in my head, get this far

00:05:39.275 --> 00:05:40.441
and make a careless mistake.

00:05:40.441 --> 00:05:46.990
4.9 divided by 4 is 1.225.

00:05:46.990 --> 00:05:48.920
So our maximum
displacement is going

00:05:48.920 --> 00:05:57.740
to be 1.225 times our total
time in the air squared,

00:05:57.740 --> 00:06:03.760
which is a pretty
straightforward calculation.

00:06:03.760 --> 00:06:07.700
So this is our max
displacement, kind

00:06:07.700 --> 00:06:11.210
of how high we get displaced.

00:06:11.210 --> 00:06:13.420
Right when that
ball is stationary,

00:06:13.420 --> 00:06:17.360
or has no net velocity, just
for a moment, and starts

00:06:17.360 --> 00:06:19.290
decelerating downwards.

00:06:19.290 --> 00:06:20.160
So we can use that.

00:06:20.160 --> 00:06:24.150
If a ball is in the air for
5 seconds-- we can verify

00:06:24.150 --> 00:06:26.480
our computation from
the last video--

00:06:26.480 --> 00:06:33.250
our maximum displacement, 1.225,
times 5 squared, which is 25,

00:06:33.250 --> 00:06:35.230
will give us 30.625.

00:06:35.230 --> 00:06:36.910
That's what we got
in the last video.

00:06:36.910 --> 00:06:39.140
If the ball's in the
air for, I don't know,

00:06:39.140 --> 00:06:45.300
2.3 seconds-- so
it's 1.225 times 2.3

00:06:45.300 --> 00:06:50.870
squared-- then that means it
went 6.48 meters in the air.

00:06:50.870 --> 00:06:55.250
So anyway, I just wanted to give
you a simple expression that

00:06:55.250 --> 00:06:57.990
gives you the maximum
displacement from the ground,

00:06:57.990 --> 00:07:00.480
assuming air resistance
is negligible,

00:07:00.480 --> 00:07:04.259
as a function of the
total time in the air.

00:07:04.259 --> 00:07:04.800
I don't know.

00:07:04.800 --> 00:07:06.290
I find that pretty fun.

00:07:06.290 --> 00:07:08.940
And it's a neat game to play.


Projectile height given time

WEBVTT
Kind: captions
Language: en

00:00:00.547 --> 00:00:02.630
Let's say you and I are
playing a game where we're

00:00:02.630 --> 00:00:05.390
trying to figure out how high a
ball is being thrown in the air

00:00:05.390 --> 00:00:08.460
or how fast that we're
throwing that ball in the air.

00:00:08.460 --> 00:00:10.540
And what we do is
one of us has a ball

00:00:10.540 --> 00:00:13.850
and the other one has
a stopwatch over here.

00:00:13.850 --> 00:00:15.400
So this is my best attempt.

00:00:15.400 --> 00:00:17.150
It looks more like a
cat than a stopwatch,

00:00:17.150 --> 00:00:18.507
but I think you get the idea.

00:00:18.507 --> 00:00:20.340
And what we do is one
of us throws the ball,

00:00:20.340 --> 00:00:22.674
and the other one times how
long the ball is in the air.

00:00:22.674 --> 00:00:24.090
And then what we
do is we're going

00:00:24.090 --> 00:00:25.890
to use that time in
the air to figure out

00:00:25.890 --> 00:00:28.380
how fast the ball was
thrown straight up,

00:00:28.380 --> 00:00:32.244
and how long it was in the
air and how high it got.

00:00:32.244 --> 00:00:34.410
And there's going to be one
assumption I make here--

00:00:34.410 --> 00:00:35.430
and frankly, this
is an assumption

00:00:35.430 --> 00:00:37.840
that we're going to make
in all of these projectile

00:00:37.840 --> 00:00:40.550
motion-type problems-- is that
air resistance is negligible.

00:00:44.760 --> 00:00:47.380
And for something like-- if
this is a baseball, or something

00:00:47.380 --> 00:00:49.669
like that, that's a
pretty good approximation.

00:00:49.669 --> 00:00:51.460
So we're not going to
get the exact answer.

00:00:51.460 --> 00:00:54.437
And I encourage you to
experiment on your own

00:00:54.437 --> 00:00:57.020
to see what air resistance does
relative to your calculations.

00:00:57.020 --> 00:00:59.785
But we're going to assume
for this projectile motion--

00:00:59.785 --> 00:01:01.160
and really all of
the future ones

00:01:01.160 --> 00:01:03.300
or at least in the
basic physics playlist--

00:01:03.300 --> 00:01:05.690
we're going to assume that
air resistance is negligible.

00:01:05.690 --> 00:01:07.910
And what that does for
us is that we can assume

00:01:07.910 --> 00:01:13.640
that the time for the ball
to go up to its peak height

00:01:13.640 --> 00:01:17.084
is the same thing as the time
that it takes to go down.

00:01:17.084 --> 00:01:18.500
If you look at
this previous video

00:01:18.500 --> 00:01:21.660
where we plotted
displacement versus time,

00:01:21.660 --> 00:01:25.330
you see after 2 seconds the ball
went from being on the ground--

00:01:25.330 --> 00:01:28.870
or I guess the thrower's hand--
all the way to its peak height.

00:01:28.870 --> 00:01:31.980
And in the next 2 seconds, it
took that same amount of time

00:01:31.980 --> 00:01:34.410
to go back down to the
ground, which makes sense.

00:01:34.410 --> 00:01:36.110
Whatever the
initial velocity is,

00:01:36.110 --> 00:01:38.130
it takes half the
time to go to 0.

00:01:38.130 --> 00:01:39.900
And it takes that
same amount of time

00:01:39.900 --> 00:01:42.460
to now be accelerating
in the downward direction

00:01:42.460 --> 00:01:45.310
back to that same
magnitude of velocity

00:01:45.310 --> 00:01:48.470
but now in the
downward direction.

00:01:48.470 --> 00:01:50.970
So let's play around
with some numbers

00:01:50.970 --> 00:01:54.040
here just so we get a little
bit more of a concrete sense.

00:01:54.040 --> 00:01:56.100
So let's say I throw
a ball in the air.

00:01:56.100 --> 00:01:59.010
And you measure,
using the stopwatch,

00:01:59.010 --> 00:02:02.020
that the ball is in
the air for 5 seconds.

00:02:02.020 --> 00:02:06.410
So how do we figure out
how fast I threw the ball?

00:02:06.410 --> 00:02:07.930
Well, the first
thing we can do is

00:02:07.930 --> 00:02:09.930
we could say, look, if
the total time in the air

00:02:09.930 --> 00:02:13.730
was 5 seconds, that means that
the time-- let me write it.

00:02:13.730 --> 00:02:15.680
That means that
the change in time

00:02:15.680 --> 00:02:19.280
to go up during the
first half of, I guess,

00:02:19.280 --> 00:02:24.490
the ball's time in the air
is going to be 2.5 seconds.

00:02:24.490 --> 00:02:27.180
Which tells us that
over this 2.5 seconds,

00:02:27.180 --> 00:02:30.650
we went from our initial
velocity, whatever it was,

00:02:30.650 --> 00:02:32.710
we went from our
initial velocity

00:02:32.710 --> 00:02:35.450
to our final velocity,
which is a velocity

00:02:35.450 --> 00:02:39.700
of 0 meters per second
in the 2 and 1/2 seconds.

00:02:39.700 --> 00:02:41.450
And this isn't the
graph for that example.

00:02:41.450 --> 00:02:45.020
This is the graph for the
previous example where

00:02:45.020 --> 00:02:47.120
we knew the initial velocity.

00:02:47.120 --> 00:02:48.710
But in whatever
that time is, you're

00:02:48.710 --> 00:02:51.900
going from your initial velocity
to being stationary at the top,

00:02:51.900 --> 00:02:52.500
right?

00:02:52.500 --> 00:02:54.083
Right when the ball
is stationary then

00:02:54.083 --> 00:02:56.060
it starts getting
increasing velocity

00:02:56.060 --> 00:02:59.410
in the downward direction.

00:02:59.410 --> 00:03:03.060
So it takes 2.5 seconds to
go from some initial velocity

00:03:03.060 --> 00:03:04.830
to 0 seconds.

00:03:04.830 --> 00:03:08.140
So we do know what the
acceleration of gravity is.

00:03:08.140 --> 00:03:12.730
We know that the acceleration
of gravity here--

00:03:12.730 --> 00:03:15.221
we're assuming it's constant,
although it's slightly not

00:03:15.221 --> 00:03:15.720
constant.

00:03:15.720 --> 00:03:16.930
But we're going to
assume it's constant,

00:03:16.930 --> 00:03:19.490
if we're just dealing close
to the surface of the earth--

00:03:19.490 --> 00:03:25.010
is negative 9.8 meters
per second squared.

00:03:25.010 --> 00:03:26.680
So let's think about it.

00:03:26.680 --> 00:03:35.020
Our change in velocity
is the final velocity

00:03:35.020 --> 00:03:38.490
minus the initial
velocity, which

00:03:38.490 --> 00:03:42.220
is the same thing as 0 minus
the initial velocity, which

00:03:42.220 --> 00:03:45.060
is the negative of
the initial velocity.

00:03:45.060 --> 00:03:47.700
And what's another way to
think about change in velocity?

00:03:47.700 --> 00:03:50.390
Well, just from the
definition of acceleration,

00:03:50.390 --> 00:03:56.650
change in velocity is
equal to acceleration--

00:03:56.650 --> 00:04:00.550
negative 9.8 meters per
second squared-- times time,

00:04:00.550 --> 00:04:03.176
or times change in time.

00:04:03.176 --> 00:04:04.800
We're just talking
about the first half

00:04:04.800 --> 00:04:07.640
of the ball's time in the air.

00:04:07.640 --> 00:04:13.630
So our change in time is 2.5
seconds-- times 2.5 seconds.

00:04:13.630 --> 00:04:15.440
So what is our
change in velocity,

00:04:15.440 --> 00:04:17.649
which is also the same
thing as the negative

00:04:17.649 --> 00:04:18.680
of our initial velocity?

00:04:21.209 --> 00:04:27.200
Let me get my calculator,
bring it onto the screen.

00:04:27.200 --> 00:04:34.280
So it is negative 9.8 meters
per second times 2.5 seconds.

00:04:36.910 --> 00:04:41.220
It gives us negative 24.5.

00:04:41.220 --> 00:04:43.440
So this gives us-- let me
write it in a new color.

00:04:43.440 --> 00:04:47.950
This gives us negative
24.5 meters per second.

00:04:47.950 --> 00:04:50.000
This second cancels out
with one of these seconds

00:04:50.000 --> 00:04:51.960
in the denominator, so we only
have one in the denominator

00:04:51.960 --> 00:04:53.670
now, so it's meters per second.

00:04:53.670 --> 00:04:59.710
And this is the same thing as
the negative initial velocity.

00:04:59.710 --> 00:05:02.490
That's the same thing as
our change in velocity.

00:05:02.490 --> 00:05:05.160
And so you multiply both
sides by a negative.

00:05:05.160 --> 00:05:06.860
We get our initial velocity.

00:05:06.860 --> 00:05:10.740
So that simply we were
able to figure out

00:05:10.740 --> 00:05:12.330
what our velocity was.

00:05:12.330 --> 00:05:15.600
So literally you take the
total time in the air,

00:05:15.600 --> 00:05:17.640
take it and divide
it by 2, and then

00:05:17.640 --> 00:05:22.530
multiply that by the
acceleration of gravity.

00:05:22.530 --> 00:05:25.180
And I guess you could take
the absolute value of that

00:05:25.180 --> 00:05:27.220
or you take the positive
version of that.

00:05:27.220 --> 00:05:31.340
And then that gives you
your initial velocity.

00:05:31.340 --> 00:05:34.040
So your initial velocity
here is literally

00:05:34.040 --> 00:05:37.460
24.5 meters per second.

00:05:37.460 --> 00:05:39.240
And since it's a
positive quantity,

00:05:39.240 --> 00:05:42.280
it is upwards in this example.

00:05:42.280 --> 00:05:43.540
So that's my initial velocity.

00:05:43.540 --> 00:05:45.700
So we already figured
out part of this game,

00:05:45.700 --> 00:05:47.810
the initial velocity
that I threw it upwards.

00:05:47.810 --> 00:05:49.440
And that's also
going to be-- we're

00:05:49.440 --> 00:05:50.940
going to have the same
magnitude of velocity

00:05:50.940 --> 00:05:52.606
when the ball's about
to hit the ground,

00:05:52.606 --> 00:05:54.740
although it's going to be
in the other direction.

00:05:54.740 --> 00:05:57.810
So what is the distance--
or let me make it clear.

00:05:57.810 --> 00:06:02.170
What is the displacement of the
ball from its lowest point--

00:06:02.170 --> 00:06:05.090
right when it leaves your
hand-- all the way to the peak?

00:06:07.325 --> 00:06:09.700
Well, we just have to remember--
and once again, all this

00:06:09.700 --> 00:06:13.750
comes from very straightforward
ideas, change in velocity

00:06:13.750 --> 00:06:16.356
is equal to acceleration
times change in time.

00:06:16.356 --> 00:06:17.730
And then the other
simple idea is

00:06:17.730 --> 00:06:26.690
that displacement is equal
to average velocity times

00:06:26.690 --> 00:06:28.410
change in time.

00:06:28.410 --> 00:06:30.490
Now, what is our
average velocity?

00:06:30.490 --> 00:06:33.360
Our average velocity is
your initial velocity

00:06:33.360 --> 00:06:36.520
plus your final
velocity divided by 2,

00:06:36.520 --> 00:06:38.450
if we assume
acceleration is constant.

00:06:38.450 --> 00:06:41.070
So it's literally just
the arithmetic mean

00:06:41.070 --> 00:06:43.800
of your initial and
final velocities.

00:06:43.800 --> 00:06:44.630
So what is that?

00:06:44.630 --> 00:06:49.460
That's going to be 24.5
meters per second plus--

00:06:49.460 --> 00:06:50.770
what's our final velocity?

00:06:50.770 --> 00:06:52.186
In this situation,
remember, we're

00:06:52.186 --> 00:06:54.020
just going over the
first 2.5 seconds.

00:06:54.020 --> 00:06:57.740
So our final velocity is once
again 0 meters per second.

00:06:57.740 --> 00:06:59.340
We're just talking
about when we get

00:06:59.340 --> 00:07:01.380
to this point right over here.

00:07:01.380 --> 00:07:04.730
So our final velocity is
just 0 meters per second.

00:07:04.730 --> 00:07:06.720
And we're just going
to divide that by 2.

00:07:06.720 --> 00:07:09.290
This will give us
the average velocity.

00:07:09.290 --> 00:07:16.930
And then we want to multiply
that times 2.5 seconds.

00:07:16.930 --> 00:07:21.580
So we get-- this part right
over here-- 24.5 divided by 2.

00:07:21.580 --> 00:07:22.570
We can ignore the 0.

00:07:22.570 --> 00:07:30.940
That still is 24.5 That
gives us 12.25 times 2.5.

00:07:30.940 --> 00:07:33.190
And remember, this right
over here is in seconds.

00:07:33.190 --> 00:07:35.180
Let me write the units down.

00:07:35.180 --> 00:07:40.525
So this is 12.25 meters per
second times 2.5 seconds.

00:07:43.690 --> 00:07:45.650
And just to remind
ourselves, we're

00:07:45.650 --> 00:07:48.450
calculating the displacement
over the first 2 and 1/2

00:07:48.450 --> 00:07:49.710
seconds.

00:07:49.710 --> 00:07:53.400
So this gives us-- I'll get
the calculator out once again.

00:07:53.400 --> 00:08:02.400
We have 12.25 times 2.5
seconds gives us 30.625.

00:08:02.400 --> 00:08:06.600
So this gives us--
so our displacement

00:08:06.600 --> 00:08:16.402
is 30.625 meters-- these
seconds cancel out-- meters.

00:08:16.402 --> 00:08:17.360
This is actually a ton.

00:08:17.360 --> 00:08:22.020
This is roughly, give or
take, about 90 feet thrown

00:08:22.020 --> 00:08:22.570
in the air.

00:08:22.570 --> 00:08:24.403
So this would be like
a nine-story building.

00:08:24.403 --> 00:08:26.410
And I, frankly, do not
have the arm for that.

00:08:26.410 --> 00:08:28.490
But if someone is
able to throw the ball

00:08:28.490 --> 00:08:30.260
for 5 seconds in
the air, they have

00:08:30.260 --> 00:08:31.857
thrown it 30 meters in the air.

00:08:31.857 --> 00:08:33.690
Well, hopefully you
found that entertaining.

00:08:33.690 --> 00:08:35.570
In the next video,
I'll generalize this.

00:08:35.570 --> 00:08:37.320
Maybe we can get a
little bit of a formula

00:08:37.320 --> 00:08:39.419
so maybe you can generalize it.

00:08:39.419 --> 00:08:41.940
So regardless of the
measurement of time,

00:08:41.940 --> 00:08:44.159
you can get the
displacement in the air.

00:08:44.159 --> 00:08:46.710
Or even better, try to
derive it yourself and we'll

00:08:46.710 --> 00:08:50.400
see how at least I tackle
it in the next video.


Plotting projectile displacement, acceleration, and velocity

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:04.000
What I want to do in this video,
now that we have displacement

00:00:04.000 --> 00:00:07.780
as a function of time,
given constant acceleration

00:00:07.780 --> 00:00:09.410
and an initial velocity.

00:00:09.410 --> 00:00:14.530
I want to plot displacement,
final velocity,

00:00:14.530 --> 00:00:17.690
and acceleration, all of
those as functions of time.

00:00:17.690 --> 00:00:19.280
Just so we really
understand what's

00:00:19.280 --> 00:00:22.870
happening as the ball is
going up and then down.

00:00:22.870 --> 00:00:25.790
So we know this is
our displacement

00:00:25.790 --> 00:00:27.270
as a function of time.

00:00:27.270 --> 00:00:29.620
We know what our final
velocity is going to be,

00:00:29.620 --> 00:00:30.679
as a function of time.

00:00:30.679 --> 00:00:32.220
We talked about it
in the last video.

00:00:32.220 --> 00:00:37.000
Our final velocity is going
to be our initial velocity

00:00:37.000 --> 00:00:41.540
plus our acceleration
times change in time.

00:00:41.540 --> 00:00:42.040
Right?

00:00:42.040 --> 00:00:44.790
If we start at some initial
velocity, and then you

00:00:44.790 --> 00:00:46.620
multiply the
acceleration times time.

00:00:46.620 --> 00:00:48.930
This part tells you how
much faster or slower

00:00:48.930 --> 00:00:51.084
you're going to go than
your initial velocity.

00:00:51.084 --> 00:00:52.750
And that will be, I
guess you could say,

00:00:52.750 --> 00:00:55.150
your current velocity,
or the final velocity

00:00:55.150 --> 00:00:57.820
at that point in time.

00:00:57.820 --> 00:01:00.550
And of course, our
acceleration, we know.

00:01:00.550 --> 00:01:03.220
Our acceleration is
pretty straightforward.

00:01:03.220 --> 00:01:05.040
The acceleration due
to gravity is just

00:01:05.040 --> 00:01:09.330
going to be negative 9.8
meters per second squared.

00:01:09.330 --> 00:01:11.200
Once again, negative
being the convention

00:01:11.200 --> 00:01:13.970
that it is in the
downward direction.

00:01:13.970 --> 00:01:15.570
Our initial velocity
is going to be

00:01:15.570 --> 00:01:19.370
in the upward direction,
19.6 meters per second.

00:01:19.370 --> 00:01:22.550
So let's plot these
out a little bit.

00:01:22.550 --> 00:01:23.650
Let's plot these out.

00:01:23.650 --> 00:01:28.810
So the first graph I want
to do, right over here,

00:01:28.810 --> 00:01:32.520
will be my displacement
versus time.

00:01:32.520 --> 00:01:36.350
So this axis right over
here is going to be time,

00:01:36.350 --> 00:01:39.315
or maybe I could call this
the change in time axis.

00:01:39.315 --> 00:01:40.690
Actually, lets
just call it time.

00:01:43.800 --> 00:01:47.550
And then this axis, right over
here, I will call displacement.

00:01:47.550 --> 00:01:49.720
And let me put
some markers here.

00:01:49.720 --> 00:01:53.740
So let's say that this is
5 meters, 10 meters, 15

00:01:53.740 --> 00:01:55.610
meters, and 20 meters.

00:01:55.610 --> 00:02:03.920
And then in the time, this is 0,
this is 1 this is 2, this is 3,

00:02:03.920 --> 00:02:06.950
and this is 4 seconds.

00:02:06.950 --> 00:02:08.460
So this is in
seconds right here.

00:02:08.460 --> 00:02:09.940
This is meters.

00:02:09.940 --> 00:02:16.986
5, 10, 5, 10, 15, 20.

00:02:16.986 --> 00:02:17.985
So this is displacement.

00:02:22.220 --> 00:02:24.070
Displacement graph.

00:02:24.070 --> 00:02:26.020
And I want to, at the
same time as that,

00:02:26.020 --> 00:02:28.300
I want to do a velocity graph.

00:02:28.300 --> 00:02:31.170
So let me draw my
velocity graph like this.

00:02:31.170 --> 00:02:32.745
I'll do it a little
bit different.

00:02:32.745 --> 00:02:36.810
So, this is because the velocity
will be going up and down.

00:02:36.810 --> 00:02:39.860
So we need to have positive
and negative values here.

00:02:39.860 --> 00:02:42.340
Time will only be positive.

00:02:42.340 --> 00:02:47.470
So once again, I care about 1
second, 2 seconds, 3 seconds,

00:02:47.470 --> 00:02:49.870
and 4 seconds in time.

00:02:49.870 --> 00:02:53.670
And velocity, I'm
going to call this.

00:02:53.670 --> 00:02:57.190
This is going to be
10 meters per second.

00:02:57.190 --> 00:02:59.350
This is 20 meters per second.

00:02:59.350 --> 00:03:01.780
This will be negative
10 meters per second.

00:03:01.780 --> 00:03:05.380
And this will be negative
20 meters per second.

00:03:05.380 --> 00:03:07.490
And so all of this is
in meters per second.

00:03:07.490 --> 00:03:08.715
This right here is velocity.

00:03:11.390 --> 00:03:12.910
This axis right here is time.

00:03:12.910 --> 00:03:14.575
So this is my velocity graph.

00:03:17.250 --> 00:03:20.080
And why don't we just throw an
acceleration graph over here,

00:03:20.080 --> 00:03:23.490
although that's, to some
degree, the easiest of them all.

00:03:23.490 --> 00:03:28.382
So the acceleration
graph, and I'll

00:03:28.382 --> 00:03:29.840
just do this right
from the get go,

00:03:29.840 --> 00:03:32.390
because we're going to assume
that acceleration is constant.

00:03:32.390 --> 00:03:36.550
So this is 1 second,
2 seconds, 3 seconds,

00:03:36.550 --> 00:03:38.500
and 4 seconds into it.

00:03:38.500 --> 00:03:42.050
And then let's call
this negative 10.

00:03:42.050 --> 00:03:44.949
And all of this is in
meters per second squared.

00:03:44.949 --> 00:03:46.740
And so we know our
acceleration is negative

00:03:46.740 --> 00:03:48.730
9.8 meters per second squared.

00:03:48.730 --> 00:03:52.350
So the acceleration
the entire time

00:03:52.350 --> 00:03:55.780
over the four seconds, the
acceleration over the four

00:03:55.780 --> 00:03:59.384
seconds is going to be
that's about negative 9.8.

00:03:59.384 --> 00:04:00.300
It's going to be that.

00:04:00.300 --> 00:04:03.520
It's going to be a constant
acceleration the entire time.

00:04:03.520 --> 00:04:05.560
But let's figure out
displacement and velocity.

00:04:05.560 --> 00:04:10.400
So let me draw a
little table here.

00:04:10.400 --> 00:04:14.980
So in one column, I
will do change in time.

00:04:14.980 --> 00:04:18.000
Or sometimes you
could do that as time.

00:04:18.000 --> 00:04:21.410
Let's figure out what
our final velocity is,

00:04:21.410 --> 00:04:24.020
or I should really say our
current velocity, or velocity

00:04:24.020 --> 00:04:25.410
at that time.

00:04:25.410 --> 00:04:27.460
And then in this
column, I'll figure out

00:04:27.460 --> 00:04:31.200
what our displacement is.

00:04:31.200 --> 00:04:37.550
And I will do it for
times 0, 1, 2, 3, 4.

00:04:37.550 --> 00:04:38.510
Or change in time.

00:04:38.510 --> 00:04:41.780
So when 0 seconds have gone
by, when 1 second has gone by,

00:04:41.780 --> 00:04:44.820
when 2 seconds, 3 seconds,
and 4 seconds have gone by.

00:04:44.820 --> 00:04:47.470
Actually let, me call this
the change in time axis,

00:04:47.470 --> 00:04:50.200
because this is essentially
how many seconds have gone by.

00:04:50.200 --> 00:04:52.816
So this is my
change in time axis.

00:04:52.816 --> 00:04:55.190
And let me make it clear that
this graph-- I didn't label

00:04:55.190 --> 00:04:59.030
it here-- this is my
acceleration graph.

00:04:59.030 --> 00:05:01.260
And I'm going off of the screen.

00:05:01.260 --> 00:05:02.030
All right.

00:05:02.030 --> 00:05:04.110
So let's fill these things out.

00:05:04.110 --> 00:05:09.760
So at time 0, what is
our what is our velocity?

00:05:09.760 --> 00:05:13.930
Well, if we use this expression
right here, time 0, or delta t

00:05:13.930 --> 00:05:14.700
is equal to 0.

00:05:14.700 --> 00:05:17.130
This expression right
here, is going to be 0.

00:05:17.130 --> 00:05:19.230
And it's just going to
be our initial velocity.

00:05:19.230 --> 00:05:22.170
And in the last video, we
gave our initial velocity,

00:05:22.170 --> 00:05:26.680
is going to be as 19.6
meters per second.

00:05:26.680 --> 00:05:31.760
So it is going to be
19.6 meters per second.

00:05:31.760 --> 00:05:33.690
And let me plot that over here.

00:05:33.690 --> 00:05:38.530
At time 0, it is going to
be 19.6 meters per second.

00:05:38.530 --> 00:05:42.890
What is our initial displacement
at time zero, or change

00:05:42.890 --> 00:05:44.080
in time 0?

00:05:44.080 --> 00:05:45.580
So you look at this up here.

00:05:45.580 --> 00:05:49.150
Well, our delta t is 0, so this
expression is going to be 0,

00:05:49.150 --> 00:05:50.730
and this expression
is going to be 0.

00:05:50.730 --> 00:05:52.800
So we haven't done
any displacement yet,

00:05:52.800 --> 00:05:54.830
when no time has gone by.

00:05:54.830 --> 00:05:56.610
So we have done no displacement.

00:05:59.300 --> 00:06:01.280
We are right over there.

00:06:01.280 --> 00:06:05.280
Now, what happens after
1 second has gone by?

00:06:05.280 --> 00:06:07.280
What is now our velocity?

00:06:07.280 --> 00:06:10.898
Well, our initial velocity,
right over here, is 19.6.

00:06:10.898 --> 00:06:14.460
19.6 meters per second,
that was a given.

00:06:14.460 --> 00:06:19.830
And our acceleration is negative
9.8 meters per second squared.

00:06:19.830 --> 00:06:21.490
So it's a negative,
right over there.

00:06:21.490 --> 00:06:24.892
And then you multiply that times
delta t in every situation.

00:06:24.892 --> 00:06:26.850
So in this situation
we're going to multiply it

00:06:26.850 --> 00:06:29.460
by 1, because delta t is 1.

00:06:29.460 --> 00:06:33.790
So we get 19.6 minus
9.8, that gives us

00:06:33.790 --> 00:06:37.429
exactly 9.8 meters per second.

00:06:37.429 --> 00:06:39.720
And the units work out,
because you multiply this times

00:06:39.720 --> 00:06:41.730
seconds, this gives
you meters per second.

00:06:41.730 --> 00:06:46.032
So 19.6 meters per second minus
9.8 meters per second-- one

00:06:46.032 --> 00:06:48.490
of these seconds goes away when
you multiply it by second--

00:06:48.490 --> 00:06:51.740
gives you 9.8 meters per second.

00:06:51.740 --> 00:06:54.210
So after 1 second,
our velocity is now

00:06:54.210 --> 00:06:56.250
half of what it was before.

00:06:56.250 --> 00:07:00.030
So we're now going
9.8 meters per second.

00:07:00.030 --> 00:07:02.790
Let me draw a line here.

00:07:02.790 --> 00:07:05.750
9.8 meters per second.

00:07:05.750 --> 00:07:08.330
Now what is our displacement?

00:07:08.330 --> 00:07:09.800
So you look up here.

00:07:09.800 --> 00:07:12.220
And let me rewrite this
displacement formula,

00:07:12.220 --> 00:07:14.400
with all of the
information that we know.

00:07:14.400 --> 00:07:16.170
So we know that
displacement is going

00:07:16.170 --> 00:07:19.747
to be equal to our initial
velocity, which is 19.6--

00:07:19.747 --> 00:07:21.330
and I won't write
the units here, just

00:07:21.330 --> 00:07:26.220
for the sake of space--
times our change in time.

00:07:26.220 --> 00:07:28.610
I'll do it in that same
color, so you what's what.

00:07:28.610 --> 00:07:32.260
Times our change
in time plus 1/2.

00:07:32.260 --> 00:07:36.650
Now let me be clear, 1/2
times negative 9.8 meters

00:07:36.650 --> 00:07:37.530
per second squared.

00:07:37.530 --> 00:07:40.830
So 1/2 times a is
going to be-- actually

00:07:40.830 --> 00:07:42.480
I can rewrite this,
right over here--

00:07:42.480 --> 00:07:44.770
because this is going to
be negative 9.8 meters

00:07:44.770 --> 00:07:51.030
per second times 1/2, so this
is going to be negative 4.9.

00:07:51.030 --> 00:07:54.600
All I did, is I took 1/2
times negative 9.8 over here.

00:07:54.600 --> 00:07:56.700
1/2 times negative 9.8.

00:07:56.700 --> 00:07:57.760
And this is important.

00:07:57.760 --> 00:08:00.110
And this is why the vector
quantities start to matter.

00:08:00.110 --> 00:08:02.340
Because if you didn't, if
you put a positive here,

00:08:02.340 --> 00:08:03.720
you wouldn't have
the object actually

00:08:03.720 --> 00:08:06.136
slowing down as it went up,
because you would have gravity

00:08:06.136 --> 00:08:07.970
somehow accelerating
it as it went up.

00:08:07.970 --> 00:08:10.560
But it's actually
slowing it down.

00:08:10.560 --> 00:08:12.567
It's accelerating it in
the downwards direction.

00:08:12.567 --> 00:08:15.150
So that's why you have to have
that negative right over there.

00:08:15.150 --> 00:08:18.130
That was our convention at the
beginning of the last video.

00:08:18.130 --> 00:08:19.280
Up is positive.

00:08:19.280 --> 00:08:21.400
Down is negative.

00:08:21.400 --> 00:08:22.120
So let's focus.

00:08:22.120 --> 00:08:25.290
So this part right over
here, negative 4.9 meters

00:08:25.290 --> 00:08:27.240
per second squared
times delta t squared.

00:08:31.090 --> 00:08:32.840
And this will make it
a little bit easier,

00:08:32.840 --> 00:08:35.090
although we'll still-- let
me get the calculator out.

00:08:35.090 --> 00:08:41.830
So when one second has passed--
I'll get my trusty TI-85 out.

00:08:41.830 --> 00:08:45.220
When one second has passed, the
displacement is 19.6 times 1.

00:08:45.220 --> 00:08:47.900
Well, that's just 19.6.

00:08:47.900 --> 00:08:50.180
Minus 4.9 times 1 squared.

00:08:50.180 --> 00:08:54.700
So that's just minus 4.9.

00:08:54.700 --> 00:09:01.480
Gives us 14.7 meters.

00:09:01.480 --> 00:09:05.900
So after 1 second, the ball
has traveled 14.7 meters

00:09:05.900 --> 00:09:08.370
in the air.

00:09:08.370 --> 00:09:10.430
So that's roughly over there.

00:09:10.430 --> 00:09:12.540
Now, what happens
after 2 seconds?

00:09:12.540 --> 00:09:14.310
I'll do this in magenta.

00:09:14.310 --> 00:09:21.740
So after 2 seconds, our velocity
is 19.6 minus 9.8 times 2.

00:09:21.740 --> 00:09:23.110
2 seconds have gone by.

00:09:23.110 --> 00:09:28.950
Well, 9.8 meters per second
squared times 2 seconds

00:09:28.950 --> 00:09:32.350
gives us 19.6 meters per second.

00:09:32.350 --> 00:09:33.770
So these just cancel out.

00:09:33.770 --> 00:09:36.420
So we get, our
velocity is now 0.

00:09:36.420 --> 00:09:40.475
So after 2 seconds,
our velocity is now 0.

00:09:40.475 --> 00:09:45.732
Actually, let me-- So let me
make it so it's-- this thing

00:09:45.732 --> 00:09:46.940
should look more like a line.

00:09:46.940 --> 00:09:50.130
I don't make you get a sense--
So this is-- So let me just

00:09:50.130 --> 00:09:51.300
draw the line like this.

00:09:51.300 --> 00:09:54.790
Our velocity is now
0 after 2 seconds.

00:09:54.790 --> 00:09:57.620
What is our displacement?

00:09:57.620 --> 00:10:01.670
So we're literally at the point
where the ball has no velocity.

00:10:01.670 --> 00:10:03.230
At exactly 2 seconds.

00:10:03.230 --> 00:10:08.520
So it's kind of gone up, and
for that exact moment in time,

00:10:08.520 --> 00:10:09.930
it is stationary.

00:10:09.930 --> 00:10:13.490
And then what do we have
going on in our displacement?

00:10:13.490 --> 00:10:17.452
We have 19.6-- let me get
the calculator out for this.

00:10:17.452 --> 00:10:19.910
I could do it by
hand, but for the sake

00:10:19.910 --> 00:10:25.280
of quickness-- 19.6
times 2-- 19.6 times

00:10:25.280 --> 00:10:32.280
2 seconds minus 4.9
times 2 seconds squared.

00:10:32.280 --> 00:10:33.950
This is 2 seconds squared.

00:10:33.950 --> 00:10:35.360
Oh, I lost the calculator.

00:10:35.360 --> 00:10:36.620
Times 2 seconds squared.

00:10:36.620 --> 00:10:38.410
So that's times 4.

00:10:38.410 --> 00:10:41.550
So that gives us 19.6 meters.

00:10:41.550 --> 00:10:44.430
So we're at 19-- let me
do that in magenta-- we

00:10:44.430 --> 00:10:48.140
are at 19.6 meters.

00:10:48.140 --> 00:10:54.130
So after 2 seconds, we are
19.6 meters in the air.

00:10:54.130 --> 00:10:57.090
Now let's go to 3 seconds.

00:10:57.090 --> 00:11:01.850
So after 3 seconds,
our velocity is now--

00:11:01.850 --> 00:11:04.980
I'll just get the-- it's
19.6 meters per second

00:11:04.980 --> 00:11:07.465
minus 9.8 times 3.

00:11:07.465 --> 00:11:08.840
And we could do
that in our head,

00:11:08.840 --> 00:11:11.740
but just to verify it for us,
let me get the calculator out.

00:11:11.740 --> 00:11:16.880
It's 19.6 minus 9.8 times 3.

00:11:16.880 --> 00:11:19.910
That gives us negative
9.8 meters per second.

00:11:24.110 --> 00:11:26.080
So after 3 seconds,
our velocity is now

00:11:26.080 --> 00:11:28.560
negative 9.8 meters per second.

00:11:28.560 --> 00:11:29.470
What does that mean?

00:11:29.470 --> 00:11:31.610
It's now going in the
downward direction

00:11:31.610 --> 00:11:34.080
at 9.8 meters per second.

00:11:34.080 --> 00:11:36.180
So this is our velocity graph.

00:11:36.180 --> 00:11:38.910
And then what is our
displacement at this point?

00:11:38.910 --> 00:11:41.400
So once again, let's
get the calculator out.

00:11:41.400 --> 00:11:43.204
If you're getting
the hang of this.

00:11:43.204 --> 00:11:45.370
At any time, I encourage
you to pause it, and try it

00:11:45.370 --> 00:11:46.770
for yourself.

00:11:46.770 --> 00:11:49.420
So now, what is-- now
this is a little, OK.

00:11:49.420 --> 00:11:51.540
So I'm looking at
my displacement,

00:11:51.540 --> 00:11:52.540
I wrote right over here.

00:11:52.540 --> 00:11:55.100
So our displacement
where delta t

00:11:55.100 --> 00:12:04.340
is 3 seconds, 19.6 times 3
minus 4.9 times-- and this

00:12:04.340 --> 00:12:06.260
is delta t, so
this is 3 seconds.

00:12:06.260 --> 00:12:08.610
We're talking about when delta
t, or our change in time,

00:12:08.610 --> 00:12:09.530
is 3 seconds.

00:12:09.530 --> 00:12:10.920
So that squared.

00:12:10.920 --> 00:12:12.760
So times 9.

00:12:12.760 --> 00:12:18.500
And that gives us 14.7 meters.

00:12:18.500 --> 00:12:22.410
So after 3 seconds, we're
at 14.7 meters again.

00:12:22.410 --> 00:12:24.660
And so we're at the same
position we were at 1 second,

00:12:24.660 --> 00:12:26.743
but the difference is, now
we're moving downwards.

00:12:26.743 --> 00:12:28.630
Over here we were
moving upwards.

00:12:28.630 --> 00:12:31.330
And then finally, what
happens after 4 seconds?

00:12:31.330 --> 00:12:32.590
Well, what's our velocity?

00:12:32.590 --> 00:12:33.860
Well let me just get
the calculator out,

00:12:33.860 --> 00:12:36.480
although you might be able to
figure this out in your head.

00:12:36.480 --> 00:12:49.530
Our velocity is going to be
19.6 minus 9.8 times 4 seconds.

00:12:49.530 --> 00:12:52.050
Which is minus 19.6
meters per second.

00:12:55.430 --> 00:12:58.160
So our magnitude of our
velocity is the same

00:12:58.160 --> 00:13:01.120
as when we initially threw
the ball, except now it's

00:13:01.120 --> 00:13:02.500
going in the opposite direction.

00:13:02.500 --> 00:13:03.720
It's now going downwards.

00:13:07.380 --> 00:13:09.520
And what is our displacement?

00:13:09.520 --> 00:13:11.660
Get the calculator out.

00:13:11.660 --> 00:13:16.840
So we have, our displacement
is 19.6 times 4-- 4 seconds

00:13:16.840 --> 00:13:21.990
have gone by-- minus
4.9 times 4 squared--

00:13:21.990 --> 00:13:26.490
which is 16-- so times
16, which is equal to 0.

00:13:26.490 --> 00:13:28.450
Our displacement is 0.

00:13:28.450 --> 00:13:30.500
We are back on the ground.

00:13:30.500 --> 00:13:32.890
We are back on the ground.

00:13:32.890 --> 00:13:35.520
So if you were to
plot its displacement

00:13:35.520 --> 00:13:38.480
you would actually get a
parabola, a downward opening

00:13:38.480 --> 00:13:40.760
parabola that looks
something like this.

00:13:40.760 --> 00:13:44.320
Doing my best to draw
it relatively neatly.

00:13:44.320 --> 00:13:46.160
Actually, I can do a
better job than that.

00:13:46.160 --> 00:13:47.076
I'll do a dotted line.

00:13:47.076 --> 00:13:49.215
Dotted lines are always
easier to adjust midstream.

00:13:53.440 --> 00:13:55.082
If you plot displacement
versus time,

00:13:55.082 --> 00:13:56.290
it looks something like this.

00:13:56.290 --> 00:13:59.180
Its velocity is just this
downward sloping line,

00:13:59.180 --> 00:14:01.260
and then the
acceleration is constant.

00:14:01.260 --> 00:14:03.680
And the whole reason
why I wanted to do this,

00:14:03.680 --> 00:14:06.560
is I wanted to show you that
the velocity the whole time

00:14:06.560 --> 00:14:08.610
is decreasing at
a constant pace.

00:14:08.610 --> 00:14:12.120
And that makes sense, because
the rate at which the velocity

00:14:12.120 --> 00:14:14.790
increases or decreases
is the acceleration.

00:14:14.790 --> 00:14:19.350
And the acceleration, based on
our convention, is downwards.

00:14:19.350 --> 00:14:20.630
So that's why it's decreasing.

00:14:20.630 --> 00:14:22.710
We have a negative slope here.

00:14:22.710 --> 00:14:27.180
We have a negative slope
of negative 9.8 meters

00:14:27.180 --> 00:14:28.890
per second squared.

00:14:28.890 --> 00:14:30.992
And so just to think
about what's happening

00:14:30.992 --> 00:14:32.450
for this ball, or
this rock-- and I

00:14:32.450 --> 00:14:34.283
know this video is
getting long-- as it goes

00:14:34.283 --> 00:14:38.460
through the air, I'm going to
draw the vectors for velocity.

00:14:38.460 --> 00:14:40.750
And I'm going to
do that in orange,

00:14:40.750 --> 00:14:42.600
or no, maybe I'll
do that in blue.

00:14:42.600 --> 00:14:44.210
So velocity in blue.

00:14:44.210 --> 00:14:47.150
So right when we start,
it has a positive velocity

00:14:47.150 --> 00:14:49.340
of 19.6 meters per second.

00:14:49.340 --> 00:14:51.560
So I'll draw a big
vector like this,

00:14:51.560 --> 00:14:54.980
19.6 meters per second,
that's its velocity.

00:14:54.980 --> 00:14:57.790
Then after 1 second, it's
9.8 meters per second.

00:14:57.790 --> 00:14:59.484
So it's half of that.

00:14:59.484 --> 00:15:01.150
Maybe it would look
something like this,

00:15:01.150 --> 00:15:03.060
9.8 meters per second.

00:15:03.060 --> 00:15:06.710
Then at this peak right over
here, it has a velocity of 0.

00:15:06.710 --> 00:15:10.580
Then as you go to 3 seconds,
the magnitude of its velocity

00:15:10.580 --> 00:15:12.360
is 9.8 meters per second.

00:15:12.360 --> 00:15:14.150
But it is now downwards.

00:15:14.150 --> 00:15:16.770
It is now downwards,
so it looks like this.

00:15:16.770 --> 00:15:20.930
And then, finally, right when
it hits the ground, right

00:15:20.930 --> 00:15:23.920
before it hits the ground,
it has a negative velocity

00:15:23.920 --> 00:15:27.170
of 19.6 meters per second.

00:15:27.170 --> 00:15:32.850
So it would look like
this, roughly like this.

00:15:32.850 --> 00:15:35.070
If I use the same
scale over here.

00:15:35.070 --> 00:15:37.992
But what was the
acceleration the entire time?

00:15:37.992 --> 00:15:40.200
Well, the entire time, the
acceleration was negative.

00:15:40.200 --> 00:15:42.900
It was negative 9.8
meters per second squared.

00:15:42.900 --> 00:15:44.250
And I'll do that in orange.

00:15:44.250 --> 00:15:46.810
So the acceleration,
over here, negative-- no,

00:15:46.810 --> 00:15:49.750
I want to do that in
orange-- the acceleration was

00:15:49.750 --> 00:15:52.580
negative 9.8 meters
per second squared.

00:15:52.580 --> 00:15:55.990
Acceleration, negative 9.8
meters per second squared.

00:15:55.990 --> 00:15:57.960
Negative 9.8 meters
per second squared.

00:15:57.960 --> 00:16:01.610
The acceleration is
constant the entire time.

00:16:01.610 --> 00:16:05.520
This last one is negative 9.8
meters per second squared.

00:16:05.520 --> 00:16:07.350
It does not change,
depending where

00:16:07.350 --> 00:16:09.090
you are in the
curve, when you're

00:16:09.090 --> 00:16:10.560
near the surface of the Earth.

00:16:10.560 --> 00:16:12.190
So hopefully that clarifies
things a little bit

00:16:12.190 --> 00:16:13.606
and gives you a
good sense of what

00:16:13.606 --> 00:16:17.480
happens when you throw a
projectile into the air.


Deriving displacement as a function of time, acceleration, and initial velocity

WEBVTT
Kind: captions
Language: en

00:00:01.333 --> 00:00:03.267
What I want to do with this video

00:00:03.267 --> 00:00:06.267
is think about what happens to some type of projectile,

00:00:06.267 --> 00:00:09.800
maybe a ball or rock, if I were to throw it

00:00:09.800 --> 00:00:12.400
straight up into the air.

00:00:12.400 --> 00:00:16.600
To do that I want to plot distance relative to time.

00:00:16.600 --> 00:00:19.800
There are a few things I am going to tell you about

00:00:19.800 --> 00:00:21.400
my throwing the rock into the air.

00:00:21.400 --> 00:00:29.667
The rock will have an initial velocity (Vi) of 19.6 meters per second (19.6m/s)

00:00:29.667 --> 00:00:31.667
I picked this initial velocity because

00:00:31.667 --> 00:00:35.133
it will make the math a little bit easier.

00:00:35.133 --> 00:00:38.667
We also know the acceleration near the surface of the earth.

00:00:38.667 --> 00:00:41.267
We know the force of gravity

00:00:41.267 --> 00:00:47.267
near the surface of the earth is the mass of the object times the acceleration.

00:00:47.267 --> 00:00:48.467
(let me write this down)

00:00:48.467 --> 00:00:54.800
The force of gravity is going to be the mass of the object times little g.

00:00:54.800 --> 00:00:59.733
little g is gravity near the surface of the earth

00:00:59.733 --> 00:01:03.800
g is 9.8 meters per second squared (9.8m/s^2)

00:01:03.800 --> 00:01:06.000
Now if you want the acceleration on earth

00:01:06.000 --> 00:01:09.800
you just take the force divided by the mass

00:01:09.800 --> 00:01:12.533
Because we have the general equation

00:01:12.533 --> 00:01:15.067
Force equals mass times acceleration (F=ma)

00:01:15.067 --> 00:01:17.133
If you want acceleration divide both sides by mass

00:01:17.133 --> 00:01:19.200
so you get force over mass

00:01:19.200 --> 00:01:21.400
So, lets just divide this by mass

00:01:21.400 --> 00:01:23.800
If you divide both sides by mass,

00:01:23.800 --> 00:01:29.733
on the left hand side you will get acceleration

00:01:29.733 --> 00:01:35.600
and on the right hand side you will get the quantity little g.

00:01:35.600 --> 00:01:37.267
The whole reason why I did this is

00:01:37.267 --> 00:01:39.467
when we look at the g it really comes from

00:01:39.467 --> 00:01:41.667
the universal law of gravitation.

00:01:41.667 --> 00:01:43.800
You can really view g as

00:01:43.800 --> 00:01:50.733
measuring the gravitational field strength near the surface of the earth.

00:01:50.733 --> 00:01:54.200
Then that helps us figure out the force

00:01:54.200 --> 00:01:56.667
when you multiply mass times g.

00:01:56.667 --> 00:01:59.000
Then you use F=ma, the second law,

00:01:59.000 --> 00:02:00.867
to come up with g again

00:02:00.867 --> 00:02:02.933
which is actually the acceleration.

00:02:02.933 --> 00:02:05.667
This is accelerating you towards the center of the earth.

00:02:05.667 --> 00:02:07.333
The other thing I want to make clear:

00:02:07.333 --> 00:02:11.533
when you talk about the Force of gravity

00:02:11.533 --> 00:02:15.333
generally the force of gravity is equal to big G

00:02:15.333 --> 00:02:17.533
Big G (which is different than little g) times

00:02:17.533 --> 00:02:20.000
the product of the masses of the two things

00:02:20.000 --> 00:02:24.067
over the square of the distance between the two things.

00:02:24.067 --> 00:02:25.867
You might be saying "Wait, clearly the force of gravity is dependent on the distance.

00:02:25.867 --> 00:02:33.867
So if I were to throw something up into the air,

00:02:33.867 --> 00:02:35.533
won't the distance change."

00:02:35.533 --> 00:02:37.400
And you would be right!

00:02:37.400 --> 00:02:38.667
That is technically right, but

00:02:38.667 --> 00:02:40.867
the reality is that when you

00:02:40.867 --> 00:02:42.333
throw something up into the air

00:02:42.333 --> 00:02:43.933
that change in distance is so small

00:02:43.933 --> 00:02:45.933
relative to the distance between the object and the center of the earth

00:02:45.933 --> 00:02:48.133
that to make the math simple,

00:02:48.133 --> 00:02:52.873
When we are at or near the surface of the earth (including in our atmosphere)

00:02:52.873 --> 00:02:55.000
we can assume that it is constant.

00:02:55.000 --> 00:03:01.557
Remember that little g over there is

00:03:01.557 --> 00:03:04.343
all of these terms combined.

00:03:04.343 --> 00:03:06.990
If we assume that mass one (m1) is

00:03:06.990 --> 00:03:13.492
the mass of the earth, and

00:03:13.492 --> 00:03:23.059
r is the radius of the earth (the distance from the center of the earth)

00:03:23.059 --> 00:03:25.938
So you would be correct in thinking that it changes a little bit.

00:03:25.938 --> 00:03:28.863
The force of gravity changes a little bit, but

00:03:28.863 --> 00:03:31.139
for the sake of throwing things up into our atmosphere

00:03:31.139 --> 00:03:34.436
we can assume that it is constant.

00:03:34.436 --> 00:03:35.876
And if we were to calculate it

00:03:35.876 --> 00:03:40.056
it is 9.8 meters per second squared

00:03:40.056 --> 00:03:42.238
and I have rounded here to the nearest tenth.

00:03:42.238 --> 00:03:44.932
I want to be clear these are vector quantities.

00:03:44.932 --> 00:03:47.904
When we start throwing things up into the air

00:03:47.904 --> 00:03:49.251
the convention is

00:03:49.251 --> 00:03:52.641
if something is moving up it is given a positive value,

00:03:52.641 --> 00:03:57.331
and if it is moving down we give it a negative value.

00:03:57.331 --> 00:04:01.650
Well, for an object that is in free fall

00:04:01.650 --> 00:04:03.833
gravity would be accelerating it downwards, or

00:04:03.833 --> 00:04:05.226
the force of gravity is downwards.

00:04:05.226 --> 00:04:07.501
So, little g over here,

00:04:07.501 --> 00:04:08.755
if you want to give it its direction,

00:04:08.755 --> 00:04:14.142
is negative. Little g is -9.8m/s2.

00:04:14.142 --> 00:04:16.975
So, we have the acceleration due to gravity.

00:04:16.975 --> 00:04:19.669
The acceleration due to gravity (ag) is

00:04:19.669 --> 00:04:25.149
negative 9.8 meters per second squared (9.8m/s^2).

00:04:25.149 --> 00:04:27.471
Now I want to plot distance relative to time.

00:04:27.471 --> 00:04:32.300
Let's think about how we can set up a formula,

00:04:32.300 --> 00:04:35.830
derive a formula that, if we input time as a variable,

00:04:35.830 --> 00:04:37.827
we can get distance.

00:04:37.827 --> 00:04:39.933
We can assume these values right over here.

00:04:39.933 --> 00:04:46.093
Well actually I want to plot displacement over time because that will be more interesting.

00:04:46.093 --> 00:04:50.667
We know that displacement is

00:04:50.667 --> 00:04:59.933
the same thing as average velocity times change in time (displacement=Vavg*(t1-t2)).

00:04:59.933 --> 00:05:01.200
Right now we have

00:05:01.200 --> 00:05:03.926
something in terms of time, distance, and average velocity

00:05:03.926 --> 00:05:07.595
but not in terms of initial velocity and acceleration.

00:05:07.595 --> 00:05:12.842
We know that average velocity is the same thing as

00:05:12.842 --> 00:05:17.626
initial velocity (vi) plus final velocity (vf) over 2. (Vavg=(vi+vf)/2)

00:05:17.626 --> 00:05:20.273
If we assume constant acceleration.

00:05:20.273 --> 00:05:26.333
We can only calculate Vavg this way assuming constant acceleration.

00:05:26.333 --> 00:05:29.467
Once again when were are dealing

00:05:29.467 --> 00:05:31.836
with objects not too far from the center of the earth

00:05:31.836 --> 00:05:33.601
we can make that assumption.

00:05:33.601 --> 00:05:36.759
Assuming that we have a constant acceleration

00:05:36.759 --> 00:05:41.031
Once again we don't have what our final velocity is.

00:05:41.031 --> 00:05:43.307
So, we need to think about this a little more.

00:05:43.307 --> 00:05:48.090
We can express our final velocity in terms of our initial velocity and time.

00:05:48.090 --> 00:05:53.733
Just dealing with this part, the average velocity.

00:05:53.733 --> 00:05:56.067
So we can rewrite this expression

00:05:56.067 --> 00:05:59.667
as the initial velocity plus something over 2.

00:05:59.667 --> 00:06:01.267
and what is final velocity?

00:06:01.267 --> 00:06:05.000
Well the final velocity is going to be

00:06:05.000 --> 00:06:16.333
your initial velocity plus your acceleration times change in time.

00:06:16.333 --> 00:06:18.000
If you are starting at 10m/s

00:06:18.000 --> 00:06:20.667
and you are accelerated at 1m/s^2

00:06:20.667 --> 00:06:24.267
then after 1 second you will be going 1 second faster than that. (11m/s)

00:06:24.267 --> 00:06:29.267
So this right here is your final velocity.

00:06:29.267 --> 00:06:35.200
Let me make sure that these are all vector quantities...(draws vector arrows)

00:06:35.200 --> 00:06:37.933
All of these are vector quantities.

00:06:37.933 --> 00:06:42.533
Hopefully it is ingrained in you that these are all vector quantities, direction matters.

00:06:42.533 --> 00:06:44.467
And let's see how we can simplify this

00:06:44.467 --> 00:06:47.467
Well these two terms

00:06:47.467 --> 00:06:49.867
(remember we are just dealing with the average velocity here)

00:06:49.867 --> 00:06:52.067
These two terms if you combine them become 2 times initial velocity (2vi).

00:06:52.067 --> 00:06:57.133
two times my initial velocity

00:06:57.133 --> 00:07:00.667
and then divided by this 2

00:07:00.667 --> 00:07:05.200
plus all of this business divided by this 2.

00:07:05.200 --> 00:07:11.200
which is my acceleration times my change in time divided by 2.

00:07:11.200 --> 00:07:13.600
All of this was another way to write average velocity.

00:07:13.600 --> 00:07:16.000
the whole reason why I did this is because we don't have final velocity

00:07:16.000 --> 00:07:18.133
but we have acceleration

00:07:18.133 --> 00:07:20.733
and we are going to use change in time as our independent variable.

00:07:20.733 --> 00:07:25.133
We still have to multiply this by this green change in time here.

00:07:25.133 --> 00:07:29.422
multiply all of this times the green change in time.

00:07:29.422 --> 00:07:31.419
All of this is what displacement is going to be.

00:07:31.419 --> 00:07:32.859
This is displacement, and lets see...

00:07:32.859 --> 00:07:34.716
we can multiply the change in time times all this

00:07:34.716 --> 00:07:37.733
actually these 2s cancel out

00:07:37.733 --> 00:07:42.867
and we get (continued over here)

00:07:42.867 --> 00:07:47.933
We get: displacement is equal to

00:07:47.933 --> 00:07:52.467
initial velocity times

00:07:52.467 --> 00:08:01.533
change in time

00:08:01.533 --> 00:08:07.400
Some physics classes or textbooks put time there but it is really change in time.

00:08:07.400 --> 00:08:13.333
change in time is a little more accurate

00:08:13.333 --> 00:08:19.800
plus 1/2 (which is the same as dividing by 2)

00:08:19.800 --> 00:08:24.268
plus one half times the acceleration

00:08:24.268 --> 00:08:27.240
times the acceleration

00:08:27.240 --> 00:08:29.748
times (we have a delta t times delta t)

00:08:29.748 --> 00:08:31.733
change in time times change in time

00:08:31.733 --> 00:08:34.333
the triangle is delta and it just means "change in"

00:08:34.333 --> 00:08:36.467
so change in time times change in time

00:08:36.467 --> 00:08:41.133
is just change in times squared.

00:08:41.133 --> 00:08:44.067
In some classes you will see this written as

00:08:44.067 --> 00:08:51.333
d is equal to vi times t plus 1/2 a t squared

00:08:51.333 --> 00:08:53.200
this is the same exact thing

00:08:53.200 --> 00:08:55.733
they are just using d for displacement

00:08:55.733 --> 00:08:58.200
and t in place of delta t.

00:08:58.200 --> 00:09:01.800
The one thing I want you to realize with this video

00:09:01.800 --> 00:09:04.533
is that this is a very straight forward thing to derive.

00:09:04.533 --> 00:09:10.400
Maybe if you were under time pressure you would want to be able to whip this out,

00:09:10.400 --> 00:09:15.491
but the important thing, so you remember how to do this when you are 30 or 40 or 50

00:09:15.491 --> 00:09:17.256
or when you are an engineer and you are trying

00:09:17.256 --> 00:09:20.367
to send a rocket into space and you don't have a physics book to look it up,

00:09:20.367 --> 00:09:25.568
is that it comes from the simple displacement is equal to average velocity times change in time

00:09:25.568 --> 00:09:27.844
and we assume constant acceleration,

00:09:27.844 --> 00:09:31.234
and you can just derive the rest of this.

00:09:31.234 --> 00:09:33.277
I am going to leave you there in this video.

00:09:33.277 --> 00:09:37.667
Let me erase this part right over here.

00:09:37.667 --> 00:09:40.243
We are going to leave it right over here.

00:09:40.243 --> 00:09:42.751
In the next video we are going to use this

00:09:42.751 --> 00:09:44.284
formula we just derived.

00:09:44.284 --> 00:09:46.141
We are going to use this to actually

00:09:46.141 --> 00:09:48.667
plot the displacement vs time

00:09:48.667 --> 00:09:50.467
because that is interesting and we are going to be thinking about

00:09:50.467 --> 00:09:52.867
what happens to the velocity and the acceleration

00:09:52.867 --> 99:59:59.999
as we move further and further in time.


Would a brick or feather fall faster?

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.140
Let's say we were to take a
little excursion to the moon.

00:00:03.140 --> 00:00:05.070
And so here we are
sitting on the surface

00:00:05.070 --> 00:00:07.740
of the moon-- that's the
surface of the moon right there.

00:00:07.740 --> 00:00:10.800
And with us to our excursion to
the moon we brought two things.

00:00:10.800 --> 00:00:15.500
We brought ourself
a concrete brick.

00:00:15.500 --> 00:00:19.940
So that's my brick
right over there,

00:00:19.940 --> 00:00:23.070
although it's orange-- we'll say
it's an orange concrete brick.

00:00:23.070 --> 00:00:25.630
And I also bought a
bird feather with us.

00:00:25.630 --> 00:00:27.150
So this is the bird feather.

00:00:31.130 --> 00:00:32.900
And then my question
to you is, if I

00:00:32.900 --> 00:00:35.210
were to hold both the
brick and the bird feather

00:00:35.210 --> 00:00:38.370
at the same time, and I were
to let go of both of them

00:00:38.370 --> 00:00:41.580
at the same time and ask
you, which one of them

00:00:41.580 --> 00:00:46.590
would hit the surface of the
moon first, what would you say?

00:00:46.590 --> 00:00:49.090
Well, if you based it on
your experience on Earth--

00:00:49.090 --> 00:00:51.740
on Earth if you were to
take a break and a feather,

00:00:51.740 --> 00:00:54.560
a brick would just
go straight down.

00:00:54.560 --> 00:00:56.716
A brick would just
immediately fall to the Earth,

00:00:56.716 --> 00:00:58.090
and it would do
it quite quickly.

00:00:58.090 --> 00:00:59.880
It would accelerate
quite quickly.

00:00:59.880 --> 00:01:01.790
While a feather would
kind of float around.

00:01:01.790 --> 00:01:07.380
If you had a feather on Earth,
it would just float around.

00:01:07.380 --> 00:01:09.420
It would go that way,
then it would go that way,

00:01:09.420 --> 00:01:13.370
and it would slowly make
its way down to the ground.

00:01:13.370 --> 00:01:16.190
So on Earth, at least
in the presence of air,

00:01:16.190 --> 00:01:19.010
it looks like the brick
will hit the ground first.

00:01:19.010 --> 00:01:20.780
But what would
happen at the moon?

00:01:20.780 --> 00:01:25.050
And what's interesting about the
moon is we have no atmosphere.

00:01:25.050 --> 00:01:29.440
We have no air to
provide resistance

00:01:29.440 --> 00:01:31.210
for either the brick
or the feather.

00:01:31.210 --> 00:01:33.550
So what do you think
is going to happen?

00:01:33.550 --> 00:01:35.350
So your first temptation
would say, well,

00:01:35.350 --> 00:01:40.120
let's just use the
universal law of gravity.

00:01:40.120 --> 00:01:42.950
So what is the force of
gravity on the brick?

00:01:46.212 --> 00:01:47.670
Well, you could
calculate that out.

00:01:47.670 --> 00:01:49.980
The force of
gravity on the brick

00:01:49.980 --> 00:01:58.210
is going to be equal to big G
times the mass of the moon--

00:01:58.210 --> 00:02:00.660
I'll say that's m
for mass and then

00:02:00.660 --> 00:02:03.850
the subscript is lowercase
m for moon-- the mass

00:02:03.850 --> 00:02:11.490
of the moon times the
mass of the brick divided

00:02:11.490 --> 00:02:14.810
by the distance
between the brick

00:02:14.810 --> 00:02:17.015
and the center of
the moon squared.

00:02:21.660 --> 00:02:22.210
Fair enough.

00:02:22.210 --> 00:02:23.460
That's the force on the brick.

00:02:23.460 --> 00:02:28.176
What's going to be the force
due to gravity on the feather?

00:02:28.176 --> 00:02:29.550
Or another way to
think about it,

00:02:29.550 --> 00:02:31.180
the weight of the
feather on the moon?

00:02:33.930 --> 00:02:35.180
We'll do the same calculation.

00:02:35.180 --> 00:02:39.870
The force on the
feather is going

00:02:39.870 --> 00:02:45.970
to be equal to big G times
the mass of the moon times

00:02:45.970 --> 00:02:52.370
the mass of the feather
divided by the distance

00:02:52.370 --> 00:02:54.150
between the center
of this feather

00:02:54.150 --> 00:02:57.120
and the center of
the moon squared.

00:02:57.120 --> 00:02:59.560
That's the distance,
and then we square it.

00:02:59.560 --> 00:03:03.834
So if you look at both
of these expressions,

00:03:03.834 --> 00:03:05.250
they both have
this quantity right

00:03:05.250 --> 00:03:07.530
over here-- G times the
mass of the moon divided

00:03:07.530 --> 00:03:09.730
by the distance
between this height

00:03:09.730 --> 00:03:12.150
and the center of
the moon squared.

00:03:12.150 --> 00:03:15.730
So they both have this
exact expression on it.

00:03:15.730 --> 00:03:17.910
So let's replace
that expression.

00:03:17.910 --> 00:03:20.130
Let's just call that
the gravitational field

00:03:20.130 --> 00:03:20.830
on the moon.

00:03:20.830 --> 00:03:22.655
So if you apply this
number by any mass,

00:03:22.655 --> 00:03:25.180
it will tell you the weight
of that object on the moon,

00:03:25.180 --> 00:03:27.420
or the gravitational
force acting downward

00:03:27.420 --> 00:03:28.990
on that object on the moon.

00:03:28.990 --> 00:03:31.310
So this is the gravitational
field of the moon.

00:03:31.310 --> 00:03:33.280
So I'll just call it g sub m.

00:03:33.280 --> 00:03:36.420
And all it is, is all of
these quantities combined.

00:03:36.420 --> 00:03:38.940
So if we simplify
that way, the force

00:03:38.940 --> 00:03:41.700
on the brick due to
the moon is going

00:03:41.700 --> 00:03:45.020
to be equal to that lowercase
g on the moon-- normally

00:03:45.020 --> 00:03:47.710
we use this lowercase g for
the gravitational constant

00:03:47.710 --> 00:03:50.887
on Earth, or the
gravitational field on Earth,

00:03:50.887 --> 00:03:52.970
or sometimes the acceleration
of gravity on Earth,

00:03:52.970 --> 00:03:54.470
but now we're
referring to the moon.

00:03:54.470 --> 00:03:57.320
That's what this lowercase
subscript m is doing for us.

00:03:57.320 --> 00:04:01.240
So it's equal to that times
the mass of the brick.

00:04:01.240 --> 00:04:05.200
For the case of the feather,
the force on the feather

00:04:05.200 --> 00:04:07.670
is equal to all
of this business.

00:04:07.670 --> 00:04:12.665
So that's the g sub m times
the mass of the feather.

00:04:23.090 --> 00:04:25.440
So we're going to assume,
which is a reasonable thing

00:04:25.440 --> 00:04:27.760
to assume, that the
mass of the brick

00:04:27.760 --> 00:04:31.810
is greater than the
mass of the feather.

00:04:31.810 --> 00:04:34.160
What's going to be
their relative forces?

00:04:34.160 --> 00:04:36.720
Well, here you have a greater
mass times the same quantity.

00:04:36.720 --> 00:04:40.327
Here you have a smaller mass
times the same quantity.

00:04:40.327 --> 00:04:41.910
So if the mass of
the brick is greater

00:04:41.910 --> 00:04:44.243
than the mass of the feather,
it's completely reasonable

00:04:44.243 --> 00:04:47.130
to say that the force
of gravity on the brick

00:04:47.130 --> 00:04:50.000
is going to be greater
than the force of gravity

00:04:50.000 --> 00:04:51.690
on the feather.

00:04:51.690 --> 00:04:53.370
So if you do all
this, and everything

00:04:53.370 --> 00:04:55.527
we've done to this point
is correct, you might say,

00:04:55.527 --> 00:04:57.610
hey, there's going to be
more force due to gravity

00:04:57.610 --> 00:05:00.090
on the brick, and that's
why the brick will

00:05:00.090 --> 00:05:03.690
be accelerated
down more quickly.

00:05:03.690 --> 00:05:05.500
But what you need to
remember is that there

00:05:05.500 --> 00:05:09.820
is more gravitational
force on this brick.

00:05:09.820 --> 00:05:12.070
But it also has greater mass.

00:05:12.070 --> 00:05:14.170
And we remember the
larger something's mass

00:05:14.170 --> 00:05:20.490
is, the less acceleration it'll
experience for a given force.

00:05:20.490 --> 00:05:23.210
So what really determines how
quickly either of these things

00:05:23.210 --> 00:05:25.282
will fall is their
accelerations.

00:05:25.282 --> 00:05:26.990
And let's figure out
their accelerations.

00:05:29.780 --> 00:05:31.660
I'll do this in a neutral color.

00:05:31.660 --> 00:05:35.102
We know that force is equal
to mass times acceleration.

00:05:35.102 --> 00:05:37.560
So if we want to figure out
the acceleration of the brick--

00:05:37.560 --> 00:05:39.470
or we could write
it the other way.

00:05:39.470 --> 00:05:42.130
If we divide both
sides by mass, we

00:05:42.130 --> 00:05:44.630
get acceleration is equal
to force divided by mass.

00:05:50.147 --> 00:05:51.730
And acceleration is
a vector quantity,

00:05:51.730 --> 00:05:53.780
and force is also
a vector quantity.

00:05:53.780 --> 00:06:02.550
And in this situation, we're
not using any actual value.

00:06:02.550 --> 00:06:04.400
But if I were using
actual values,

00:06:04.400 --> 00:06:07.250
I would use negative numbers for
downwards and positive values

00:06:07.250 --> 00:06:08.000
for upwards.

00:06:08.000 --> 00:06:09.720
But we're not using
any signs here.

00:06:09.720 --> 00:06:12.470
But you could assume that
the direction is implicitly

00:06:12.470 --> 00:06:13.700
being given.

00:06:13.700 --> 00:06:15.690
So what's the
acceleration of the brick?

00:06:18.610 --> 00:06:20.780
That's a lowercase
b I was writing.

00:06:20.780 --> 00:06:23.160
The acceleration of
the brick is going

00:06:23.160 --> 00:06:30.720
to be equal to the force
applied to the brick divided

00:06:30.720 --> 00:06:32.174
by the mass of the brick.

00:06:32.174 --> 00:06:34.340
But the force applied to
the brick, we already know,

00:06:34.340 --> 00:06:36.300
is this business
right over here.

00:06:36.300 --> 00:06:40.640
It is little g on the moon,
the gravitational field

00:06:40.640 --> 00:06:44.100
on the moon, times
the mass of the brick,

00:06:44.100 --> 00:06:46.930
and we're dividing that
by the mass of the brick.

00:06:46.930 --> 00:06:50.260
So the acceleration on
the brick on the moon--

00:06:50.260 --> 00:06:52.530
the acceleration that the
brick will experience--

00:06:52.530 --> 00:06:56.750
is the same thing as that
gravitational field expression.

00:06:56.750 --> 00:06:58.370
It is g sub m.

00:06:58.370 --> 00:07:01.080
This is how quickly it would
accelerate on the moon.

00:07:01.080 --> 00:07:02.980
Now let's do the same
thing for the feather.

00:07:02.980 --> 00:07:04.480
I think you see
where this is going.

00:07:04.480 --> 00:07:08.780
The acceleration
of the feather is

00:07:08.780 --> 00:07:12.520
going to be the force
on the feather divided

00:07:12.520 --> 00:07:14.240
by the mass of the feather.

00:07:14.240 --> 00:07:18.530
The force on the
feather is g sub m--

00:07:18.530 --> 00:07:24.022
g with the subscript m--
times the mass of the feather,

00:07:24.022 --> 00:07:25.480
and then we're
going to divide that

00:07:25.480 --> 00:07:27.210
by the mass of the feather.

00:07:27.210 --> 00:07:29.490
And so, once again,
its acceleration

00:07:29.490 --> 00:07:31.660
is going to be
the same quantity.

00:07:31.660 --> 00:07:34.660
So they are both going to
accelerate at the same rate

00:07:34.660 --> 00:07:36.600
downwards, which
tells us that they'll

00:07:36.600 --> 00:07:38.440
both hit the ground
at the same rate.

00:07:38.440 --> 00:07:41.406
They'll both accelerate from
the same point at the same time,

00:07:41.406 --> 00:07:43.030
and they'll both have
the same velocity

00:07:43.030 --> 00:07:44.071
when they hit the ground.

00:07:44.071 --> 00:07:46.640
And they'll both hit it
at the exact same time,

00:07:46.640 --> 00:07:48.940
despite one having
a larger mass.

00:07:48.940 --> 00:07:55.640
So the reality is, because
it has a larger mass,

00:07:55.640 --> 00:07:59.470
it has a larger gravitational
attraction to the moon.

00:07:59.470 --> 00:08:01.840
But because of its
mass, that attraction

00:08:01.840 --> 00:08:05.730
gives it the same acceleration
as something with a smaller

00:08:05.730 --> 00:08:06.380
mass.

00:08:06.380 --> 00:08:09.790
So any mass at the same level
on the surface of the moon

00:08:09.790 --> 00:08:12.920
would experience the
same acceleration.

00:08:12.920 --> 00:08:14.780
So now the quite
natural question

00:08:14.780 --> 00:08:17.150
is, wait, Sal, if that's true
on the moon it should also

00:08:17.150 --> 00:08:18.130
be true on Earth.

00:08:18.130 --> 00:08:19.380
And it would be true on Earth.

00:08:19.380 --> 00:08:21.500
If you did this
exact same experiment

00:08:21.500 --> 00:08:23.800
and you evacuated all
the air from the room,

00:08:23.800 --> 00:08:25.425
so that you didn't
have air resistance,

00:08:25.425 --> 00:08:26.850
and you took a
brick and a feather

00:08:26.850 --> 00:08:28.360
and let them go
at the same time,

00:08:28.360 --> 00:08:31.230
they will both hit the ground
at the exact same time, which

00:08:31.230 --> 00:08:33.877
is a little unintuitive,
to imagine a feather just

00:08:33.877 --> 00:08:35.460
plummeting the same
way a brick would.

00:08:35.460 --> 00:08:38.100
But it would if you
evacuated all the air.

00:08:38.100 --> 00:08:40.502
And so the reason why
we see this over here,

00:08:40.502 --> 00:08:42.460
and I think you get the
sense because I already

00:08:42.460 --> 00:08:44.482
talked about
evacuating the air, is

00:08:44.482 --> 00:08:46.690
that the difference between
the brick and the feather

00:08:46.690 --> 00:08:49.224
is all due to air resistance.

00:08:49.224 --> 00:08:51.390
If you took the same brick,
or if you took something

00:08:51.390 --> 00:08:53.170
that had the same
mass as the brick,

00:08:53.170 --> 00:08:59.647
and you were to flatten it out
so it has more air resistance--

00:08:59.647 --> 00:09:01.980
but let's say it has the same
mass, let's say this thing

00:09:01.980 --> 00:09:03.350
and this thing have
the same mass--

00:09:03.350 --> 00:09:04.770
this thing would
fall slower than

00:09:04.770 --> 00:09:08.110
that because it'll have
more air resistance.

00:09:08.110 --> 00:09:11.470
It has more air to collide
into, to provide resistance

00:09:11.470 --> 00:09:13.180
as it falls.

00:09:13.180 --> 00:09:15.900
And if you took a feather and
if you compacted it really,

00:09:15.900 --> 00:09:18.400
really, really, really, really,
really small-- the same mass

00:09:18.400 --> 00:09:21.370
as a feather, but you made
it so small that it could

00:09:21.370 --> 00:09:25.340
cut through the air-- you'll see
that it will drop a lot faster.

00:09:25.340 --> 00:09:28.980
So the real difference between
how things fall on Earth--

00:09:28.980 --> 00:09:30.960
if you had no air,
they would all

00:09:30.960 --> 00:09:33.220
fall at the exact same rate.

00:09:33.220 --> 00:09:36.464
It's only because of air that
they fall at different rates.

00:09:36.464 --> 00:09:37.630
And the air does two things.

00:09:37.630 --> 00:09:39.280
For constant
pressure-- so if you

00:09:39.280 --> 00:09:41.210
have two objects that
have the same shape,

00:09:41.210 --> 00:09:44.430
the object that is heavier,
that has more weight,

00:09:44.430 --> 00:09:46.910
will fall faster
because it'll overcome--

00:09:46.910 --> 00:09:49.930
it'll be able to provide more
net force against the air

00:09:49.930 --> 00:09:50.560
pressure.

00:09:50.560 --> 00:09:52.540
If you have something
that has the same weight,

00:09:52.540 --> 00:09:54.430
the object that is
more aerodynamic

00:09:54.430 --> 00:09:56.520
will fall faster-- the
one that cuts through,

00:09:56.520 --> 00:09:58.670
the one that has the
least air resistance.

00:09:58.670 --> 00:10:00.140
And as a little
experiment that you

00:10:00.140 --> 00:10:04.900
can try in the comfort of
your own room right now,

00:10:04.900 --> 00:10:10.490
take a book like this.

00:10:10.490 --> 00:10:11.452
And you could drop it.

00:10:11.452 --> 00:10:13.410
And then you could take
another piece of paper,

00:10:13.410 --> 00:10:15.989
or even a little postcard or
something, and you drop it.

00:10:15.989 --> 00:10:17.530
And you'll see, of
course, a postcard

00:10:17.530 --> 00:10:19.980
will fall much slower
than this book.

00:10:19.980 --> 00:10:23.730
But what you do is put the
postcard on top of the book

00:10:23.730 --> 00:10:25.830
so that the book is
essentially breaking

00:10:25.830 --> 00:10:28.110
all of the air resistance
for the postcard.

00:10:28.110 --> 00:10:30.812
And what you'll see is, if
you put it on top of the book

00:10:30.812 --> 00:10:32.270
and you were to
drop it, you'll see

00:10:32.270 --> 00:10:35.040
that they fall at
the exact same rate.


Mass and weight clarification

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.040
In this video, I want
to clarify two ideas

00:00:03.040 --> 00:00:04.940
that we talk about
on a regular basis,

00:00:04.940 --> 00:00:08.200
but are really muddled up
in our popular language.

00:00:08.200 --> 00:00:13.380
And these are the ideas
of mass and weight.

00:00:13.380 --> 00:00:15.890
And first, I'll tell
you what they are.

00:00:15.890 --> 00:00:18.860
And then, we'll talk about
how they are muddled up.

00:00:18.860 --> 00:00:22.170
So mass is literally-- there's
a couple of ways to view mass.

00:00:22.170 --> 00:00:24.050
One way to view mass
is-- and this is not

00:00:24.050 --> 00:00:25.883
a technical definition,
but it will give you

00:00:25.883 --> 00:00:29.550
a sense of it-- is how
much the stuff there is.

00:00:29.550 --> 00:00:32.509
So this is similar
to saying matter.

00:00:32.509 --> 00:00:37.560
So if I have more
molecules of a given mass,

00:00:37.560 --> 00:00:39.120
I will have a
total of more mass.

00:00:39.120 --> 00:00:41.560
Or if I have more atoms,
I will have more mass.

00:00:41.560 --> 00:00:45.005
So how much stuff
there is of something.

00:00:45.005 --> 00:00:47.130
And I want to be careful
with this definition right

00:00:47.130 --> 00:00:50.096
here, because there are
other things that aren't what

00:00:50.096 --> 00:00:51.970
we would traditionally
associate with matter,

00:00:51.970 --> 00:00:54.360
once we start going into
more fancy physics, that

00:00:54.360 --> 00:00:56.310
still will exhibit mass.

00:00:56.310 --> 00:01:02.200
So another way to define
mass is, how does something

00:01:02.200 --> 00:01:04.129
react to a specific force?

00:01:04.129 --> 00:01:06.410
And we already learned
from Newton's second law

00:01:06.410 --> 00:01:09.800
that if you have a given
force and you have more mass,

00:01:09.800 --> 00:01:11.180
you'll accelerate less.

00:01:11.180 --> 00:01:13.540
If you have less mass,
you'll accelerate more.

00:01:13.540 --> 00:01:24.050
So how something responds
to a given force.

00:01:24.050 --> 00:01:26.630
Something with lower
mass will accelerate more

00:01:26.630 --> 00:01:27.380
for a given force.

00:01:27.380 --> 00:01:29.820
Something with higher
mass will accelerate less.

00:01:29.820 --> 00:01:35.930
Now weight is the force
of gravity on a mass,

00:01:35.930 --> 00:01:37.890
or on an object.

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So this is the force of
gravity on an object.

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And just to think about
the difference here,

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let's think about, I guess,
myself sitting on Earth.

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So if I'm on Earth, my
mass is 70 kilograms.

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My mass-- let me do
this in a new color--

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so my mass is 70 kilograms.

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There's 70 kilograms of
stuff that constitute Sal.

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But my weight is
not 70 kilograms.

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I mean, you'll often hear people
say, I weighs 70 kilograms.

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And that's all right in
just conversational usage,

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but that is not
technically correct.

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Because weight is the force
that Earth is pulling down--

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or I should say-- the force
of gravity on my mass.

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And so my weight-- let me think
about the weight for a second--

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the weight is going to be equal
to the gravitational field

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at Earth.

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Hopefully you've watched
the video on gravity.

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Or if you haven't,
feel free to watch it.

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But the gravitational
attraction between two objects,

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so the force of gravity
between two objects

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is going to big G, the universal
gravitational constant,

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times the mass of
the first object--

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let me actually-- times the mass
of the second object, divided

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by the distance that separates
the two objects squared.

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And if you're on the Earth, and
if you take all of this stuff

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right over here combined-- so
if you say that this right here

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is the mass of Earth.

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If you say this, right
here, is the distance

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from the center to
the surface of Earth,

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because that's where
I'm sitting right now.

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So distance from center
to surface of Earth.

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Then all of this stuff over here
simplifies to what's sometimes

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called as lower case g.

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And lower case g
is-- and I'm just

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rounding it here-- 9.8
meters per second squared.

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So the force of
gravity for something

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near the surface of
the Earth is going

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to be this quantity right
over here times the mass.

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So my weight on the
surface of the Earth

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is this 9.8 meters
per second squared

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times my mass,
times 70 kilograms.

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And so this is going
to be-- I won't do,

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well, I could get
my calculator out.

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Why don't I just get my
calculator out and do the math?

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I was going to round it to
10 and say it's about 700,

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but let's just
actually calculate it.

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So we have 9.8
times 70 kilograms.

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So we have 686.

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So this gives me 686.

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And then the units are kilogram
meters per second squared.

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And these units, kilogram
meters per second squared,

00:05:04.760 --> 00:05:08.040
are the same thing as a newton.

00:05:08.040 --> 00:05:09.750
So my weight-- and
you'll never hear

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people say this-- but my weight
on the surface of the Earth

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is 686 newtons.

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And notice, I just
said that is my weight

00:05:21.610 --> 00:05:24.100
on the surface of the Earth.

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Because as you could
imagine, weight

00:05:25.630 --> 00:05:29.370
is the force due to gravity
on an object, on a mass.

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So if I go someplace else, if
I go to the moon, for example,

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my weight will change.

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But my mass will not.

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So let's write this.

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This is the weight on Earth.

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If I were to take my
weight on the moon-- and I

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haven't looked this
up before the video.

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And you can verify this
for me if you like.

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But I've been told that
the gravitational force

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on the moon, or the
gravitational attraction

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at the surface of the
moon, is about 1/6 that

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of the surface of the Earth.

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So my weight on the
moon will be roughly 1/6

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of my weight on the Earth.

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Times 686 newtons.

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So that gives me a
little bit over-- what

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is that-- 114 maybe?

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I'll just get the
calculator out.

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My brain operates
a little bit slower

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while I'm recording videos.

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Yeah.

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I got it right.

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114.

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So that gets us 114,
approximately 114 newtons.

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So this is the thing I really
want to emphasize then.

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Weight is a force due
to gravity on an object.

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Your weight changes from
planet to planet you go on.

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Your weight would
actually even change

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if you went to a
very high altitude

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because you're getting
slightly further--

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it would be immeasurably
small-- but you're

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getting slightly further
from the center of the Earth.

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Your weight would change
an imperceptible amount.

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In fact, because the Earth
is not a perfect sphere--

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it's often referred to
as an oblique spheroid--

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your weight is actually slightly
different on different parts

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of the Earth.

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If you went to the poles
verses the equator,

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you would have a slightly
different weight.

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Your mass does not change.

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It doesn't matter
where you go, assuming

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that you don't have some type
of nuclear reaction going

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on inside of you.

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Your mass does not change.

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So your mass does not change
depending on where you are.

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Now, you might be
saying, hey, look,

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I don't deal with kilograms,
and newtons, and all of this.

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I operate in America.

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And in America, we
talk about pounds.

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Is pounds appropriate?

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And yes, pounds is
a unit of weight.

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So if I say that I weigh 160
pounds, this is indeed weight.

00:08:06.510 --> 00:08:13.110
I'm saying that the force of
gravity on me is 160 pounds.

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But then you might say,
well, what is mass then

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if you're talking about
the English system

00:08:18.120 --> 00:08:20.410
or sometimes called
the imperial system?

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And here, I will
introduce you to a concept

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that very few people know.

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It's kind of a good
trivial concept.

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The unit of mass--
so let's just be

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clear here-- the unit of mass
in the imperial system, mass

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is called the slug.

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So if you wanted to
figure out how many

00:08:41.679 --> 00:08:45.420
slugs you are, so your weight--
the force of gravity on you

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is 160 pounds.

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This is going to be
equal to-- if you

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were to calculate all of this
stuff-- the force of gravity

00:08:56.450 --> 00:08:57.450
on the surface of Earth.

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But if you were to do
it in imperial units,

00:09:00.960 --> 00:09:07.990
instead of getting 9.8
meters per second squared,

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you would get 32 feet
per second squared, which

00:09:12.010 --> 00:09:15.290
is also the acceleration
near the surface of the Earth

00:09:15.290 --> 00:09:17.530
due to gravity in
feet and seconds,

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as opposed to
meters and seconds.

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And then, this is times
your mass in slugs.

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So to figure it out,
you divide both sides

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by 32 feet per second squared.

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So let's do that.

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Let's divide both sides by
32 feet per second squared.

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It cancels out.

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And then let me get
my calculator out.

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So I have 160 pounds divided
by 32 feet per second squared.

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And I get exactly 5.

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I should have been able
to do that in my head.

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So I get 5.

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And the units here-- in the
numerator, I have pounds.

00:10:02.840 --> 00:10:04.980
And then I'm dividing by
feet per second squared.

00:10:04.980 --> 00:10:07.110
That's the same
thing as multiplying

00:10:07.110 --> 00:10:10.570
by second squared per feet.

00:10:10.570 --> 00:10:15.420
And these units, 5 pounds
second squareds over feet, this

00:10:15.420 --> 00:10:18.160
is the same thing as a slug.

00:10:18.160 --> 00:10:22.320
So if I weigh 160
pounds, my mass

00:10:22.320 --> 00:10:28.210
is going to be equal to 5 slugs.

00:10:28.210 --> 00:10:37.520
If my mass is 70 kilograms,
my weight is 686 newtons.

00:10:37.520 --> 00:10:40.780
So hopefully that clarifies
things a little bit.