0:00:00.000,0:00:00.670


0:00:00.670,0:00:05.440
This right here is a picture[br]of an Airbus A380 aircraft.

0:00:05.440,0:00:08.109
And I was curious[br]how long would it

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take this aircraft to take off?

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And I looked up its[br]takeoff velocity.

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And the specs I got were[br]280 kilometers per hour.

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And to make this a velocity we[br]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[br]direction of the runway.

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So that would be the positive[br]direction right over there.

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So when we're talking about[br]acceleration or velocity

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in this, we're[br]going to assume it's

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in this direction, the direction[br]of going down the runway.

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And I also looked up[br]its specs, and this,

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

0:00:47.120,0:00:49.214
going to have a purely[br]constant acceleration.

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But let's just say[br]from the moment

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

0:00:53.490,0:00:55.690
takes off it has a[br]constant acceleration.

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Its engines are able to[br]provide a constant acceleration

0:01:02.280,0:01:09.960
of 1.0 meters per[br]second per second.

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So after every second it[br]can go one meter per second

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

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

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this way-- meters[br]per second per second

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can also be written as[br]meters per second squared.

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I find this a little[br]bit more intuitive.

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This is a little[br]bit neater to write.

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So let's figure this out.

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So the first thing[br]we're trying to answer

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is, how long does take off last?

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That is the question[br]we will try to answer.

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And to answer this,[br]at least my brain

0:01:52.400,0:01:54.289
wants to at least[br]get the units right.

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So over here we have[br]our acceleration

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in terms of meters and[br]seconds, or seconds squared.

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And over here we have[br]our takeoff velocity

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in terms of[br]kilometers and hours.

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So let's just convert[br]this takeoff velocity

0:02:06.030,0:02:07.170
into meters per second.

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And then it might simplify[br]answering this question.

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So if we have 280[br]kilometers per hour,

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how do we convert that[br]to meters per second?

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

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So we want to get[br]rid of this hours.

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And the best way[br]to do that, if we

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have an hour in[br]the denominator, we

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want an hour in the[br]numerator, and we

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want a second in[br]the denominator.

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And so what do we[br]multiply this by?

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

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So one hour, in one hour[br]there are 3,600 seconds,

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60 seconds in a minute,[br]60 minutes in an hour.

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And so you have one[br]of the larger unit

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is equal to 3,600[br]of the smaller unit.

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And that we can[br]multiply by that.

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

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

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But I want to do[br]all my math at once.

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

0:03:04.550,0:03:08.770
So once again, we have[br]kilometers in the numerator.

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So we want the kilometers[br]in the denominator now.

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So it cancels out.

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And we want meters[br]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[br]for every 1 kilometer.

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And so when you multiply[br]this out the kilometers

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are going to cancel out.

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

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

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

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are meters per-- and the[br]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[br]out and actually calculate this.

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So we have 280 times 1000, which[br]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[br]repeating indefinitely.

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And it looks like I had[br]two significant digits

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in each of these[br]original things.

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I had 1.0 over[br]here, not 100% clear

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how many significant[br]digits over here.

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Was the spec rounded to[br]the nearest 10 kilometers?

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Or is it exactly 280[br]kilometers per hour?

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Just to be safe I'll[br]assume that it's

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rounded to the[br]nearest 10 kilometers.

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So we only have two[br]significant digits here.

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So we should only have[br]two significant digits

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in our answer.

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

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So this is going to be[br]78 meters per second,

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which is pretty fast.

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

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has to travel 78[br]meters, roughly 3/4

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of the length of a football[br]field in every second.

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

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

0:05:05.950,0:05:09.650
Well we could just do this in[br]our head if you think about it.

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The acceleration is 1 meter[br]per second, per second.

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Which tells us[br]after every second

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it's going 1 meter[br]per second faster.

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

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it'll be going 1[br]meter per second.

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After 2 seconds it will be[br]going 2 meters per second.

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

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

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Well, it will take 78[br]seconds, or roughly a minute

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and 18 seconds.

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And just to verify this with our[br]definition of our acceleration,

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so to speak, just[br]remember acceleration,

0:05:46.877,0:05:48.960
which is a vector quantity,[br]and all the directions

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we're talking about now[br]are in the direction

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of this direction of the runway.

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The acceleration is equal[br]to change in velocity

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over change in time.

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

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or the change in time.

0:06:08.740,0:06:09.520
So let's do that.

0:06:09.520,0:06:12.040
So let's multiply both[br]sides by change in time.

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You get change in time[br]times acceleration

0:06:17.780,0:06:20.785
is equal to change in velocity.

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And to solve for change[br]in time, divide both sides

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by the acceleration.

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So divide both sides by[br]the acceleration you get

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a change in time.

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I could go down[br]here, but I just want

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to use all this real[br]estate I have over here.

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I have change in time[br]is equal to change

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in velocity divided[br]by acceleration.

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And in this situation, what[br]is our change in velocity?

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Well, we're starting[br]off with the velocity,

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or we're assuming[br]we're starting off

0:06:54.980,0:06:57.880
with a velocity of[br]0 meters per second.

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

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

0:07:04.650,0:07:09.230


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So this is equal,[br]in our situation,

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78 meters per second is[br]our change in velocity.

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

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and subtract from that[br]the initial velocity,

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which is 0 meters per second.

0:07:20.590,0:07:22.000
And you just get this.

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Divided by the[br]acceleration, divided

0:07:24.230,0:07:28.990
by 1 meter per[br]second per second,

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or 1 meter per second squared.

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So the numbers part[br]are pretty easy.

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You have 78 divided by[br]1, which is just 78.

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And then the units you[br]have meters per second.

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And then if you divide by[br]meters per second squared,

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that's the same[br]thing as multiplying

0:07:44.230,0:07:46.760
by seconds squared per meter.

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Right?

0:07:48.440,0:07:49.940
Dividing by something[br]the same thing

0:07:49.940,0:07:51.940
as multiplying by[br]its reciprocal.

0:07:51.940,0:07:54.130
And you can do the[br]same thing with units.

0:07:54.130,0:07:57.110
And then we see the[br]meters cancel out.

0:07:57.110,0:07:59.060
And then seconds squared[br]divided by seconds,

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you're just left with seconds.

0:08:00.660,0:08:04.340
So once again, we[br]get 78 seconds,

0:08:04.340,0:08:07.910
a little over a minute for[br]this thing to take off.