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.

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

## Contact Form

Before contacting support, search for your issue or question in the Alison help section.