# The Naked Scientists Forum

### Author Topic: How much further to the horizon for every storey upwards?  (Read 3936 times)

#### thedoc

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##### How much further to the horizon for every storey upwards?
« on: 05/12/2010 00:13:55 »
How much further can you see beyond the horizon as you go up each floor of a skyscraper building?

Go to the show page.

« Last Edit: 05/12/2010 00:13:55 by _system »

#### thedoc

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##### How much further to the horizon for every storey upwards?
« Reply #1 on: 05/12/2010 00:13:55 »
Chris -   Well let’s assume your skyscraper is standing in isolation, so it’s not going to get your view blocked by an adjacent building for instance, so there’s no get out clauses like that. It’s a building in isolation in the middle of nowhere.

There’s a sort of approximation for distance to the horizon in miles which is 1.23 times the square root of the height of your eyes above the ground in feet.

So you could work out, using that little rule, how much further you're going to see if you go up the height, in feet, of one storey in a building, and then you could keep doing that to work out how much further you're going to be able to see to the horizon from the top of the building than the bottom, for example.
« Last Edit: 05/12/2010 22:26:16 by chris »

#### Bored chemist

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##### How much further to the horizon for every storey upwards?
« Reply #2 on: 05/12/2010 12:33:41 »
That rule of thumb would be better for the addition of a unit or two.

#### Geezer

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##### How much further to the horizon for every storey upwards?
« Reply #3 on: 05/12/2010 20:00:10 »
That rule of thumb would be better for the addition of a unit or two.

Which part of "in feet" did you not understand?
« Last Edit: 05/12/2010 20:15:01 by Geezer »

#### granpa

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##### How much further to the horizon for every storey upwards?
« Reply #4 on: 05/12/2010 22:12:40 »
r=radius of earth = 6.3675×10^6 meters = 2.0891×10^7 feet
h=height above the ground
d=distance to horizon

r^2 + d^2 = (r + h)^2
r^2 + d^2 = r^2 + 2rh + h^2
d^2 = 2rh + h^2

since h^2 is so much smaller than the others it can be ignored

d = √2 * √r * √h
d = 1.414 * 2523.39 * √h (all in meters)
d = 3568.61 * √h (all in meters)

d = 1.414 * 4570.67 * √h (all in feet)
d = 6463.90 * √h (all in feet)
« Last Edit: 05/12/2010 22:26:20 by granpa »

#### chris

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##### How much further to the horizon for every storey upwards?
« Reply #5 on: 05/12/2010 22:27:35 »
That rule of thumb would be better for the addition of a unit or two.
Which part of "in feet" did you not understand?

In BC's defense, I omitted the word "MILES" - which I've now added.

C

#### Geezer

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##### How much further to the horizon for every storey upwards?
« Reply #6 on: 05/12/2010 23:35:45 »
In BC's defense, I omitted ......

LOL. I'm quite sure BC is entirely capable of defending himself

#### Bored chemist

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##### How much further to the horizon for every storey upwards?
« Reply #7 on: 06/12/2010 19:24:27 »
That rule of thumb would be better for the addition of a unit or two.

Which part of "in feet" did you not understand?

None.
What variety of esp let you know what units he was using for the distance to the horizon.

I suspect the rule breaks down for silly distances like a skyscraper that reaches the moon.

While I'm at it, what the OP asked for is actually the first derivative of that estimate with respect to height, measured in storeys.
That's a mess of a set of units.

#### Geezer

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##### How much further to the horizon for every storey upwards?
« Reply #8 on: 06/12/2010 19:40:16 »
Which part of "in feet" did you not understand?

None.
What variety of esp let you know what units he was using for the distance to the horizon.

[/quote]

I confess was deliberatley misinterpreting your point to mean that thare were no units mentioned at all .

You were quite correct though. Additional units were required.

#### The Naked Scientists Forum

##### How much further to the horizon for every storey upwards?
« Reply #8 on: 06/12/2010 19:40:16 »