Maths tools

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Maths tools

Post by Xah » Thu Feb 11, 2016 3:29 pm

Want to get a realistic perspective for population growth?
Interested to see just how much space your population takes up?
Just how tall is that megastructure anyway?

Well, wonder no more. Throw these figures into a handy spreadsheet, or just use pen, paper and a calculator if you're stuck in the 80's and get all the answers you need.

Population growth

The simple formula for population growth is this:

(((p1-p2)/p2)*100)/T = % growth rate per year

p1 = your present population
p2 = the population at a point in the past
T = Number of years elapsed since p2

Modern day growth rates are less than 1%, developing nations might reach 3 or 4%, anything more than that would require some narrative explanations.

For example, back in 2016 your population was 10 million, in 2322, you want it to be 80 million. Doing the sum gives you an answer of 2%, an easily obtainable growth rate.

Say you wanted 500 million people; now your growth rate is 16%, steep, but probably explainable with cloning or something or other.


Population space requirements

An easy one, work out how much space you want to give each of your inhabitants; the UK rules for HMOs (home with multiple occupancy - flat share) is that 10 m² per person is a reasonable amount of living space (a square just over 3 m on each side). Times this by 5 to allow for non-living space (ultilities, factories, etc. More if you like) then multiply that up by the population involved. A square root of that number will give you the side of the area in metres (it's assuming its a square and only one dimensional).

Three-dimensional structures

Get your area figures from the previous sum, multiple this by the height of your storey to get the total volume of a single storied building. Work out how long you want your sides to be. Your height is then volume/(length x width).

Example, you've got 20,000 people you want housed in a mega-structure. Doing the quick calculations, you work out they need 1,000,000 m² of space (20,000 x 50). You make each floor of your structure 5 m in height, so a single storied structure would occupy 5,000,000 m³ (1,000,000 x 5). You want it to be 80 m square, so the total height of your tower is 781 meters, or 157 floors.

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Re: Maths tools

Post by Xah » Fri Mar 11, 2016 2:36 pm

So, I've been working on shields and energy and so on and I've came up with all of this:

To make things simpler, I've converted all my energies into joules, it's not that simple in real life, as watts to joules needs a time component as well, but it kind of works.

Weapons

A standard 5.56 mm NATO round, fired from an M16 rifle has the following energy:
mass (kg): 0.0046
velocity (m/s): 980
energy (J): 1/2 mass * velocity² = 2209 J or 2.2 kJ

Not very much really, but at 1 round a second, and 1000 men, that's 2200 kJ per second

M60 machine gun
mass (kg): 0.01
velocity (m/s): 853
energy (J): 3,638 J or 3.6 kJ

At 10 rounds a second, that's 36 kJ per second, you need a hundred of these to come close to equalling the tank shell

A standard kinetic penetrator tank shell:
mass (kg): 4.6
velocity (m/s): 1475
Energy (J): 5,003,938 J or 5,003 kJ

As you can see, a tank shell has much, much more kinetic energy than a bullet but only hits once, so is comparable on a second's cycle.

For energy weapons, it gets more complex, as you need the wattage of the power source and the time of the cycle.
A 30 MW laser on a 0.1 second cycle
energy (W): 30,000,000
duration of cycle (s): 0.1
energy (J): energy * duration = 3,000,000 J or 3,000 kJ


Comparable to a tank shell per shot, but a good laser could pump out 5 cycles per second, increasing the joules per second to 15,000 kJ

Shields

Shields are a little easier to calculate as they will have a direct power measurement in watts. They should also have an area coverage in meters square. I'm making the assumption that the shield is active for the second that the shot hits, but working in time isn't overly difficult. Watts to joules is easy in this case as joules = watts * time. With 1 second, that equals joule = watt

A 5 GW shield, covering 10 m²:
Energy (W) = 5,000,000
Area = 10 m²
Resistance (watts) = energy / area = 5,000,000 / 10 = 500,000 W / m² or 500 kW / m² or 500 kJ /m² per second


A 5 GW shield, covering ten square meters (about the size of a car) could withstand 500 kJ of energy per second - enough for bullets, but no where near enough for tanks or lasers.

Increase the size of the shield, and increase the factor of energy required too.

Bear in mind, a good sized nuclear reactor can put out 2 or 3 GW of power....

So unless the energy of a shield can convert up the energy given to it by a power supply (breaks the laws of thermodynamics), or power sources are phenomenally efficient, shields work on a small-scale against smaller weapons, but definately not against large calibre weapons or big lasers.

I should also add, that any shield that can stop bullets would also stop air for breathing, and any shield that can stop lasers would stop visible light, so you'd be blind.

Thoughts?

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Re: Maths tools

Post by LordSylveon » Fri Mar 11, 2016 3:17 pm

The Shield math seems to all make sense.

However, this is all assuming that the shield is a bubble, right?
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Re: Maths tools

Post by Xah » Fri Mar 11, 2016 3:23 pm

Potentially. But not essential

Actually, a bubble poses more issues; like what happens as it passes through the ground?

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Re: Maths tools

Post by Kleo Kira Azu » Sat Mar 12, 2016 12:14 am

I'm going to assume that explosions are higher in Kilojoules but will be further spreadout over a larger area because it's releasing energy towards the air, yes?

Shaped Charges is what's got me wondering at the moment. I'm going to assume it'll work the same way as reactive armor.

As for stopping air for breathing... I know it'll take more energy to do so, but wouldn't a few sensors that can detect if a projectile is going to impact [Tank] then start up the shield?

And since the example for vehicles, I'm going to assume space is going to be used for oxygen ... recycling... thingamajigs.
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Re: Maths tools

Post by Gibraltarica » Sat Mar 12, 2016 2:02 am

*Math
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Re: RE: Re: Maths tools

Post by Xah » Sat Mar 12, 2016 11:27 am

Gibraltarica wrote:*Math
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