Acceleration
Airbus A380 Take-off Time Figuring how long it takes an a380 to take off given a constant acceleration
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- This right here is a picture of an Airbus A380 aircraft,
- and I was curious
- How long would it take this aircraft to take-off?
- And I looked up its take-off velocity,
- and the specs I got were 280 km/h.
- And to make this a velocity,
- we have to specify a direction as well,
- not just a magnitude.
- So the direction is in the direction of the runway.
- So that would be the positive direction.
- So when we're talking about acceleration,
- we're going to assume it's in this direction,
- the direction of going down the runway.
- And I also looked up its specs,
- and this I'm simplifying a little bit,
- because it's not going to have a purely
- constant acceleration.
- But let's just say:
- from the moment that the pilot says,
- "We're taking off" to when it actually takes off,
- it has a constant acceleration.
- Its engines are able to provide a constant acceleration.
- Acceleration of 1.0 m/s per second
- So after every second,
- it can go one m/s faster
- than it was going
- at the beginning of that second.
- Or, another way to write this is
- 1.0 m/s per second,
- which can also be written as:
- 1.0 m/s^2
- I find this a little bit more intuitive,
- a little bit neater to write.
- So let's figure this out.
- So the first thing
- that we're trying to answer is:
- How long does take-off last?
- That is the question we will try to answer.
- And to answer this,
- at least my brain,
- wants to at least get the units right.
- So over here,
- we have our acceleration
- in terms of meters and seconds,
- or seconds squared.
- And over here,
- we have our take-off velocity
- in terms of kilometers and hours.
- So let's just convert
- this take-off velocity into m/s,
- and then it might simplify
- answering this question.
- So if we have 280 km/h,
- how do we convert that to m/s?
- So let's convert it to km/s first.
- So we want to get rid of this 'hour'.
- And the best way to do that:
- if we have an 'hour'
- in the denominator,
- we want an 'hour' in the numerator,
- and we want a 'second' in the denominator.
- And so, what do we multiply this by?
- Or what do we put in front of
- the 'hours' and 'seconds'?
- So in 1 hour there are 3600 seconds.
- 60 seconds in a minute,
- 60 minutes in an hour
- And so 1 of the larger unit
- is equal to 3600 of the smaller unit.
- And so we can multiply by that,
- And if we do that,
- The 'hours' will cancel out.
- And we'll get 280 divided by 3600
- kilometers per second.
- But I want to do all my math at once,
- so let's also do the conversion from
- kilometers to meters.
- So once again,
- we have kilometers in the numerator,
- so we want kilometers in the denominator now.
- So it cancels out.
- And we want meters in the numerator.
- And what's the smaller unit?
- It's meters, and we have 1,000 meters
- for every 1 kilometer.
- And when you multiply this out,
- the kilometers are going to cancel out,
- and you're going to be left with
- 280 times 1,000 all over 3600,
- And the units we have left are:
- meters per second.
- So let's get my trusty TI-85 out
- and actually calculate this.
- So we have 280 * 1,000,
- which is obviously 280,000,
- but let me just divide that by 3600.
- And it gives me 77.7 repeating.
- And it looks like I had 2 significant digits
- in each of these original things,
- I had 1.0 over here,
- not 100% clear how many
- significant digits I have over here.
- Was the spec rounded
- to the nearest 10 kilometers,
- or was it exactly 280 km/h?
- Just to be safe,
- I'll assume that it's rounded
- to the nearest 10 kilometers,
- so we only have 2 significant digits here.
- So we should only have 2 significant digits
- in our answer,
- so we're gonna round this to 78 m/s.
- So this is going to be 78 m/s,
- which is pretty fast!
- For this thing to take off,
- every second that goes by,
- it has to travel 78 meters,
- roughly 3/4 the length of a football field
- in every second.
- But that's not what we're trying to answer,
- we're trying to say how long
- will take-off last?
- Well we could just do this in our head,
- if you think about it.
- The acceleration is 1 m/s per second,
- which tells us:
- after every second,
- it's going 1 m/s faster.
- So, if you start at a velocity of 0,
- and then after 1 second,
- it will be going 1 m/s.
- After 2 seconds,
- it will be going 2 m/s.
- After 3 seconds,
- it will be going 3 m/s.
- So how long will it get to 78 m/s?
- Well, it will take 78 seconds.
- It'll take 78 seconds, or roughly
- a minute and 18 seconds.
- And just to verify this
- with our definition of acceleration,
- so to speak,
- just remember acceleration,
- which is a vector quantity,
- and all the directions
- we're talking about now
- are in the direction of
- this direction of the runway.
- The acceleration is equal to
- change in velocity over change in time.
- And we're trying to solve for:
- how much time does it take,
- or the change in time.
- So let's do that.
- So let's multiply both sides by
- change in time.
- You get Δt * acceleration
- is equal to
- change in velocity.
- And to solve for change in time,
- divide both sides by the acceleration,
- you get change in time.
- I could go down here,
- but I just want to use all this
- real-estate I have over here.
- I have change in time
- is equal to
- change in velocity
- divided by acceleration.
- And in this situation,
- what is our change in velocity?
- Well, we're starting off with the velocity,
- or we're assuming we're starting off
- with the velocity of 0 m/s,
- and we're getting up to 78 m/s,
- so our change in velocity is
- the 78 m/s.
- So this is equal,
- in our situation.
- 78 m/s is our change in velocity.
- I'm taking the final velocity,
- 78 m/s,
- and subtracting from that
- the initial velocity,
- which is 0 m/s,
- and you just get this
- divided by the acceleration,
- divided by 1 m/s per second,
- or 1 m/s^2.
- So the numbers part is pretty easy.
- You have 78 divided by 1,
- which is just 78,
- and then the units, you have:
- meters per second,
- and then if you divide by m/s^2,
- that's the same thing as multiplying by
- seconds squared per meter.
- Right?
- Dividing by something is the same thing
- as multiplying by its reciprocal,
- and you can do the same thing with units.
- And then we see
- the meters cancel out,
- and then sec^2 divided by seconds
- you're just left with seconds.
- So once again, we get 78 seconds.
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