Newton's third law of motion

WEBVTT
Kind: captions
Language: en

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We're now ready for Newton's
third law of motion.

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And something, once again,
you've probably heard,

00:00:06.480 --> 00:00:07.302
people talk about.

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But in this video, I want to
make sure we really understand

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what Newton is talking
about when he says--

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this is a translation of
the Latin version of it--

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to every action, there's
always-- and just

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to be clear--
Newton was English,

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but he wrote it in Latin
because at that point in time,

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people wrote things
in Latin because it

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was viewed as a more
serious language.

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But anyway-- to every
action, there is always

00:00:28.910 --> 00:00:32.360
an equal and opposite
reaction, or the forces

00:00:32.360 --> 00:00:35.210
of two bodies on each
other are always equal

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and are directed in
opposite directions.

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So what Newton is saying
is that you can't just

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have a force acting
on some object

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without that object also
having an opposite force acting

00:00:47.350 --> 00:00:49.400
on the thing that's
trying to act on it.

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And just to make it clear,
let's say that we have a--

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and we'll talk about these
examples in the second.

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Let's say that I have some
type of block right over here.

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And that I move, and
I press on the block

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and I try to push it forward.

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So this is my hand.

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This is my hand trying
to press on the block

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and exert a force, a net
force in that direction.

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So that the block
moves to the right.

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Maybe this block is
sitting on some type of ice

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so that it can move.

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So let's say that I have some--
that doesn't look like ice--

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I'll give it a more
ice-like color.

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So the block is sitting on,
maybe, some ice like that.

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So Newton's third law
is saying, look, I

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can press on this
block, and sure,

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I'll exert a net
force on this block

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and that net force will
accelerate the block assuming

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that I can overcome
friction, and if it's on ice

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I can do that.

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But that block is going to exert
an equal and opposite force

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on me.

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And for direct evidence--
this is something,

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even though it might not
be so intuitive, when

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it's said-- this equal
and opposite force.

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But direct evidence that it's
exerting an equal and opposite

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force-- is that my hand
will get compressed.

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I could actually feel the
block exerting pressure on me.

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Take your hand right now and
push it against your desk

00:02:12.740 --> 00:02:15.420
or whatever you have
nearby and you are clearly

00:02:15.420 --> 00:02:17.340
exerting a force on the desk.

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So let me draw-- so let's
say I have a desk right here.

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And if I try to push on the
desk-- so once again that's

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my hand right here,
pushing on the desk.

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If I push on the
desk, and I'm actually

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doing it right now while
I record this video.

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You'll see.

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So I'm clearly exerting
a force on the desk,

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if I do it hard
enough, I might even

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get the desk to shake
or tilt a little bit.

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And I'm actually
doing that right now.

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But at the same time, you'll
see that your hand is getting

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compressed, the palm of your
hand is being pressed down.

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And that's because
the desk is exerting

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an equal and opposite
force on you.

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If it wasn't, you
actually wouldn't even

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feel it, because you wouldn't
even feel the pressure.

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Your hand would be
completely uncompressed.

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Another example of that-- say
you're walking in the beach,

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and you have some
sand right here.

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If you were to step on the sand.

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So let's say that
this is your shoe.

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I'll do my best
attempt to draw a shoe.

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So this is the shoe.

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If you were to step on
the sand, clearly you

00:03:20.970 --> 00:03:25.340
are exerting a
force on the sand.

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The force that you're
exerting on the sand

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is the force of your weight.

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The gravitational attraction
between you and the Earth.

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You are exerting
that on the sand.

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The sand is also-- and
another evidence of that

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is that the sand is
going to be displaced.

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You're going to
create a footprint.

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The sand is going to
move out of the way

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because it's being
pushed down so hard.

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So clearly you are exerting
a force on the sand.

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But the sand is also exerting
an equal and opposite force

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on you.

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And what's the evidence of that?

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Well, if you believe,
Newton's second law--

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if you have this
gravitational force on you,

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you should be
accelerating downwards

00:04:09.450 --> 00:04:13.180
unless there is some other
force that balances it out.

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And the force that
balances it out

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is the force that the beach,
or the sand is exerting on you

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upwards.

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And so when you
net them out there

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is a zero net force on you.

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And that's why you
get to stay there.

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Why you don't start
accelerating down

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towards the center of the Earth.

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Other examples of this- this is
maybe the most famous example

00:04:33.585 --> 00:04:37.240
of Newton's third law--
is just how rockets work.

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When you're in a
rocket, either trying

00:04:39.130 --> 00:04:41.700
to escape the atmosphere,
or maybe you're in space,

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there's nothing to push off
of, nothing to push off,

00:04:44.280 --> 00:04:47.090
that lets you accelerate.

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So what you do is you keep stuff
to push off in your fuel tanks,

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and when you allow the
proper chemical reactions

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or the proper combustion
to take place,

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what it does is it expels gases
at ultra high velocities out

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the back of your rockets.

00:05:03.500 --> 00:05:05.450
And each of those
particles you're

00:05:05.450 --> 00:05:06.954
exerting a force on them.

00:05:06.954 --> 00:05:09.370
Enough force even though they're
super small mass for each

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of them, they're going
at super high velocity.

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So they're being
accelerated tremendously.

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So there's an equal
and opposite force

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on the rocket, the thing that
is actually expelling the gas.

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And so that's what
allows a rocket

00:05:23.120 --> 00:05:26.690
to accelerate even when there's
nothing in this direct vicinity

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to push off of.

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It just expels a bunch
of things or accelerates

00:05:31.210 --> 00:05:33.300
a bunch of things at
a super fast rate.

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It exerts a force on
all these particles,

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and that allows an
equal and opposite force

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to accelerate the rocket ahead.

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And another example of
this is, if you ever

00:05:42.580 --> 00:05:44.035
find yourself drifting in space.

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And this is an actually
useful example,

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so that you don't end up
drifting in space forever.

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Let's say, we don't ever
want this to happen.

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This astronaut,
by some chance he

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loses his connection
to this little tool arm

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right here in the space shuttle,
and he starts drifting away.

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What can that
astronaut do to change

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the direction of his motion
so that he drifts back

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to the space shuttle?

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Well you look around, there's
nothing to push off of.

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He doesn't have any
wall to push off of.

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Let's just assume he
doesn't have any rocket jets

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or anything like that.

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What could he do?

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Well the one thing
you could do--

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and this is the situation if
you're ever drifting in space--

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is you should find
the heaviest or I

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should say the most
massive thing on you--

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and we'll explain the difference
between mass and weight

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in a future video-- but
you should find the most

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massive thing that you can
carry, that you can take off

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of you, that you could throw.

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And you should throw it in a
direction opposite yourself.

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So let me put it this way.

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If I throw-- let's say I'm
in space and I'm floating.

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I'll make it look like the glove
of a-- so let's say that this

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the glove of the astronaut.

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There you go.

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That's his hand, that's
the astronaut's hand

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right over here.

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And let's say he find some
piece of equipment on his,

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or she finds some piece
of equipment on them

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that they can throw.

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They can take off
of their tool set.

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And they could find
the most massive object

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that they could throw.

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So what's going to happen
is-- for some period of time,

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while they push
this object away,

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they will be exerting a force
on that object for some period

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of time while they have
contact with the object.

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And that entire
time, that object,

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while it is accelerating--
while the astronaut is exerting

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a force on it-- will be exerting
an equal and opposite force

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on the hand of the astronaut,
or on the astronaut itself.

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So the object will
accelerate in that direction,

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and while the
astronaut is pushing,

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the astronaut will
accelerate in this direction.

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So what you do is you throw
in the opposite direction

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and that'll allow the astronaut
to accelerate towards the space

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shuttle and hopefully
grab on to something.


Newton's second law of motion

WEBVTT
Kind: captions
Language: en

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Newton's First Law tells
us that an object at rest

00:00:03.560 --> 00:00:07.096
will stay at rest, and object
with a constant velocity

00:00:07.096 --> 00:00:08.720
will keep having that
constant velocity

00:00:08.720 --> 00:00:12.420
unless it's affected by
some type of net force.

00:00:12.420 --> 00:00:16.620
Or you actually could say an
object with constant velocity

00:00:16.620 --> 00:00:19.130
will stay having a
constant velocity

00:00:19.130 --> 00:00:20.930
unless it's affected
by net force.

00:00:20.930 --> 00:00:23.140
Because really, this
takes into consideration

00:00:23.140 --> 00:00:25.040
the situation where
an object is at rest.

00:00:25.040 --> 00:00:26.750
You could just have
a situation where

00:00:26.750 --> 00:00:28.700
the constant velocity is zero.

00:00:28.700 --> 00:00:30.220
So Newton's First
Law, you're going

00:00:30.220 --> 00:00:31.600
to have your constant velocity.

00:00:31.600 --> 00:00:32.360
It could be zero.

00:00:32.360 --> 00:00:34.318
It's going to stay being
that constant velocity

00:00:34.318 --> 00:00:35.930
unless it's affected,
unless there's

00:00:35.930 --> 00:00:38.600
some net force that acts on it.

00:00:38.600 --> 00:00:40.430
So that leads to the
natural question,

00:00:40.430 --> 00:00:44.210
how does a net force affect
the constant velocity?

00:00:44.210 --> 00:00:46.930
Or how does it affect of
the state of an object?

00:00:46.930 --> 00:00:49.920
And that's what Newton's
Second Law gives us.

00:00:49.920 --> 00:00:57.010
So Newton's Second
Law of Motion.

00:00:59.650 --> 00:01:03.198
And this one is maybe
the most famous.

00:01:03.198 --> 00:01:04.739
They're all kind of
famous, actually.

00:01:04.739 --> 00:01:07.220
I won't pick favorites here.

00:01:07.220 --> 00:01:09.970
But this one gives us
the famous formula force

00:01:09.970 --> 00:01:14.830
is equal to mass
times acceleration.

00:01:14.830 --> 00:01:17.030
And acceleration is
a vector quantity,

00:01:17.030 --> 00:01:19.487
and force is a vector quantity.

00:01:19.487 --> 00:01:21.320
And what it tells us--
because we're saying,

00:01:21.320 --> 00:01:22.694
OK, if you apply
a force it might

00:01:22.694 --> 00:01:24.090
change that constant velocity.

00:01:24.090 --> 00:01:26.990
But how does it change
that constant velocity?

00:01:26.990 --> 00:01:29.030
Well, let's say I have
a brick right here,

00:01:29.030 --> 00:01:32.250
and it is floating in space.

00:01:32.250 --> 00:01:35.725
And it's pretty nice for us
that the laws of the universe--

00:01:35.725 --> 00:01:37.850
or at least in the classical
sense, before Einstein

00:01:37.850 --> 00:01:39.975
showed up-- the laws of
the universe actually dealt

00:01:39.975 --> 00:01:42.030
with pretty simple mathematics.

00:01:42.030 --> 00:01:47.675
What it tells us is if
you apply a net force,

00:01:47.675 --> 00:01:49.300
let's say, on this
side of the object--

00:01:49.300 --> 00:01:52.120
and we talk about net force,
because if you apply two forces

00:01:52.120 --> 00:01:54.940
that cancel out and that
have zero net force, then

00:01:54.940 --> 00:01:58.100
the object won't change
its constant velocity.

00:01:58.100 --> 00:01:59.670
But if you have a
net force applied

00:01:59.670 --> 00:02:01.950
to one side of this
object, then you're

00:02:01.950 --> 00:02:04.920
going to have a net acceleration
going in the same direction.

00:02:04.920 --> 00:02:06.950
So you're going to have
a net acceleration going

00:02:06.950 --> 00:02:08.710
in that same direction.

00:02:08.710 --> 00:02:10.330
And what Newton's
Second Law of Motion

00:02:10.330 --> 00:02:14.070
tells us is that acceleration
is proportional to the force

00:02:14.070 --> 00:02:16.370
applied, or the force
applied is proportional

00:02:16.370 --> 00:02:17.840
to that acceleration.

00:02:17.840 --> 00:02:19.870
And the constant
of proportionality,

00:02:19.870 --> 00:02:23.745
or to figure out what you have
to multiply the acceleration by

00:02:23.745 --> 00:02:26.120
to get the force, or what you
have to divide the force by

00:02:26.120 --> 00:02:29.440
to get the acceleration,
is called mass.

00:02:29.440 --> 00:02:31.600
That is an object's mass.

00:02:34.170 --> 00:02:35.670
And I'll make a
whole video on this.

00:02:35.670 --> 00:02:38.670
You should not confuse
mass with weight.

00:02:38.670 --> 00:02:40.680
And I'll make a whole
video on the difference

00:02:40.680 --> 00:02:42.690
between mass and weight.

00:02:42.690 --> 00:02:45.980
Mass is a measure of
how much stuff there is.

00:02:45.980 --> 00:02:47.445
Now, that we'll
see in the future.

00:02:47.445 --> 00:02:49.320
There are other things
that we don't normally

00:02:49.320 --> 00:02:51.810
consider stuff that
does start to have mass.

00:02:51.810 --> 00:02:54.429
But for our classical, or at
least a first year physics

00:02:54.429 --> 00:02:55.970
course, you could
really just imagine

00:02:55.970 --> 00:02:57.650
how much stuff there is.

00:02:57.650 --> 00:02:59.820
Weight, as we'll see
in a future video,

00:02:59.820 --> 00:03:02.980
is how much that stuff
is being pulled down

00:03:02.980 --> 00:03:04.530
by the force of gravity.

00:03:04.530 --> 00:03:06.190
So weight is a force.

00:03:06.190 --> 00:03:09.230
Mass is telling you how
much stuff there is.

00:03:09.230 --> 00:03:11.930
And this is really neat that
this formula is so simple,

00:03:11.930 --> 00:03:16.050
because maybe we could have
lived in a universe where force

00:03:16.050 --> 00:03:21.250
is equal to mass squared
times acceleration times

00:03:21.250 --> 00:03:23.750
the square root of acceleration,
which would've made all

00:03:23.750 --> 00:03:25.437
of our math much
more complicated.

00:03:25.437 --> 00:03:26.020
But it's nice.

00:03:26.020 --> 00:03:28.070
It's just this constant
of proportionality right

00:03:28.070 --> 00:03:29.090
over here.

00:03:29.090 --> 00:03:31.300
It's just this nice
simple expression.

00:03:31.300 --> 00:03:35.200
And just to get our feet wet
a little bit with computations

00:03:35.200 --> 00:03:39.350
involving force, mass,
and acceleration,

00:03:39.350 --> 00:03:42.470
let's say that I have a force.

00:03:42.470 --> 00:03:44.640
And the unit of force
is appropriately

00:03:44.640 --> 00:03:47.350
called the newton.

00:03:47.350 --> 00:03:54.050
So let's say I have a
force of 10 newtons.

00:03:54.050 --> 00:03:57.130
And just to be clear, a
newton is the same thing

00:03:57.130 --> 00:04:02.302
as 10 kilogram meters
per second squared.

00:04:02.302 --> 00:04:04.760
And that's good that a newton
is the same thing as kilogram

00:04:04.760 --> 00:04:07.176
meters per second squared,
because that's exactly what you

00:04:07.176 --> 00:04:09.780
get on this side of the formula.

00:04:09.780 --> 00:04:11.620
So let's say I have a
force of 10 newtons,

00:04:11.620 --> 00:04:15.400
and it is acting on a mass.

00:04:15.400 --> 00:04:18.646
Let's say that the
mass is 2 kilograms.

00:04:18.646 --> 00:04:20.145
And I want to know
the acceleration.

00:04:24.860 --> 00:04:27.750
And once again, in this video,
these are vector quantities.

00:04:27.750 --> 00:04:30.270
If I have a positive
value here, we're

00:04:30.270 --> 00:04:32.900
going to make the assumption
that it's going to the right.

00:04:32.900 --> 00:04:35.483
If I had a negative value, then
it would be going to the left.

00:04:35.483 --> 00:04:37.270
So implicitly I'm
giving you not only

00:04:37.270 --> 00:04:39.029
the magnitude of the
force, but I'm also

00:04:39.029 --> 00:04:40.070
giving you the direction.

00:04:40.070 --> 00:04:43.200
I'm saying it is to the
right, because it is positive.

00:04:43.200 --> 00:04:44.590
So what would be acceleration?

00:04:44.590 --> 00:04:46.740
Well we just use f equals ma.

00:04:46.740 --> 00:04:50.952
You have, on the
left hand side, 10.

00:04:50.952 --> 00:04:52.410
I could write 10
newtons here, or I

00:04:52.410 --> 00:04:56.830
could write 10 kilogram
meters per second squared.

00:04:56.830 --> 00:04:58.830
And that is going to be
equal to the mass, which

00:04:58.830 --> 00:05:03.293
is 2 kilograms times
the acceleration.

00:05:06.990 --> 00:05:08.620
And then to solve
for the acceleration,

00:05:08.620 --> 00:05:12.200
you just divide both
sides by 2 kilograms.

00:05:12.200 --> 00:05:14.310
So let's divide the
left by 2 kilograms.

00:05:16.890 --> 00:05:18.370
Let me do it this way.

00:05:18.370 --> 00:05:20.810
Let's divide the
right by 2 kilograms.

00:05:20.810 --> 00:05:22.180
That cancels out.

00:05:22.180 --> 00:05:29.040
The 10 and the 2, 10
divided by 2 is 5.

00:05:29.040 --> 00:05:31.260
And then you have kilograms
canceling with kilograms.

00:05:31.260 --> 00:05:36.430
Your left hand side, you get
5 meters per second squared.

00:05:36.430 --> 00:05:40.450
And then that's equal
to your acceleration.

00:05:40.450 --> 00:05:46.250
Now just for fun, what happens
if I double that force?

00:05:46.250 --> 00:05:48.790
Well then I have 20 newtons.

00:05:48.790 --> 00:05:50.410
Well, I'll actually work it out.

00:05:50.410 --> 00:05:56.870
Then I have 20 kilogram
meters per second squared

00:05:56.870 --> 00:05:59.910
is equal to-- I'll
have to color code--

00:05:59.910 --> 00:06:02.960
2 kilograms times
the acceleration.

00:06:02.960 --> 00:06:09.160
Divide both sides by 2
kilograms, and what do we get?

00:06:09.160 --> 00:06:10.330
Cancels out.

00:06:10.330 --> 00:06:12.410
20 divided by 2 is 10.

00:06:12.410 --> 00:06:14.520
Kilograms cancel kilograms.

00:06:14.520 --> 00:06:17.300
And so we have the
acceleration, in this situation,

00:06:17.300 --> 00:06:20.140
is equal to 10 meters
per second squared

00:06:20.140 --> 00:06:22.100
is equal to the acceleration.

00:06:22.100 --> 00:06:24.500
So when we doubled the force--
we went from 10 newtons

00:06:24.500 --> 00:06:26.780
to 20 newtons-- the
acceleration doubled.

00:06:26.780 --> 00:06:29.500
We went from 5 meters
per second squared

00:06:29.500 --> 00:06:31.150
to 10 meters per second squared.

00:06:31.150 --> 00:06:33.440
So we see that they are
directly proportional,

00:06:33.440 --> 00:06:37.210
and the mass is that how
proportional they are.

00:06:37.210 --> 00:06:41.660
And so you could imagine what
happens if we double the mass.

00:06:41.660 --> 00:06:46.150
If we double the mass in this
situation with 20 newtons,

00:06:46.150 --> 00:06:48.430
then we won't be dividing
by 2 kilograms anymore.

00:06:48.430 --> 00:06:50.640
We'll be dividing
by 4 kilograms.

00:06:50.640 --> 00:06:54.060
And so then we'll have 20
divided by 4, which would be 5

00:06:54.060 --> 00:06:55.990
and would be meters
per second squared.

00:06:55.990 --> 00:06:59.100
So if you make the mass
larger, if you double it,

00:06:59.100 --> 00:07:01.250
then your acceleration
would be half as much.

00:07:01.250 --> 00:07:03.967
So the larger the mass
you have, the more force

00:07:03.967 --> 00:07:05.050
you need to accelerate it.

00:07:05.050 --> 00:07:08.820
Or for a given force, the less
that it will accelerate it,

00:07:08.820 --> 00:07:12.923
the harder it is to change
its constant velocity.


More on Newton's first law of motion

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:02.200
In this video, I want
to talk a little bit

00:00:02.200 --> 00:00:06.060
about Newton's
First Law of Motion.

00:00:06.060 --> 00:00:09.730
And this is a translation from
Newton's Principia from Latin

00:00:09.730 --> 00:00:10.710
into English.

00:00:10.710 --> 00:00:13.600
So the First Law,
"Every body persists

00:00:13.600 --> 00:00:20.090
in a state of being at
rest, or moving uniformly

00:00:20.090 --> 00:00:24.190
straightforward, except
insofar as it is compelled

00:00:24.190 --> 00:00:27.910
to change its state
by force impressed."

00:00:27.910 --> 00:00:30.240
So another way to rephrase
what they're saying

00:00:30.240 --> 00:00:33.610
is, that if there's something--
every body persists--

00:00:33.610 --> 00:00:35.860
so everything will
stay at rest, or

00:00:35.860 --> 00:00:39.240
moving with a constant
velocity, unless it

00:00:39.240 --> 00:00:42.050
is compelled to change
its state by force.

00:00:42.050 --> 00:00:45.420
Unless it's acted on by a force,
especially an unbalanced force.

00:00:45.420 --> 00:00:47.000
and I'll explain
that in a second.

00:00:47.000 --> 00:00:50.720
So if I have something that's
at rest, so completely at rest.

00:00:50.720 --> 00:00:53.840
So I have-- and
this is something

00:00:53.840 --> 00:00:56.840
that we've seen before.

00:00:56.840 --> 00:00:59.150
Let's say that I have a rock.

00:00:59.150 --> 00:01:03.720
Let's say that I
have a rock someplace

00:01:03.720 --> 00:01:08.210
and it's laying on
a field of grass,

00:01:08.210 --> 00:01:10.005
I can keep observing that rock.

00:01:10.005 --> 00:01:11.966
And it is unlikely
to move, assuming

00:01:11.966 --> 00:01:13.090
that nothing happens to it.

00:01:13.090 --> 00:01:15.310
If there's no force
applied to that rock,

00:01:15.310 --> 00:01:17.310
that rock will just stay there.

00:01:17.310 --> 00:01:20.260
So the first part
is pretty obvious.

00:01:20.260 --> 00:01:23.599
So, "Every body persists in
a state of being at rest"--

00:01:23.599 --> 00:01:25.140
I'm not going to do
the second part--

00:01:25.140 --> 00:01:28.150
"except insofar as there's some
force being applied to it."

00:01:28.150 --> 00:01:30.710
So clearly a rock
will be at rest,

00:01:30.710 --> 00:01:33.330
unless there's some force
applied to it, unless someone

00:01:33.330 --> 00:01:37.110
here tries to push it or roll
it or do something to it.

00:01:37.110 --> 00:01:41.730
What's less intuitive about the
first law is the second part.

00:01:41.730 --> 00:01:44.270
"Every body persists in,"
either, "being in a state

00:01:44.270 --> 00:01:48.300
of rest or moving
uniformly straight forward,

00:01:48.300 --> 00:01:50.390
except insofar as
it is compelled

00:01:50.390 --> 00:01:52.620
to change its state
by force impressed."

00:01:52.620 --> 00:01:55.057
So this Newton's
first law-- and I

00:01:55.057 --> 00:01:56.640
think I should do a
little aside here,

00:01:56.640 --> 00:02:00.450
because, this right
here is Newton.

00:02:00.450 --> 00:02:01.890
And if this is
Newton's first law,

00:02:01.890 --> 00:02:04.610
why do I have this huge
picture of this guy over here?

00:02:04.610 --> 00:02:07.370
Well, the reason is is because
Newton's first law is really

00:02:07.370 --> 00:02:10.600
just a restatement of
this guy's law of inertia.

00:02:10.600 --> 00:02:14.490
And this guy, another
titan of civilization

00:02:14.490 --> 00:02:19.170
really, this is Galileo Galilei.

00:02:19.170 --> 00:02:22.880
And he is the first person to
formulate the law of inertia.

00:02:22.880 --> 00:02:25.460
And Newton just rephrased it
a little bit and packaged it

00:02:25.460 --> 00:02:26.630
with his other laws.

00:02:26.630 --> 00:02:28.500
But he did many, many,
many other things.

00:02:28.500 --> 00:02:30.520
So you really have
to give Galileo

00:02:30.520 --> 00:02:32.130
credit for Newton's first law.

00:02:32.130 --> 00:02:34.460
So that's why I made
him bigger than here.

00:02:34.460 --> 00:02:36.260
But I was in the
midst of a thought.

00:02:36.260 --> 00:02:38.140
So we understand if
something is at rest,

00:02:38.140 --> 00:02:39.889
it's going to stay at
rest, unless there's

00:02:39.889 --> 00:02:41.120
some force that acts on it.

00:02:41.120 --> 00:02:42.578
And in some
definitions, you'll see

00:02:42.578 --> 00:02:44.270
unless there's some
unbalanced force.

00:02:44.270 --> 00:02:46.380
And the reason why
they say unbalanced

00:02:46.380 --> 00:02:49.210
is, because you could have two
forces that act on something

00:02:49.210 --> 00:02:50.930
and they might balance out.

00:02:50.930 --> 00:02:53.710
For example, I could push
on this side of the rock

00:02:53.710 --> 00:02:55.094
with a certain amount of force.

00:02:55.094 --> 00:02:56.760
And if you push on
this side of the rock

00:02:56.760 --> 00:03:00.630
with the exact same amount of
force, the rock won't move.

00:03:00.630 --> 00:03:03.470
And the only way that it would
move if there's a lot more

00:03:03.470 --> 00:03:05.510
force on one side than
on the other side,

00:03:05.510 --> 00:03:07.380
so if you have an
unbalanced force.

00:03:07.380 --> 00:03:09.760
So if you have a ton
of-- and maybe the rock

00:03:09.760 --> 00:03:10.680
is a bad analogy.

00:03:10.680 --> 00:03:14.030
Let's take ice, because ice is
easier to move, or ice on ice.

00:03:14.030 --> 00:03:16.099
So there's ice right here.

00:03:16.099 --> 00:03:17.640
And then, I have
another block of ice

00:03:17.640 --> 00:03:19.820
sitting on top of that ice.

00:03:19.820 --> 00:03:21.790
So once again, we're
familiar with the idea,

00:03:21.790 --> 00:03:25.760
if there's no force acting
on it that ice won't move.

00:03:25.760 --> 00:03:29.177
But what happens if
I'm pushing on the ice

00:03:29.177 --> 00:03:31.010
with a certain amount
of force on that side,

00:03:31.010 --> 00:03:32.950
and you're pushing on
the ice on that side

00:03:32.950 --> 00:03:34.620
with the same amount of force?

00:03:34.620 --> 00:03:36.690
The ice will still not move.

00:03:36.690 --> 00:03:39.120
So this right here, this
would be a balanced force.

00:03:41.920 --> 00:03:47.630
So the only way for the ice to
change its condition, to change

00:03:47.630 --> 00:03:50.980
its restful condition is
if the force is unbalanced.

00:03:50.980 --> 00:03:53.200
So if we add a little bit
of force on this side,

00:03:53.200 --> 00:03:57.920
so it more than compensates
the force pushing it this way,

00:03:57.920 --> 00:04:00.730
then you're going to see
the ice block start to move,

00:04:00.730 --> 00:04:04.160
start to really accelerate
in that direction.

00:04:04.160 --> 00:04:05.584
But I think this
part is obvious.

00:04:05.584 --> 00:04:07.250
This, you know,
something that's at rest

00:04:07.250 --> 00:04:09.300
will stay at rest, unless
it's being acted on

00:04:09.300 --> 00:04:11.050
by an unbalanced force.

00:04:11.050 --> 00:04:14.200
What's less obvious is
the idea that something

00:04:14.200 --> 00:04:16.690
moving uniformly
straightforward, which

00:04:16.690 --> 00:04:18.135
is another way of
saying something

00:04:18.135 --> 00:04:19.259
having a constant velocity.

00:04:22.587 --> 00:04:24.170
What he's saying is,
is that something

00:04:24.170 --> 00:04:26.240
that has a constant
velocity will continue

00:04:26.240 --> 00:04:29.560
to have that constant
velocity indefinitely,

00:04:29.560 --> 00:04:33.290
unless it is acted on
by an unbalanced force.

00:04:33.290 --> 00:04:34.590
And that's less intuitive.

00:04:34.590 --> 00:04:37.970
Because everything in
our human experience--

00:04:37.970 --> 00:04:41.430
even if I were to push
this block of ice,

00:04:41.430 --> 00:04:42.700
eventually it'll stop.

00:04:42.700 --> 00:04:44.510
It won't just keep
going forever, even

00:04:44.510 --> 00:04:47.460
assuming that this ice field
is infinitely long, that ice

00:04:47.460 --> 00:04:48.890
will eventually stop.

00:04:48.890 --> 00:04:54.020
Or if I throw a tennis ball.

00:04:54.020 --> 00:04:56.380
That tennis ball
will eventually stop.

00:04:59.030 --> 00:05:01.080
It'll eventually
grind to a halt.

00:05:01.080 --> 00:05:04.336
Or if I roll a bowling
ball, or if I, anything.

00:05:04.336 --> 00:05:06.460
We've never seen, at least
in our human experience,

00:05:06.460 --> 00:05:09.510
it looks like everything
will eventually stop.

00:05:09.510 --> 00:05:11.350
So this is a very
unintuitive thing to say,

00:05:11.350 --> 00:05:12.820
that something in
motion will just

00:05:12.820 --> 00:05:14.940
keep going in
motion indefinitely.

00:05:14.940 --> 00:05:17.430
Everything in human intuition
says if you want something

00:05:17.430 --> 00:05:19.910
to keep going in
motion, you have to keep

00:05:19.910 --> 00:05:22.595
putting more force, keep
putting more energy into it

00:05:22.595 --> 00:05:23.470
for it to keep going.

00:05:23.470 --> 00:05:25.490
Your car won't go
forever, unless you keep,

00:05:25.490 --> 00:05:29.810
unless the engine keeps burning
fuel to drive and consuming

00:05:29.810 --> 00:05:30.310
energy.

00:05:30.310 --> 00:05:31.927
So what are they talking about?

00:05:31.927 --> 00:05:33.510
Well, in all of these
examples-- and I

00:05:33.510 --> 00:05:36.040
think this is actually a pretty
brilliant insight from all

00:05:36.040 --> 00:05:39.630
of these fellows is
that-- all of these things

00:05:39.630 --> 00:05:41.090
would have gone on forever.

00:05:41.090 --> 00:05:42.540
The ball would
keep going forever.

00:05:42.540 --> 00:05:44.970
This ice block would
be going on forever,

00:05:44.970 --> 00:05:47.200
except for the fact that
there are unbalanced

00:05:47.200 --> 00:05:50.000
forces acting on
them to stop them.

00:05:50.000 --> 00:05:52.489
So in the case of ice,
even though ice on ice

00:05:52.489 --> 00:05:54.030
doesn't have a lot
of friction, there

00:05:54.030 --> 00:05:56.460
is some friction
between these two.

00:05:56.460 --> 00:05:58.050
And so you have,
in this situation,

00:05:58.050 --> 00:06:00.150
the force of friction
is going to be

00:06:00.150 --> 00:06:03.560
acting against the direction
of the movement of the ice.

00:06:03.560 --> 00:06:06.930
And friction really comes
from, at an atomic level--

00:06:06.930 --> 00:06:10.570
so if you have the actual
water molecules in a lattice

00:06:10.570 --> 00:06:12.775
structure in the ice
cube, and then here are

00:06:12.775 --> 00:06:17.130
the water molecules in a
lattice structure on the ice,

00:06:17.130 --> 00:06:20.810
on the actual kind of sea of
ice that it's traveling on--

00:06:20.810 --> 00:06:22.769
they do kind of bump and
grind into each other.

00:06:22.769 --> 00:06:25.226
Although they're both smooth,
there are imperfections here.

00:06:25.226 --> 00:06:26.140
They bump and grind.

00:06:26.140 --> 00:06:28.080
They generate a
little bit of heat.

00:06:28.080 --> 00:06:32.310
And they'll, essentially, be
working against the movement.

00:06:32.310 --> 00:06:34.810
So there's a force of friction
that's being applied to here.

00:06:34.810 --> 00:06:35.620
And that's why it's stopping.

00:06:35.620 --> 00:06:37.203
Not only a force of
friction, you also

00:06:37.203 --> 00:06:38.480
have some air resistance.

00:06:38.480 --> 00:06:40.560
The ice block is
going to be bumping

00:06:40.560 --> 00:06:42.570
into all sorts of air particles.

00:06:42.570 --> 00:06:44.660
It might not be
noticeable at first,

00:06:44.660 --> 00:06:47.390
but it's definitely going to
keep it from going on forever.

00:06:47.390 --> 00:06:49.460
Same thing with the ball
being tossed to the air.

00:06:49.460 --> 00:06:51.380
Obviously, at some
point, it hits

00:06:51.380 --> 00:06:52.810
the ground because of gravity.

00:06:52.810 --> 00:06:54.200
So that's one
force acting on it.

00:06:54.200 --> 00:06:55.700
But even once it
hits the ground,

00:06:55.700 --> 00:06:57.990
it doesn't keep rolling
forever, once again,

00:06:57.990 --> 00:07:01.670
because of the friction,
especially if there's grass

00:07:01.670 --> 00:07:02.260
here.

00:07:02.260 --> 00:07:04.290
The grass is going to
stop it from going.

00:07:04.290 --> 00:07:06.779
And even while it's in the
air, it's going to slow down.

00:07:06.779 --> 00:07:08.570
It's not going to have
a constant velocity.

00:07:08.570 --> 00:07:10.710
Because you have all
of these air particles

00:07:10.710 --> 00:07:15.750
that are going to
bump into it and exert

00:07:15.750 --> 00:07:17.646
force to slow it down.

00:07:17.646 --> 00:07:19.520
So what was really
brilliant about these guys

00:07:19.520 --> 00:07:22.060
is that they could
imagine a reality where

00:07:22.060 --> 00:07:24.610
you didn't have gravity,
where you did not

00:07:24.610 --> 00:07:27.850
have air slowing things down.

00:07:27.850 --> 00:07:30.100
And they could imagine
that in that reality,

00:07:30.100 --> 00:07:33.960
something would just keep
persisting in its motion.

00:07:33.960 --> 00:07:36.416
And the reason why Galileo,
frankly, was probably good

00:07:36.416 --> 00:07:37.790
at thinking about
that is that he

00:07:37.790 --> 00:07:41.130
studied the orbits of planets.

00:07:41.130 --> 00:07:43.050
And he could, or at
least he's probably

00:07:43.050 --> 00:07:45.431
theorized that, hey, maybe
there's no air out there.

00:07:45.431 --> 00:07:47.430
And that maybe that's why
these planets can just

00:07:47.430 --> 00:07:49.780
keep going round
and round in orbit.

00:07:49.780 --> 00:07:54.500
And I should say their speed,
because their direction is

00:07:54.500 --> 00:07:56.540
changing, but their
speed never slows down,

00:07:56.540 --> 00:07:58.810
because there's
nothing in the space

00:07:58.810 --> 00:08:01.572
to actually slow
down those planets.

00:08:01.572 --> 00:08:04.030
So anyway, hopefully you found
that as fascinating as I do.

00:08:04.030 --> 00:08:06.410
Because on some level,
it's super-duper obvious.

00:08:06.410 --> 00:08:08.743
But on a whole other level,
it's completely not obvious,

00:08:08.743 --> 00:08:10.910
especially this moving
uniformly straightforward.

00:08:10.910 --> 00:08:12.470
And just to make
the point clear,

00:08:12.470 --> 00:08:16.270
if gravity disappeared,
and you had no air,

00:08:16.270 --> 00:08:18.900
and you threw a ball,
that ball literally

00:08:18.900 --> 00:08:20.950
would keep going
in that direction

00:08:20.950 --> 00:08:24.980
forever, unless some other
unbalanced force acted

00:08:24.980 --> 00:08:26.390
to stop it.

00:08:26.390 --> 00:08:28.790
And another way to think
about it-- and this

00:08:28.790 --> 00:08:32.270
is an example that you might
see in everyday life-- is,

00:08:32.270 --> 00:08:38.600
if I'm in an airplane
that's going at a completely

00:08:38.600 --> 00:08:40.880
constant velocity and
there's absolutely

00:08:40.880 --> 00:08:42.960
no turbulence in the airplane.

00:08:42.960 --> 00:08:45.790
So if I'm sitting in the
airplane right over here.

00:08:45.790 --> 00:08:51.250
And it's going at a constant
velocity, completely smooth,

00:08:51.250 --> 00:08:52.490
no turbulence.

00:08:52.490 --> 00:08:55.647
There's really no way for me to
tell whether that airplane is

00:08:55.647 --> 00:08:57.230
moving without looking
out the window.

00:08:57.230 --> 00:08:59.480
Let's assume that there's
no windows in that airplane.

00:08:59.480 --> 00:09:01.110
It's going at a
constant velocity.

00:09:01.110 --> 00:09:02.390
And there's no turbulence.

00:09:02.390 --> 00:09:06.380
And let's say, I
can't hear anything.

00:09:06.380 --> 00:09:07.780
So I can't even
hear the engines.

00:09:07.780 --> 00:09:10.870
There's no way for me to sense
that the plane is moving.

00:09:10.870 --> 00:09:12.950
Because from my
frame of reference,

00:09:12.950 --> 00:09:15.290
it looks completely
identical to if I

00:09:15.290 --> 00:09:18.622
was in that same plane that
was resting on the ground.

00:09:18.622 --> 00:09:20.330
And that's another
way to think about it.

00:09:20.330 --> 00:09:22.110
That it's actually
very intuitive

00:09:22.110 --> 00:09:25.920
that they're similar states,
moving at a constant velocity

00:09:25.920 --> 00:09:27.480
or being at rest.

00:09:27.480 --> 00:09:29.290
And you really can't
tell whether you

00:09:29.290 --> 00:09:31.477
are one or the other.


Acceleration of aircraft carrier take-off

WEBVTT
Kind: captions
Language: en

00:00:00.744 --> 00:00:03.600
So I'm curious about how much acceleration

00:00:03.600 --> 00:00:05.600
does a pilot, or the pilot and the plane,

00:00:05.600 --> 00:00:08.336
experience when they need to take off

00:00:08.336 --> 00:00:09.854
from an aircraft carrier?

00:00:09.854 --> 00:00:11.400
So I looked up a few statistics

00:00:11.400 --> 00:00:13.166
on the Internet, this right here is

00:00:13.166 --> 00:00:16.417
a picture of an F/A-18 Hornet right over here.

00:00:16.417 --> 00:00:21.400
It has a take-off speed of 260 kilometers per hour.

00:00:21.400 --> 00:00:24.067
If we want that to be a velocity, 260 km/hour

00:00:24.067 --> 00:00:27.330
in this direction, if it's taking off from

00:00:27.330 --> 00:00:30.667
this Nimitz class carrier right over here.

00:00:30.667 --> 00:00:32.867
And I also looked it up, and I found

00:00:32.867 --> 00:00:34.867
the runway length, or I should say

00:00:34.867 --> 00:00:36.467
the catapult length, because these planes

00:00:36.467 --> 00:00:38.824
don't take off just with their own power.

00:00:38.824 --> 00:00:40.658
They have their own thrusters going,

00:00:40.658 --> 00:00:42.400
but they also are catapulted off,

00:00:42.400 --> 00:00:44.164
so they can be really accelerated quickly

00:00:44.164 --> 00:00:46.467
off of the flight deck of this carrier.

00:00:46.467 --> 00:00:49.000
And the runway length of a Nimitz class carrier

00:00:49.000 --> 00:00:51.084
is about 80 meters. So this is where

00:00:51.084 --> 00:00:52.933
they take off from. This right over here

00:00:52.933 --> 00:00:54.533
is where they take off from.

00:00:54.533 --> 00:00:56.587
And then they come in and they land over here.

00:00:56.587 --> 00:00:58.421
But I'm curious about the take-off.

00:00:58.421 --> 00:01:01.667
So to do this, let's figure out, well let's just

00:01:01.667 --> 00:01:03.333
figure out the acceleration, and from that

00:01:03.333 --> 00:01:04.933
we can also figure out how long it takes

00:01:04.933 --> 00:01:07.400
them to be catapulted off the flight deck.

00:01:07.400 --> 00:01:10.067
So, let me get the numbers in one place,

00:01:10.067 --> 00:01:11.933
so the take-off velocity, I could say,

00:01:11.933 --> 00:01:16.733
is 260 km/hour, so let me write this down.

00:01:16.733 --> 00:01:18.414
So that has to be your final velocity

00:01:18.414 --> 00:01:20.200
when you're getting off, of the plane,

00:01:20.200 --> 00:01:21.757
if you want to be flying.

00:01:21.757 --> 00:01:24.867
So your initial velocity is going to be 0,

00:01:24.867 --> 00:01:26.667
and once again I'm going to use the convention

00:01:26.667 --> 00:01:30.800
that the direction of the vector is implicit.

00:01:30.800 --> 00:01:33.200
Positive means going in the direction of take-off,

00:01:33.200 --> 00:01:35.749
negative would mean going the other way.

00:01:35.749 --> 00:01:39.933
My initial velocity is 0, I'll denote it as a vector

00:01:39.933 --> 00:01:43.886
right here. My final velocity over here

00:01:43.886 --> 00:01:48.739
has to be 260 km/hour.

00:01:48.739 --> 00:01:50.667
And I want to convert everything to meters

00:01:50.667 --> 00:01:53.000
and seconds, just so that I can get my,

00:01:53.000 --> 00:01:56.333
at least for meters, so that I can use my runway length

00:01:56.333 --> 00:01:58.200
in meters. So let's just do it in meters per second,

00:01:58.200 --> 00:02:00.000
I have a feeling it'll be a little bit easier

00:02:00.000 --> 00:02:02.137
to understand when we talk about acceleration

00:02:02.137 --> 00:02:04.067
in those units as well.

00:02:04.067 --> 00:02:06.090
So if we want to convert this into seconds,

00:02:06.090 --> 00:02:08.639
we have, we'll put hours in the numerator,

00:02:08.639 --> 00:02:10.728
1 hour, so it cancels out with this hour,

00:02:10.728 --> 00:02:14.857
is equal to 3600 seconds.

00:02:14.857 --> 00:02:17.323
I'll just write 3600 s. And then

00:02:17.323 --> 00:02:20.200
if we want to convert it to meters,

00:02:20.200 --> 00:02:27.400
we have 1000 meters is equal to 1 km,

00:02:27.400 --> 00:02:29.133
and this 1 km will cancel out with those kms

00:02:29.133 --> 00:02:30.667
right over there.

00:02:30.667 --> 00:02:32.000
And whenever you're doing any type of

00:02:32.000 --> 00:02:33.667
this dimensional analysis, you really should see

00:02:33.667 --> 00:02:35.318
whether it makes sense.

00:02:35.318 --> 00:02:37.067
If I'm going 260 km in an hour,

00:02:37.067 --> 00:02:40.078
I should go much fewer km in a second

00:02:40.078 --> 00:02:41.402
because a second is so much shorter

00:02:41.402 --> 00:02:43.074
amount of time, and that's why we're dividing

00:02:43.074 --> 00:02:45.233
by 3600. If I can go a certain

00:02:45.233 --> 00:02:47.400
number of km in an hour a second,

00:02:47.400 --> 00:02:48.333
I should be able to go a lot,

00:02:48.333 --> 00:02:50.867
many many more meters in that same amount

00:02:50.867 --> 00:02:52.547
of time, and that's why we're multiplying

00:02:52.547 --> 00:02:54.600
by 1000. When you multiply these out,

00:02:54.600 --> 00:02:59.400
the hours cancel out, you have km canceling out,

00:02:59.400 --> 00:03:01.800
and you have 260 times 1000

00:03:01.800 --> 00:03:05.295
divided by 3600 meters per second.

00:03:05.295 --> 00:03:07.733
So let me get my trusty TI-85 out,

00:03:07.733 --> 00:03:12.052
and actually calculate that.

00:03:12.052 --> 00:03:26.754
So I have 260 times 1000 divided by 3600

00:03:26.754 --> 00:03:29.467
gets me, I'll just round it to 72, because

00:03:29.467 --> 00:03:31.133
that's about how many significant digits

00:03:31.133 --> 00:03:34.645
I can assume here. 72 meters per second.

00:03:34.645 --> 00:03:37.641
So all I did here is I converted the take-off velocity,

00:03:37.641 --> 00:03:41.667
so this is 72 m/s, this has to be the final velocity

00:03:41.667 --> 00:03:44.560
after accelerating. So let's think about

00:03:44.560 --> 00:03:46.867
what that acceleration could be, given that we know

00:03:46.867 --> 00:03:49.267
the length of the runway, and we're going to assume

00:03:49.267 --> 00:03:51.667
constant acceleration here, just to simplify things

00:03:51.667 --> 00:03:52.933
a little bit. But what does that

00:03:52.933 --> 00:03:54.667
constant acceleration have to be?

00:03:54.667 --> 00:03:56.890
So let's think a little bit about it.

00:03:56.890 --> 00:03:59.630
The total displacement, I'll do that in purple,

00:03:59.630 --> 00:04:02.467
the total displacement is going to be

00:04:02.467 --> 00:04:07.966
equal to our average velocity while we're accelerating,

00:04:07.966 --> 00:04:13.800
times the difference in time, or the amount of time

00:04:13.800 --> 00:04:16.046
it takes us to accelerate.

00:04:16.046 --> 00:04:19.467
Now, what is the average velocity here?

00:04:19.467 --> 00:04:23.600
It's going to be our final velocity, plus our initial velocity,

00:04:23.600 --> 00:04:26.144
over 2. It's just the average of the initial and final.

00:04:26.144 --> 00:04:28.608
And we can only do that because we are dealing

00:04:28.608 --> 00:04:30.733
with a constant acceleration.

00:04:30.733 --> 00:04:33.133
And what is our change in time over here?

00:04:33.133 --> 00:04:35.067
What is our change in time?

00:04:35.067 --> 00:04:37.800
Well our change in time is how long does it take

00:04:37.800 --> 00:04:42.133
us to get to that velocity? Or another way to think about it is:

00:04:42.133 --> 00:04:45.884
it is our change in velocity divided by our acceleration.

00:04:45.884 --> 00:04:50.505
If we're trying to get to 10 m/s, or we're trying to get

00:04:50.505 --> 00:04:54.267
10 m/s faster, and we're accelerating at 2 m/s squared,

00:04:54.267 --> 00:04:55.467
it'll take us 5 seconds.

00:04:55.467 --> 00:04:57.610
Or if you want to see that explicitly written in a formula,

00:04:57.610 --> 00:05:01.467
we know that acceleration is equal to

00:05:01.467 --> 00:05:05.760
change in velocity over change in time.

00:05:05.760 --> 00:05:08.267
You multiply both sides by change in time,

00:05:08.267 --> 00:05:10.000
and you divide both sides by acceleration,

00:05:10.000 --> 00:05:12.400
so let's do that, multiply both sides by change in time

00:05:12.400 --> 00:05:14.259
and divide by acceleration.

00:05:14.259 --> 00:05:18.200
Multiply by change in time and divide by acceleration.

00:05:18.200 --> 00:05:22.267
And you get, that cancels out, and then you have

00:05:22.267 --> 00:05:24.499
that cancels out, and you have change in time

00:05:24.499 --> 00:05:29.667
is equal to change in velocity divided by acceleration.

00:05:29.667 --> 00:05:32.800
Change in velocity divided by acceleration.

00:05:32.800 --> 00:05:34.507
So what's the change in velocity?

00:05:34.507 --> 00:05:36.733
Change in velocity, so this is going to be

00:05:36.733 --> 00:05:41.357
change in velocity divided by acceleration.

00:05:41.357 --> 00:05:42.867
Change in velocity is the same thing as your

00:05:42.867 --> 00:05:46.133
final velocity minus your initial velocity,

00:05:46.133 --> 00:05:49.467
all of that divided by acceleration.

00:05:49.467 --> 00:05:52.267
So this delta t part we can re-write as

00:05:52.267 --> 00:06:00.867
our final velocity minus our initial velocity, over acceleration.

00:06:00.867 --> 00:06:03.400
And just doing this simple little derivation here

00:06:03.400 --> 00:06:05.459
actually gives us a pretty cool result!

00:06:05.459 --> 00:06:08.872
If we just work through this math, and I'll try to

00:06:08.872 --> 00:06:11.667
write a little bigger, I see my writing is getting smaller,

00:06:11.667 --> 00:06:13.733
our displacement can be expressed as

00:06:13.733 --> 00:06:15.954
the product of these two things.

00:06:15.954 --> 00:06:17.626
And what's cool about this, well let me just

00:06:17.626 --> 00:06:21.620
write it this way: so this is our final velocity

00:06:21.620 --> 00:06:29.027
plus our initial velocity, times our final velocity

00:06:29.027 --> 00:06:32.067
minus our initial velocity,

00:06:32.067 --> 00:06:36.783
all of that over 2 times our acceleration.

00:06:36.783 --> 00:06:40.733
Our assumed constant acceleration.

00:06:40.733 --> 00:06:43.133
And you probably remember from algebra class

00:06:43.133 --> 00:06:47.200
this takes the form: a plus b times a minus b.

00:06:47.200 --> 00:06:50.800
And so this equal to -- and you can multiply it out

00:06:50.800 --> 00:06:52.667
and you can review in our algebra playlist

00:06:52.667 --> 00:06:54.801
how to multiply out two binomials like this,

00:06:54.801 --> 00:06:56.933
but this numerator right over here,

00:06:56.933 --> 00:06:59.027
I'll write it in blue, is going to be equal to

00:06:59.027 --> 00:07:03.867
our final velocity squared minus our initial velocity squared.

00:07:03.867 --> 00:07:06.272
This is a difference of squares, you can factor it out

00:07:06.272 --> 00:07:10.267
into the sum of the two terms times the difference

00:07:10.267 --> 00:07:13.067
of the two terms, so that when you multiply these two out

00:07:13.067 --> 00:07:21.452
you just get that over there, over 2 times the acceleration.

00:07:21.452 --> 00:07:24.667
Now what's really cool here is we were able to derive

00:07:24.667 --> 00:07:27.267
a formula that just deals with the displacement,

00:07:27.267 --> 00:07:31.600
our final velocity, our initial velocity, and the acceleration.

00:07:31.600 --> 00:07:35.067
And we know all of those things except for the acceleration.

00:07:35.067 --> 00:07:38.176
We know that our displacement is 80 meters.

00:07:38.176 --> 00:07:40.521
We know that this is 80 meters.

00:07:40.521 --> 00:07:42.774
We know that our final velocity, just before

00:07:42.774 --> 00:07:45.267
we square it, we know that our final velocity is

00:07:45.267 --> 00:07:50.133
72 meters per second. And we know that

00:07:50.133 --> 00:07:53.362
our initial velocity is 0 meters per second.

00:07:53.362 --> 00:07:55.133
And so we can use all of this information

00:07:55.133 --> 00:07:59.267
to solve for our acceleration.

00:07:59.267 --> 00:08:02.267
And you might see this formula, displacement,

00:08:02.267 --> 00:08:03.933
sometimes called distance, if you're just using

00:08:03.933 --> 00:08:07.271
the scalar version, and really we are thinking only

00:08:07.271 --> 00:08:08.757
in the scalar, we're thinking about the magnitudes

00:08:08.757 --> 00:08:10.684
of all of these things for the sake of this video.

00:08:10.684 --> 00:08:12.200
We're only dealing in one dimension.

00:08:12.200 --> 00:08:14.200
But sometimes you'll see it written like this,

00:08:14.200 --> 00:08:18.022
sometimes you'll multiply both sides times the 2 a,

00:08:18.022 --> 00:08:19.800
and you'll get something like this, where you have

00:08:19.800 --> 00:08:23.400
2 times, really the magnitude of the acceleration,

00:08:23.400 --> 00:08:26.067
times the magnitude of the displacement,

00:08:26.067 --> 00:08:27.133
which is the same thing as the distance,

00:08:27.133 --> 00:08:31.600
is equal to the final velocity, the magnitude of

00:08:31.600 --> 00:08:41.845
the final velocity, squared, minus the initial velocity squared.

00:08:41.845 --> 00:08:44.733
Or sometimes, in some books, it'll be written as 2 a d

00:08:44.733 --> 00:08:46.933
is equal to v f squared minus v i squared.

00:08:46.933 --> 00:08:48.602
And it seems like a super mysterious thing,

00:08:48.602 --> 00:08:50.600
but it's not that mysterious. We just very simply

00:08:50.600 --> 00:08:54.268
derived it from displacement, or if you want to say distance,

00:08:54.268 --> 00:08:55.847
if you're just thinking about the scalar quantity,

00:08:55.847 --> 00:08:59.533
is equal to average velocity times the change in time.

00:08:59.533 --> 00:09:02.023
So, so far we've just derived ourselves a kind of a

00:09:02.023 --> 00:09:05.274
neat formula that is often not derived in

00:09:05.274 --> 00:09:07.267
physics class, but let's use it to actually

00:09:07.267 --> 00:09:10.127
figure out the acceleration that a pilot experiences

00:09:10.127 --> 00:09:14.267
when they're taking off of a Nimitz class carrier.

00:09:14.267 --> 00:09:18.867
So we have 2 times the acceleration

00:09:18.867 --> 00:09:21.467
times the distance, that's 80 meters,

00:09:21.467 --> 00:09:25.267
times 80 meters, is going to be equal to

00:09:25.267 --> 00:09:27.400
our final velocity squared.

00:09:27.400 --> 00:09:30.421
What's our final velocity? 72 meters per second.

00:09:30.421 --> 00:09:35.667
So 72 meters per second, squared,

00:09:35.667 --> 00:09:37.867
minus our initial velocity.

00:09:37.867 --> 00:09:40.467
So our initial velocity in this situation is just 0.

00:09:40.467 --> 00:09:42.200
So it's just going to be minus 0 squared,

00:09:42.200 --> 00:09:43.933
which is just going to be 0, so we don't even have to

00:09:43.933 --> 00:09:46.838
write it down. And so to solve for acceleration,

00:09:46.838 --> 00:09:49.578
to solve for acceleration, you just divide,

00:09:49.578 --> 00:09:52.086
so this is the same thing as 160 meters,

00:09:52.086 --> 00:09:55.667
well, let's just divide both sides by 2 times 80,

00:09:55.667 --> 00:10:03.708
so we get acceleration is equal to 72 m/s squared

00:10:03.708 --> 00:10:11.361
over 2 times 80 meters.

00:10:11.361 --> 00:10:12.267
And what we're gonna get is,

00:10:12.267 --> 00:10:13.733
I'll just write this in one color,

00:10:13.733 --> 00:10:16.867
it's going to be 72 divided by 160,

00:10:16.867 --> 00:10:20.832
times, we have in the numerator,

00:10:20.832 --> 00:10:22.400
meters squared over seconds squared,

00:10:22.400 --> 00:10:26.173
we're squaring the units, and then we're

00:10:26.173 --> 00:10:28.680
going to be dividing by meters.

00:10:28.680 --> 00:10:30.533
So times, I'll do this in blue,

00:10:30.533 --> 00:10:33.733
times one over meters. Right?

00:10:33.733 --> 00:10:35.000
Because we have a meters in the denominator.

00:10:35.000 --> 00:10:37.341
And so what we're going to get is this

00:10:37.341 --> 00:10:39.431
meters squared divided by meters,

00:10:39.431 --> 00:10:40.940
that's going to cancel out, we're going to get

00:10:40.940 --> 00:10:42.267
meters per second squared. Which is cool

00:10:42.267 --> 00:10:44.841
because that's what acceleration should be in.

00:10:44.841 --> 00:10:46.933
And so let's just get the calculator out,

00:10:46.933 --> 00:10:48.766
to calculate this exact acceleration.

00:10:48.766 --> 00:10:53.267
So we have to take, oh sorry, this is 72 squared,

00:10:53.267 --> 00:10:57.600
let me write that down. So this is, this is going to be

00:10:57.600 --> 00:10:59.200
72 squared, don't want to forget about this part

00:10:59.200 --> 00:11:02.001
right over here. 72 squared divided by 160.

00:11:02.001 --> 00:11:06.267
So we have, and we can just use the original number

00:11:06.267 --> 00:11:07.867
right over here that we calculated,

00:11:07.867 --> 00:11:12.845
so let's just square that, and then divide that by 160,

00:11:12.845 --> 00:11:17.140
divided by 160. And if we go to 2 significant digits,

00:11:17.140 --> 00:11:22.551
we get 33, we get our acceleration is, our acceleration

00:11:22.551 --> 00:11:28.286
is equal to 33 meters per second squared.

00:11:28.286 --> 00:11:31.800
And just to give you an idea of how much acceleration

00:11:31.800 --> 00:11:35.467
that is, is if you are in free fall over Earth,

00:11:35.467 --> 00:11:40.709
the force of gravity will be accelerating you,

00:11:40.709 --> 00:11:46.630
so g is going to be equal to 9.8 meters per second squared.

00:11:46.630 --> 00:11:52.400
So this is accelerating you 3 times more than what

00:11:52.400 --> 00:11:54.667
Earth is making you accelerate if you were to

00:11:54.667 --> 00:11:56.034
jump off of a cliff or something.

00:11:56.034 --> 00:11:59.470
So another way to think about this is that the force,

00:11:59.470 --> 00:12:00.800
and we haven't done a lot on force yet,

00:12:00.800 --> 00:12:02.048
we'll talk about this in more depth,

00:12:02.048 --> 00:12:05.533
is that this pilot would be experiencing

00:12:05.533 --> 00:12:08.400
more than 3 times the force of gravity,

00:12:08.400 --> 00:12:11.800
more than 3 g's. 3 g's would be about

00:12:11.800 --> 00:12:15.533
30 meters per second squared, this is more than that.

00:12:15.533 --> 00:12:19.625
So an analogy for how the pilot would feel

00:12:19.625 --> 00:12:22.867
is when he's, you know, if this is the chair right here,

00:12:22.867 --> 00:12:26.707
his pilot's chair, that he's in, so this is the chair,

00:12:26.707 --> 00:12:29.133
and he's sitting on the chair, let me do my best

00:12:29.133 --> 00:12:32.733
to draw him sitting on the chair, so this is him

00:12:32.733 --> 00:12:37.133
sitting on the chair, flying the plane, and this is the pilot,

00:12:37.133 --> 00:12:41.000
the force he would feel, or while this thing is accelerating

00:12:41.000 --> 00:12:44.401
him forward at 33 meters per second squared,

00:12:44.401 --> 00:12:48.067
it would feel very much to him like if he was lying down

00:12:48.067 --> 00:12:52.467
on the surface of the planet, but he was 3 times heavier,

00:12:52.467 --> 00:12:56.475
or more than 3 times heavier. Or if he was lying down,

00:12:56.475 --> 00:12:59.958
or if you were lying down, like this, let's say this is you,

00:12:59.958 --> 00:13:04.667
this is your feet, and this is your face, this is your hands,

00:13:04.667 --> 00:13:07.040
let me draw your hands right here, and if you had

00:13:07.040 --> 00:13:11.475
essentially two more people stacked above you,

00:13:11.475 --> 00:13:14.067
roughly, I'm just giving you the general sense of it,

00:13:14.067 --> 00:13:16.467
that's how it would feel, a little bit more than two people,

00:13:16.467 --> 00:13:19.733
that squeezing sensation. So his entire body

00:13:19.733 --> 00:13:22.876
is going to feel 3 times heavier than it would

00:13:22.876 --> 00:13:25.600
if he was just laying down on the beach or something

00:13:25.600 --> 00:13:29.518
like that. So it's very very very interesting, I guess,

00:13:29.518 --> 00:13:32.733
idea, at least to me. Now the other question

00:13:32.733 --> 00:13:34.486
that we can ask ourselves is how long will it take

00:13:34.486 --> 00:13:39.067
to get catapulted off of this carrier? And if he's

00:13:39.067 --> 00:13:45.930
accelerating at 33 meters per second squared,

00:13:45.930 --> 00:13:47.675
how long would it take him to get from 0

00:13:47.675 --> 00:13:50.133
to 72 meters per second?

00:13:50.133 --> 00:13:53.067
So after 1 second, he'll be going 33 meters per second,

00:13:53.067 --> 00:13:56.533
after 2 seconds, he'll be going 66 meters per second,

00:13:56.533 --> 00:13:58.800
so it's going to take, and so it's a little bit more

00:13:58.800 --> 00:14:00.067
than 2 seconds. So it's going to take him

00:14:00.067 --> 00:14:01.533
a little bit more than 2 seconds.

00:14:01.533 --> 00:14:04.161
And we can calculate it exactly if you take

00:14:04.161 --> 00:14:07.733
72 meters per second, and you divide it by 33,

00:14:07.733 --> 00:14:13.600
it'll take him 2.18 seconds, roughly, to be catapulted

00:14:13.600 --> 00:14:15.267
off of that carrier.


Airbus A380 take-off distance

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:02.500
In the last video,
we figured out

00:00:02.500 --> 00:00:06.875
that given a takeoff velocity
of 280 kilometers per hour--

00:00:06.875 --> 00:00:09.250
and if we have a positive
value for any of these vectors,

00:00:09.250 --> 00:00:12.580
we assume it's in the forward
direction for the runway--

00:00:12.580 --> 00:00:16.860
given this takeoff velocity,
and a constant acceleration of 1

00:00:16.860 --> 00:00:20.260
meter per second per second,
or 1 meter per second squared,

00:00:20.260 --> 00:00:22.380
we figured out that it
would take an Airbus

00:00:22.380 --> 00:00:28.240
A380 about 78
seconds to take off.

00:00:28.240 --> 00:00:30.020
What I want to figure
out in this video

00:00:30.020 --> 00:00:33.110
is, given all of these
numbers, how long of a runaway

00:00:33.110 --> 00:00:36.080
does it need, which is a very
important question if you want

00:00:36.080 --> 00:00:39.020
to build a runway that
can at least allow

00:00:39.020 --> 00:00:40.699
Airbus A380s to take off.

00:00:40.699 --> 00:00:42.990
And you probably want it to
be a little bit longer than

00:00:42.990 --> 00:00:45.300
that just in case it takes a
little bit longer than expected

00:00:45.300 --> 00:00:45.890
to take off.

00:00:45.890 --> 00:00:48.350
But what is the minimum
length of the runway

00:00:48.350 --> 00:00:51.450
given these numbers?

00:00:51.450 --> 00:00:53.870
So we want to figure
out the displacement,

00:00:53.870 --> 00:00:57.230
or how far does
this plane travel

00:00:57.230 --> 00:01:00.220
as it is accelerating at
1 meter per second squared

00:01:00.220 --> 00:01:05.150
to 280 kilometers per
hour, or to 78-- or where

00:01:05.150 --> 00:01:07.630
did I write it
over here-- to 78.

00:01:07.630 --> 00:01:10.840
I converted it right over here.

00:01:14.930 --> 00:01:17.410
As it accelerates to
78 meters per second,

00:01:17.410 --> 00:01:21.430
how much land does
this thing cover?

00:01:21.430 --> 00:01:23.940
So let's call this,
the displacement

00:01:23.940 --> 00:01:26.760
is going to be equal
to-- So displacement

00:01:26.760 --> 00:01:31.760
is equal to-- You could view
it as velocity times time.

00:01:31.760 --> 00:01:34.490
But the velocity
here is changing.

00:01:34.490 --> 00:01:37.110
If we just had a constant
velocity for this entire time,

00:01:37.110 --> 00:01:39.230
we could just multiply
that times however

00:01:39.230 --> 00:01:41.710
long it's traveling, and it
would give us the displacement.

00:01:41.710 --> 00:01:44.000
But here our
velocity is changing.

00:01:44.000 --> 00:01:46.740
But lucky for us,
we learned-- and I

00:01:46.740 --> 00:01:50.880
encourage you to watch the video
on why distance, or actually

00:01:50.880 --> 00:01:54.370
the video on average velocity
for constant acceleration--

00:01:54.370 --> 00:01:57.510
but if you have
constant acceleration,

00:01:57.510 --> 00:02:02.600
and that is what we are
assuming in this example--

00:02:02.600 --> 00:02:06.572
so if you assume that your
acceleration is constant,

00:02:06.572 --> 00:02:08.030
then you can come
up with something

00:02:08.030 --> 00:02:10.030
called an average velocity.

00:02:10.030 --> 00:02:13.210
And the average velocity, if
your acceleration is constant,

00:02:13.210 --> 00:02:16.420
if and only if your
acceleration is constant, then

00:02:16.420 --> 00:02:20.020
your average velocity
will be the average

00:02:20.020 --> 00:02:23.740
of your final velocity
and your initial velocity.

00:02:23.740 --> 00:02:27.234
And so in this situation,
what is our average velocity?

00:02:27.234 --> 00:02:28.650
Well, our average
velocity-- let's

00:02:28.650 --> 00:02:30.320
do it in meters per
second-- is going

00:02:30.320 --> 00:02:32.880
to be our final velocity,
which is-- let me calculate it

00:02:32.880 --> 00:02:34.450
down here.

00:02:34.450 --> 00:02:38.814
So our average velocity
in this example

00:02:38.814 --> 00:02:40.480
is going to be our
final velocity, which

00:02:40.480 --> 00:02:45.444
is 78 meters per second,
plus our initial velocity.

00:02:45.444 --> 00:02:46.860
Well, what's our
initial velocity?

00:02:46.860 --> 00:02:48.870
We're assuming we're
starting at a standstill.

00:02:48.870 --> 00:02:52.330
Plus 0, all of that over 2.

00:02:52.330 --> 00:02:56.180
So our average velocity in this
situation, 78 divided by 2,

00:02:56.180 --> 00:02:59.480
is 39 meters per second.

00:02:59.480 --> 00:03:03.780
And the value of an average
velocity in this situation--

00:03:03.780 --> 00:03:06.590
actually, average velocity
in any situation--

00:03:06.590 --> 00:03:08.800
but in this situation, we
can calculate it this way.

00:03:08.800 --> 00:03:10.300
But the value of
an average velocity

00:03:10.300 --> 00:03:12.400
is we can figure
out our displacement

00:03:12.400 --> 00:03:21.020
by multiplying our average
velocity times the time that

00:03:21.020 --> 00:03:24.340
goes by, times the
change in time.

00:03:24.340 --> 00:03:28.190
So we know the change
in time is 78 seconds.

00:03:28.190 --> 00:03:30.260
We know our average
velocity here

00:03:30.260 --> 00:03:37.100
is 39 meters per second,
just the average of 0 and 78,

00:03:37.100 --> 00:03:38.770
39 meters per second.

00:03:38.770 --> 00:03:41.020
Another way to think
about it, if you want

00:03:41.020 --> 00:03:42.580
think about the
distance traveled,

00:03:42.580 --> 00:03:44.590
this plane is
constantly accelerating.

00:03:44.590 --> 00:03:48.350
So let me draw a
little graph here.

00:03:48.350 --> 00:03:52.370
This plane's velocity time graph
would look something like this.

00:03:52.370 --> 00:03:56.480
So if this is time and this
is velocity right over here,

00:03:56.480 --> 00:03:58.280
this plane has a
constant acceleration

00:03:58.280 --> 00:03:59.510
starting with 0 velocity.

00:03:59.510 --> 00:04:01.540
It has a constant acceleration.

00:04:01.540 --> 00:04:03.962
This slope right here is
constant acceleration.

00:04:03.962 --> 00:04:05.420
It should actually
be a slope of 1,

00:04:05.420 --> 00:04:07.260
given the numbers
in this example.

00:04:07.260 --> 00:04:10.180
And the distance traveled
is the distance that

00:04:10.180 --> 00:04:13.300
is the area under this
curve up to 78 seconds,

00:04:13.300 --> 00:04:16.170
because that's how long it
takes for it to take off.

00:04:16.170 --> 00:04:19.700
So the distance traveled is
this area right over here, which

00:04:19.700 --> 00:04:22.630
we cover in another video, or
we give you the intuition of why

00:04:22.630 --> 00:04:26.640
that works and why distance is
area under a velocity timeline.

00:04:26.640 --> 00:04:31.350
But what an average velocity
is, is some velocity,

00:04:31.350 --> 00:04:34.930
and in this case, it's exactly
right in between our final

00:04:34.930 --> 00:04:36.700
and our initial
velocities, that if you

00:04:36.700 --> 00:04:39.520
take that average velocity
for the same amount of time,

00:04:39.520 --> 00:04:44.540
you would get the exact
same area under the curve,

00:04:44.540 --> 00:04:47.060
or you would get the
exact same distance.

00:04:47.060 --> 00:04:48.980
So our average
velocity is 39 meters

00:04:48.980 --> 00:04:51.290
per second times 78 seconds.

00:04:51.290 --> 00:04:55.640
And let's just get our
calculator out for this.

00:04:55.640 --> 00:05:04.220
We have 39 times
78 gives us 3,042.

00:05:04.220 --> 00:05:07.600
So this gives us 3,042.

00:05:07.600 --> 00:05:10.010
And then meters per second
times second just leaves us

00:05:10.010 --> 00:05:11.580
with meters.

00:05:11.580 --> 00:05:16.030
So you need a runway
of over 3,000 meters

00:05:16.030 --> 00:05:18.000
for one of these
suckers to take off,

00:05:18.000 --> 00:05:24.030
or over 3 kilometers, which is
like about 1.8 or 1.9 miles,

00:05:24.030 --> 00:05:25.990
just for this guy
to take off, which

00:05:25.990 --> 00:05:28.525
I think is pretty fascinating.


Average velocity for constant acceleration

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:03.620
The goal of this video is to
explore some of the concepts,

00:00:03.620 --> 00:00:06.590
or formulas you might see in
a traditional physics class.

00:00:06.590 --> 00:00:09.460
But even more importantly to see
that they're really just common

00:00:09.460 --> 00:00:11.540
sense ideas.

00:00:11.540 --> 00:00:14.000
So let's just start
with a simple example.

00:00:14.000 --> 00:00:16.100
Let's say that-- and for
the sake of this video

00:00:16.100 --> 00:00:17.947
just so I don't
have to keep saying

00:00:17.947 --> 00:00:19.530
this is the magnitude
of the velocity,

00:00:19.530 --> 00:00:22.029
this is the direction of the
velocity, et cetera, et cetera.

00:00:22.029 --> 00:00:24.870
Let's just assume that if I
have a positive number that it

00:00:24.870 --> 00:00:27.580
means, for example, if I
have a positive velocity,

00:00:27.580 --> 00:00:29.994
it means that I am
going to the right.

00:00:29.994 --> 00:00:32.369
And let's say if I have a
negative number, which we won't

00:00:32.369 --> 00:00:34.987
see in this video, let's assume
that I'm going to the left.

00:00:34.987 --> 00:00:36.820
And that way I can just
write a number down,

00:00:36.820 --> 00:00:38.700
are we're only operating
in one dimension.

00:00:38.700 --> 00:00:40.158
You know that it
is specifying what

00:00:40.158 --> 00:00:41.720
the magnitude and
the direction is.

00:00:41.720 --> 00:00:44.072
If I said the velocity is
five meters per second,

00:00:44.072 --> 00:00:46.030
that means five meters
per second to the right.

00:00:46.030 --> 00:00:48.120
If I say it's negative
five meters per second,

00:00:48.120 --> 00:00:51.480
that means it's five meters
per second to the left.

00:00:51.480 --> 00:00:53.220
Now let's just say
just for simplicity,

00:00:53.220 --> 00:00:57.430
let's just say we start
with an initial velocity,

00:00:57.430 --> 00:00:59.317
of five meters per second.

00:00:59.317 --> 00:01:01.400
And once again, I am
specifying both the magnitude

00:01:01.400 --> 00:01:02.150
and the direction.

00:01:02.150 --> 00:01:05.370
Because of this convention here,
we know it is to the right.

00:01:05.370 --> 00:01:08.050
And let's say that we have
a constant acceleration.

00:01:08.050 --> 00:01:11.010
We have a constant
acceleration at play

00:01:11.010 --> 00:01:14.310
of two meters per
second, per second,

00:01:14.310 --> 00:01:16.730
or two meters per
second squared.

00:01:16.730 --> 00:01:20.470
And once again, since this is
positive, it is to the right.

00:01:20.470 --> 00:01:23.340
And let's say that we
do this for a duration.

00:01:23.340 --> 00:01:25.810
So my change in
time is- let's say

00:01:25.810 --> 00:01:31.560
we do this for a duration
of four-- I'll just use s.

00:01:31.560 --> 00:01:33.520
So I don't use seconds
and then sec, and then

00:01:33.520 --> 00:01:34.580
s in different places.

00:01:34.580 --> 00:01:37.150
So s for this video is seconds.

00:01:37.150 --> 00:01:42.170
So I want to do is think
about how far do we travel.

00:01:42.170 --> 00:01:43.860
Well there's two things.

00:01:43.860 --> 00:01:46.320
How fast are we going
after four seconds,

00:01:46.320 --> 00:01:50.490
and how far have we traveled
over the course of those four

00:01:50.490 --> 00:01:51.610
seconds.

00:01:51.610 --> 00:01:55.960
So let's draw ourselves a
little bit of a diagram here.

00:01:55.960 --> 00:02:00.840
So this is my velocity axis and
this over here is my time axis.

00:02:00.840 --> 00:02:03.490
I can draw a straighter
line than that.

00:02:03.490 --> 00:02:07.010
So that is my time axis.

00:02:07.010 --> 00:02:07.940
This is velocity.

00:02:11.050 --> 00:02:13.250
That is my velocity
right over there.

00:02:13.250 --> 00:02:15.890
And I'm starting off at
five meters per second.

00:02:15.890 --> 00:02:18.820
So this is five meters per
second right over here.

00:02:18.820 --> 00:02:23.650
So v i is equal to
five meters per second.

00:02:23.650 --> 00:02:25.530
And then every second
that goes by, I'm

00:02:25.530 --> 00:02:27.380
going two meters
per second faster.

00:02:27.380 --> 00:02:30.060
Because it's two meters
per second, per second.

00:02:30.060 --> 00:02:33.710
So every second that goes
by-- so after one second,

00:02:33.710 --> 00:02:35.740
I'm going to go two
meters per second faster.

00:02:35.740 --> 00:02:36.876
So I'll be at seven.

00:02:36.876 --> 00:02:38.250
Or another way to
think about it,

00:02:38.250 --> 00:02:41.080
is the slope of
this velocity line

00:02:41.080 --> 00:02:42.645
is my constant acceleration.

00:02:42.645 --> 00:02:44.310
Or I have a constant slope here.

00:02:44.310 --> 00:02:48.440
So it might look something
look something like that.

00:02:48.440 --> 00:02:51.290
So what has happened
after four seconds?

00:02:51.290 --> 00:02:53.860
So one, two, three, four.

00:02:53.860 --> 00:02:55.470
So this is my delta t.

00:02:55.470 --> 00:02:58.380
So my final velocity is
going to be right over there.

00:03:02.112 --> 00:03:03.820
I'll write it down
here just because it's

00:03:03.820 --> 00:03:06.080
getting in the way of
this word velocity.

00:03:06.080 --> 00:03:09.830
And so this is v. This
is my final velocity.

00:03:09.830 --> 00:03:11.150
And what would it be?

00:03:11.150 --> 00:03:13.690
Well I'm starting at
five meters per second.

00:03:13.690 --> 00:03:15.780
So I'm going to do it
both, using the variables,

00:03:15.780 --> 00:03:17.800
and using the concrete numbers.

00:03:17.800 --> 00:03:21.940
So starting at some
initial velocity.

00:03:21.940 --> 00:03:24.750
The subscript i
says i for initial.

00:03:24.750 --> 00:03:27.720
And then each second
that goes by, I'm

00:03:27.720 --> 00:03:29.830
getting this much faster.

00:03:29.830 --> 00:03:32.470
So if I want to see how
much faster have I gone,

00:03:32.470 --> 00:03:34.640
I multiply the
number of seconds--

00:03:34.640 --> 00:03:37.180
I'll just multiply the
number of seconds that go by,

00:03:37.180 --> 00:03:38.265
times my acceleration.

00:03:42.300 --> 00:03:44.800
And once again, this
right here, I just

00:03:44.800 --> 00:03:46.400
wrote the subscript
c saying that it's

00:03:46.400 --> 00:03:48.120
a constant acceleration.

00:03:48.120 --> 00:03:50.650
And so that will tell
me how fast I've gone.

00:03:50.650 --> 00:03:53.310
If I start at this
point, and I multiply

00:03:53.310 --> 00:03:56.950
the duration times my
slope, I will get this high.

00:03:56.950 --> 00:03:58.475
I will get to my final velocity.

00:04:01.365 --> 00:04:03.455
And just to make it
clear with the numbers,

00:04:03.455 --> 00:04:05.080
these numbers really
could be anything,

00:04:05.080 --> 00:04:07.990
I'm just picking these to
make it concrete in your mind.

00:04:07.990 --> 00:04:12.330
You have five meters per
second, plus four seconds,

00:04:12.330 --> 00:04:14.760
plus-- I want to
do that in yellow--

00:04:14.760 --> 00:04:19.680
plus four seconds times
our acceleration of two

00:04:19.680 --> 00:04:21.500
meters per second squared.

00:04:24.900 --> 00:04:26.814
And what is this
going to be equal to?

00:04:26.814 --> 00:04:29.230
You have the seconds canceling
out with one of the seconds

00:04:29.230 --> 00:04:31.450
down here.

00:04:31.450 --> 00:04:32.640
So let me write it.

00:04:32.640 --> 00:04:35.210
So we have five
meters per second,

00:04:35.210 --> 00:04:40.100
plus four times two is eight.

00:04:40.100 --> 00:04:40.996
This second's gone.

00:04:40.996 --> 00:04:42.620
We're just left with
meters per second.

00:04:42.620 --> 00:04:44.580
Eight meters per second.

00:04:44.580 --> 00:04:49.490
Or this is the same thing as
13 meters per second, which

00:04:49.490 --> 00:04:51.837
is going to be our
final velocity.

00:04:51.837 --> 00:04:53.170
And I want to take a pause here.

00:04:53.170 --> 00:04:54.961
And you can pause and
think about yourself.

00:04:54.961 --> 00:04:56.710
This hopefully
should be intuitive.

00:04:56.710 --> 00:04:59.780
We were starting at going
five meters per second.

00:04:59.780 --> 00:05:01.590
Every second that
goes by we're going

00:05:01.590 --> 00:05:03.920
to go two meters
per second faster.

00:05:03.920 --> 00:05:06.630
So after one second we will
be at seven meters per second.

00:05:06.630 --> 00:05:09.870
After two seconds we will be
at nine meters per second.

00:05:09.870 --> 00:05:14.142
After three seconds we will
be 11 meters per second.

00:05:14.142 --> 00:05:15.600
And then after four
seconds we will

00:05:15.600 --> 00:05:17.420
be at 13 meters per second.

00:05:17.420 --> 00:05:20.220
So you multiply how much time
passed times acceleration.

00:05:20.220 --> 00:05:22.460
This is how much faster,
we're going to be going.

00:05:22.460 --> 00:05:24.520
If we're already going
five meters per second.

00:05:24.520 --> 00:05:26.220
Five plus how much faster?

00:05:26.220 --> 00:05:28.250
13 meters per second.

00:05:28.250 --> 00:05:34.150
So this right up here
is 13 meters per second.

00:05:34.150 --> 00:05:36.840
So, I'll take a
little pause here.

00:05:36.840 --> 00:05:38.340
Hopefully that's intuitive.

00:05:38.340 --> 00:05:40.090
And the whole point
of that is to show you

00:05:40.090 --> 00:05:42.867
that this formula that you'll
often see in many physics

00:05:42.867 --> 00:05:45.200
books-- It's not just something
that randomly popped out

00:05:45.200 --> 00:05:48.330
of the air-- it just makes
complete common sense.

00:05:48.330 --> 00:05:50.190
Now the next thing
I want to talk about

00:05:50.190 --> 00:05:54.250
is what is the total distance
that we would have traveled.

00:05:54.250 --> 00:05:56.630
Now we know from the last
video that distance is just

00:05:56.630 --> 00:05:59.970
the area under this
curve right over here.

00:05:59.970 --> 00:06:03.440
So it's just the area
under this curve.

00:06:03.440 --> 00:06:05.870
You say wait, this is kind of
a strange shape right here.

00:06:05.870 --> 00:06:07.540
How do I calculate its area?

00:06:07.540 --> 00:06:09.470
And we can just use a
little simple geometry

00:06:09.470 --> 00:06:11.790
to break it down into
two different areas, that

00:06:11.790 --> 00:06:14.740
are very easy to
calculate their areas.

00:06:14.740 --> 00:06:16.000
Or two simple shapes.

00:06:16.000 --> 00:06:18.950
You can break it down into
this blue part, this rectangle

00:06:18.950 --> 00:06:20.510
right over here.

00:06:20.510 --> 00:06:22.670
Easy to figure out the
area of a rectangle.

00:06:22.670 --> 00:06:25.030
And we could break it down
into this purple part.

00:06:25.030 --> 00:06:26.490
This triangle right here.

00:06:26.490 --> 00:06:28.920
Easy to figure out the
area of a triangle.

00:06:28.920 --> 00:06:30.890
And that will be the
total distance we travel.

00:06:30.890 --> 00:06:33.330
And even this will hopefully
make some intuition.

00:06:33.330 --> 00:06:36.000
Because this blue
area is how far would

00:06:36.000 --> 00:06:38.470
we have traveled if we
were not accelerating.

00:06:38.470 --> 00:06:42.330
If we just went five meters
per second, for four seconds.

00:06:42.330 --> 00:06:44.192
So if you go five meters
per second times--

00:06:44.192 --> 00:06:46.400
so this is one second, two
second, three second, four

00:06:46.400 --> 00:06:47.320
second.

00:06:47.320 --> 00:06:51.020
So you're going from
times zero to time four,

00:06:51.020 --> 00:06:54.020
your change in time
is four seconds.

00:06:54.020 --> 00:06:57.330
So if you go five meters per
second, for four seconds,

00:06:57.330 --> 00:06:59.610
you're going to go 20 meters.

00:06:59.610 --> 00:07:02.360
This right here is 20 meters.

00:07:02.360 --> 00:07:04.040
That is the area
of this right here.

00:07:04.040 --> 00:07:06.840
Five times four.

00:07:06.840 --> 00:07:09.750
This purple or
magenta area tells you

00:07:09.750 --> 00:07:12.040
how further than
this, are you going,

00:07:12.040 --> 00:07:14.050
because you are
accelerating, because you

00:07:14.050 --> 00:07:16.840
kept going faster and
faster and faster.

00:07:16.840 --> 00:07:19.400
And it's pretty easy
to calculate this area.

00:07:19.400 --> 00:07:22.850
The base here is still
five, because that's

00:07:22.850 --> 00:07:24.090
five seconds have gone by.

00:07:24.090 --> 00:07:25.750
And what's the height here?

00:07:25.750 --> 00:07:28.740
The height here is
my final velocity

00:07:28.740 --> 00:07:30.275
minus my initial velocity.

00:07:33.590 --> 00:07:38.240
Or it's the change in velocity
due to the acceleration.

00:07:38.240 --> 00:07:40.860
And the change in velocity
due to the acceleration.

00:07:40.860 --> 00:07:43.200
13 minus five is eight.

00:07:43.200 --> 00:07:44.810
Or it's this 8 right over here.

00:07:44.810 --> 00:07:48.300
It is eight meters per second.

00:07:48.300 --> 00:07:51.290
So this height right over here
is eight meters per second.

00:07:51.290 --> 00:07:54.720
The base right over
here is four seconds.

00:07:54.720 --> 00:07:56.320
That's the time that past.

00:07:56.320 --> 00:07:59.230
So what's the area
of this triangle?

00:07:59.230 --> 00:08:02.760
Area of a triangle is
one half, times the base,

00:08:02.760 --> 00:08:07.000
which is four seconds,
times the height, which

00:08:07.000 --> 00:08:10.570
is eight meters per second.

00:08:10.570 --> 00:08:11.940
Seconds cancel out.

00:08:11.940 --> 00:08:14.540
One half times four
is two, times eight

00:08:14.540 --> 00:08:21.130
is equal to 16 meters.

00:08:21.130 --> 00:08:24.850
So the total distance we
traveled is 20 plus 16,

00:08:24.850 --> 00:08:27.090
is 36 meters.

00:08:27.090 --> 00:08:30.250
That is the total, or I could
say the total displacement,

00:08:30.250 --> 00:08:33.010
and once again, it's to the
right, since it is positive.

00:08:33.010 --> 00:08:35.590
So this is our displacement.

00:08:35.590 --> 00:08:38.054
Now what I want to do is do
this exact same calculation,

00:08:38.054 --> 00:08:39.720
but keep it in variable
form and that'll

00:08:39.720 --> 00:08:42.500
give another formula that many
people, many often memorize.

00:08:42.500 --> 00:08:44.000
But I want you to
understand it it's

00:08:44.000 --> 00:08:46.100
a completely intuitive
formula, and it just

00:08:46.100 --> 00:08:49.400
comes out of the logical
flow of reasoning

00:08:49.400 --> 00:08:51.410
that we went through
in this video.

00:08:51.410 --> 00:08:52.670
What is the area?

00:08:52.670 --> 00:08:56.320
Once again, if we just
think about the variables.

00:08:56.320 --> 00:08:58.790
Well, the area of
this rectangle right

00:08:58.790 --> 00:09:03.100
here is our initial velocity,
times our change in time.

00:09:06.900 --> 00:09:11.460
So this is the blue
rectangle right over here.

00:09:11.460 --> 00:09:13.770
And then plus-- what
do we have to do--

00:09:13.770 --> 00:09:15.960
we have the change
in time, once again,

00:09:15.960 --> 00:09:19.730
we have the change in
time, times this height.

00:09:19.730 --> 00:09:22.560
Which is our final velocity.

00:09:22.560 --> 00:09:24.220
Which is times
our final velocity

00:09:24.220 --> 00:09:28.160
minus our initial velocity.

00:09:28.160 --> 00:09:29.822
These are all vectors.

00:09:29.822 --> 00:09:32.280
They're just positive, telling
us we're going to the right.

00:09:32.280 --> 00:09:34.340
And we don't just multiply
the base times the height.

00:09:34.340 --> 00:09:36.270
That would give us the area
of this entire rectangle.

00:09:36.270 --> 00:09:38.690
We have to take it by half
because a triangle is only half

00:09:38.690 --> 00:09:40.020
of that rectangle.

00:09:40.020 --> 00:09:41.280
So times one half.

00:09:44.590 --> 00:09:46.320
And so this is the area.

00:09:46.320 --> 00:09:47.560
This is the purple area.

00:09:47.560 --> 00:09:49.160
That's not purple.

00:09:49.160 --> 00:09:53.810
This is the purple
area, right over here.

00:09:53.810 --> 00:09:55.140
This is the area of this.

00:09:55.140 --> 00:09:57.479
This is the area of that.

00:09:57.479 --> 00:09:59.020
And let's simplify
this a little bit.

00:09:59.020 --> 00:10:00.890
Let's factor out a delta t.

00:10:00.890 --> 00:10:02.850
So if you factor
out a delta t, you

00:10:02.850 --> 00:10:07.965
get delta t times, a
bunch of stuff, v sub i.

00:10:07.965 --> 00:10:11.580
So your initial velocity.

00:10:11.580 --> 00:10:12.780
We factored this out.

00:10:12.780 --> 00:10:14.710
Plus this stuff.

00:10:14.710 --> 00:10:17.070
Plus this thing right over here.

00:10:17.070 --> 00:10:18.790
And we can distribute
the one half.

00:10:18.790 --> 00:10:24.330
We factored the delta t's out.

00:10:24.330 --> 00:10:26.910
And let's multiply the one half
times each of these things.

00:10:26.910 --> 00:10:30.810
So it's going to be
plus, one half times v f,

00:10:30.810 --> 00:10:32.670
times our final velocity.

00:10:32.670 --> 00:10:35.040
That's not the right
color, let me actually

00:10:35.040 --> 00:10:38.040
do it in the right color so
you understand what I'm doing.

00:10:38.040 --> 00:10:39.750
So this is the one half.

00:10:39.750 --> 00:10:42.870
So plus one half times
our final velocity.

00:10:45.640 --> 00:10:49.845
Minus one half times
our initial velocity.

00:10:53.480 --> 00:10:55.271
I want to do that in blue.

00:10:55.271 --> 00:10:57.270
Sorry, I'm having trouble
changing colors today.

00:10:57.270 --> 00:10:59.635
Minus one half times
our initial velocity.

00:11:03.572 --> 00:11:05.030
And now what does
this simplify to?

00:11:05.030 --> 00:11:07.450
We have something plus one
half times something else,

00:11:07.450 --> 00:11:10.230
minus one half times
that original something.

00:11:10.230 --> 00:11:14.490
So what is v i minus
one half times v i.

00:11:14.490 --> 00:11:16.230
So anything minus
half of it, you're

00:11:16.230 --> 00:11:19.070
just going to have a
positive half left.

00:11:19.070 --> 00:11:21.490
So these two terms,
this term and this term

00:11:21.490 --> 00:11:27.300
will simplify to one half v i.

00:11:27.300 --> 00:11:30.980
One half the initial
velocity plus one half times

00:11:30.980 --> 00:11:31.780
the final velocity.

00:11:35.480 --> 00:11:37.730
And all of that is
being multiplied

00:11:37.730 --> 00:11:42.740
by our change in time.

00:11:42.740 --> 00:11:45.410
Or the time that has gone by.

00:11:45.410 --> 00:11:49.359
And this tells us the distance.

00:11:49.359 --> 00:11:50.650
The distance that we travelled.

00:11:50.650 --> 00:11:53.050
Or another way to
think about it-- let's

00:11:53.050 --> 00:11:56.090
factor out this one half.

00:11:56.090 --> 00:12:00.200
You get distance is equal
to change in time, times--

00:12:00.200 --> 00:12:06.930
factoring out the one
half-- v i plus v f.

00:12:06.930 --> 00:12:09.260
No, that's not the right color.

00:12:09.260 --> 00:12:13.150
v i plus v f.

00:12:16.160 --> 00:12:17.220
So this is interesting.

00:12:17.220 --> 00:12:21.020
The distance we traveled
is equal to one half

00:12:21.020 --> 00:12:24.340
of the initial velocity
plus the final velocity.

00:12:24.340 --> 00:12:26.350
So this is really--
if you just took,

00:12:26.350 --> 00:12:28.330
this quantity right
over here is just

00:12:28.330 --> 00:12:32.140
the arithmetic-- I always
have trouble saying that word.

00:12:32.140 --> 00:12:35.829
It's the arithmetic mean
of these two numbers.

00:12:35.829 --> 00:12:37.870
And so I'm going to define
this as something new.

00:12:37.870 --> 00:12:39.920
I'm going to call this
the average velocity.

00:12:39.920 --> 00:12:42.000
But we have to be very
careful with this.

00:12:42.000 --> 00:12:47.130
This right here is
the average velocity.

00:12:47.130 --> 00:12:48.830
But the only reason
why I can just

00:12:48.830 --> 00:12:52.137
take the starting velocity and
the ending velocity and, adding

00:12:52.137 --> 00:12:53.720
them together, and
then divide by two.

00:12:53.720 --> 00:12:55.803
Essentially taking the
average of these two things

00:12:55.803 --> 00:12:57.370
which would be
someplace over here.

00:12:57.370 --> 00:12:59.220
And I take that as
the average velocity,

00:12:59.220 --> 00:13:02.280
is because my
acceleration is constant.

00:13:02.280 --> 00:13:05.771
Which is usually an assumption
in most introductory physics

00:13:05.771 --> 00:13:06.270
classes.

00:13:06.270 --> 00:13:07.770
But it's not always
the assumption.

00:13:07.770 --> 00:13:11.090
But if you do have a constant
acceleration like this,

00:13:11.090 --> 00:13:13.900
you can assume that
the average velocity

00:13:13.900 --> 00:13:17.890
is going to be the average
of the initial velocity

00:13:17.890 --> 00:13:19.647
and the final velocity.

00:13:19.647 --> 00:13:21.980
If this was a curve or if the
acceleration was changing,

00:13:21.980 --> 00:13:23.720
you could not do that.

00:13:23.720 --> 00:13:25.430
But what's useful
about this-- is

00:13:25.430 --> 00:13:28.270
if you want to figure out the
distance that was traveled

00:13:28.270 --> 00:13:30.530
you just need to know
the initial velocity

00:13:30.530 --> 00:13:32.350
and the final velocity.

00:13:32.350 --> 00:13:34.980
Average the two, and
then multiply that

00:13:34.980 --> 00:13:37.310
times the time that goes by.

00:13:37.310 --> 00:13:39.210
So in this situation,
our final velocity

00:13:39.210 --> 00:13:41.150
is 13 meters per second.

00:13:41.150 --> 00:13:44.180
Our initial velocity was
five meters per second.

00:13:44.180 --> 00:13:47.600
So you have 13 plus
5 is equal to 18.

00:13:47.600 --> 00:13:49.230
You divide that by 2.

00:13:49.230 --> 00:13:53.350
Your average velocity
is 9 meters per second,

00:13:53.350 --> 00:13:55.480
if you take the
average of 13 and 5.

00:13:55.480 --> 00:13:58.440
And then 9 meters per
second times 4 seconds

00:13:58.440 --> 00:13:59.977
gives you 36 meters.

00:13:59.977 --> 00:14:01.560
So hopefully that
doesn't confuse you.

00:14:01.560 --> 00:14:03.200
I just want to show you
where some of these things

00:14:03.200 --> 00:14:04.360
that you'll see in
your physics class,

00:14:04.360 --> 00:14:06.693
some of these formulas, why
you shouldn't memorize them,

00:14:06.693 --> 00:14:08.607
and they can all be deduced.