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This right here is a picture
of an Airbus A380 aircraft.

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And I was curious
how long would it

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take this aircraft to take off?

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And I looked up its
takeoff velocity.

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And the specs I got were
280 kilometers per hour.

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And to make this a velocity we
have to specify a direction as

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well, not just a magnitude.

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So the direction is in the
direction of the runway.

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So that would be the positive
direction right over there.

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So when we're talking about
acceleration or velocity

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in this, we're
going to assume it's

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in this direction, the direction
of going down the runway.

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And I also looked up
its specs, and this,

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I'm simplifying a little
bit, because it's not

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going to have a purely
constant acceleration.

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But let's just say
from the moment

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that the pilot says we're
taking off to when it actually

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takes off it has a
constant acceleration.

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Its engines are able to
provide a constant acceleration

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of 1.0 meters per
second per second.

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So after every second it
can go one meter per second

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faster than it was going at
the beginning of that second.

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Or another way to write this
is 1.0-- let me write it

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this way-- meters
per second per second

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can also be written as
meters per second squared.

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I find this a little
bit more intuitive.

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This is a little
bit neater to write.

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So let's figure this out.

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So the first thing
we're trying to answer

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is, how long does take off last?

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That is the question
we will try to answer.

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And to answer this,
at least my brain

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wants to at least
get the units right.

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So over here we have
our acceleration

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in terms of meters and
seconds, or seconds squared.

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And over here we have
our takeoff velocity

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in terms of
kilometers and hours.

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So let's just convert
this takeoff velocity

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into meters per second.

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And then it might simplify
answering this question.

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So if we have 280
kilometers per hour,

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how do we convert that
to meters per second?

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So let's convert it to
kilometers per second first.

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So we want to get
rid of this hours.

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And the best way
to do that, if we

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have an hour in
the denominator, we

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want an hour in the
numerator, and we

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want a second in
the denominator.

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And so what do we
multiply this by?

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Or what do we put in front
of the hours and seconds?

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So one hour, in one hour
there are 3,600 seconds,

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60 seconds in a minute,
60 minutes in an hour.

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And so you have one
of the larger unit

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is equal to 3,600
of the smaller unit.

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And that we can
multiply by that.

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And if we do that, the
hours will cancel out.

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And we'll get 280 divided by
3,600 kilometers per second.

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But I want to do
all my math at once.

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So let's also do the conversion
from kilometers to meters.

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So once again, we have
kilometers in the numerator.

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So we want the kilometers
in the denominator now.

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So it cancels out.

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And we want meters
in the numerator.

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And what's the smaller unit?

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It's meters.

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And we have 1,000 meters
for every 1 kilometer.

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And so when you multiply
this out the kilometers

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are going to cancel out.

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And you are going to be
left with 280 times 1,

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so we don't have to write
it down, times 1,000,

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all of that over 3,600,
and the units we have left

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are meters per-- and the
only unit we have left here

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is second-- meters per second.

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So let's get my trusty TI-85
out and actually calculate this.

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So we have 280 times 1000, which
is obviously 280,000, but let

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me just divide that by 3,600.

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And it gives me 77.7
repeating indefinitely.

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And it looks like I had
two significant digits

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in each of these
original things.

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I had 1.0 over
here, not 100% clear

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how many significant
digits over here.

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Was the spec rounded to
the nearest 10 kilometers?

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Or is it exactly 280
kilometers per hour?

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Just to be safe I'll
assume that it's

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rounded to the
nearest 10 kilometers.

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So we only have two
significant digits here.

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So we should only have
two significant digits

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in our answer.

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So we're going to round this
to 78 meters per second.

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So this is going to be
78 meters per second,

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which is pretty fast.

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For this thing to take off
every second that goes by it

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has to travel 78
meters, roughly 3/4

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of the length of a football
field in every second.

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But that's not what
we're trying to answer.

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We're trying to say how
long will take off last?

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Well we could just do this in
our head if you think about it.

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The acceleration is 1 meter
per second, per second.

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Which tells us
after every second

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it's going 1 meter
per second faster.

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So if you start at a velocity
of 0 and then after 1 second

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it'll be going 1
meter per second.

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After 2 seconds it will be
going 2 meters per second.

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After 3 seconds it'll be
going 3 meters per second.

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So how long will it take to
get to 78 meters per second?

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Well, it will take 78
seconds, or roughly a minute

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and 18 seconds.

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And just to verify this with our
definition of our acceleration,

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so to speak, just
remember acceleration,

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which is a vector quantity,
and all the directions

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we're talking about now
are in the direction

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of this direction of the runway.

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The acceleration is equal
to change in velocity

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over change in time.

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And we're trying to solve for
how much time does it take,

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or the change in time.

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So let's do that.

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So let's multiply both
sides by change in time.

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You get change in time
times acceleration

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is equal to change in velocity.

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And to solve for change
in time, divide both sides

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by the acceleration.

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So divide both sides by
the acceleration you get

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a change in time.

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I could go down
here, but I just want

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to use all this real
estate I have over here.

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I have change in time
is equal to change

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in velocity divided
by acceleration.

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And in this situation, what
is our change in velocity?

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Well, we're starting
off with the velocity,

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or we're assuming
we're starting off

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with a velocity of
0 meters per second.

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And we're getting up to
78 meters per second.

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So our change in velocity
is the 78 meters per second.

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So this is equal,
in our situation,

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78 meters per second is
our change in velocity.

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I'm taking the final velocity,
78 meters per second,

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and subtract from that
the initial velocity,

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which is 0 meters per second.

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And you just get this.

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Divided by the
acceleration, divided

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by 1 meter per
second per second,

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or 1 meter per second squared.

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So the numbers part
are pretty easy.

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You have 78 divided by
1, which is just 78.

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And then the units you
have meters per second.

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And then if you divide by
meters per second squared,

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that's the same
thing as multiplying

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by seconds squared per meter.

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Right?

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Dividing by something
the same thing

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as multiplying by
its reciprocal.

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And you can do the
same thing with units.

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And then we see the
meters cancel out.

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And then seconds squared
divided by seconds,

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you're just left with seconds.

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So once again, we
get 78 seconds,

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a little over a minute for
this thing to take off.