Some more problems...

[corky]

Here are some more problems I was thinking about.  Some would be pretty quick calculations, others would require a fair amount of thought.    

1.)  How big are water droplets in clouds?  Alternative:  Why don't clouds fall out of the sky?

2.)  How big a computer do you need to make The Matrix?  How much memory do you need to store a simulated reality for all the world's population? 

3.)  How much is the statue of liberty worth in pennies and copper?  That is, compare the "worth" of the statue of liberty if you chopped it up and used the copper to make pennies to the worth of all the actual copper.  Assume it is after 1982. 

4.)  How much of the moon falls on the earth each year?  So the moon gets hit with meteorites and all that jazz a lot (I assume).  What I want to know is how much of the dust on the moon gets ejected from the moon and lands on earth.   

5.)  Give everyone on earth a cell phone, how much does the mean global temperature increase?  Just curious. 

6.)  The moon is moving away from the earth at 22 mm/yr.  What implications does this have?  On, like, anything.

Just wanted to get them written down so I don't forget.  Doesn't mean I'll ever get to do them.  Anyone who gives a solution gets 400 quatloos.

[Alemi]

Alright, I'll bite.  This is real quick and by no means entirely correct.

1)  Lower bound, they scatter light as white, so must be larger than the wavelength of light, so we'll guess order micron.  The terminal velocity for a micron water droplet would be about 1/4 m/s, which is slow.  Why don't they fall, I've always heard its small air currents.  What kind of current?  Well, the earth is hot, and space is cold, so there should be a small thermal current.  We should estimate that thermal current.

2)  We principally use our vision to see the world.  Let's assume this is the highest capacity channel we have.  Our angular resolution, we can estimate as 550 nm / 1 mm which is about 10^-3 radians or 0.05 degrees. Good.  if I think about the visual field I can see, I'd say it is roughly 180 degrees across and 100 degrees up down.  Half a sphere is 2 pi sterideans and we have some fraction of that, say 100/180 of it or 3.5 sterideans.  Our visual acuity is 10^-3 radians or 10^-6 sterideans,  So, our visual is equivalent to roughly a 3 or 4 Megapixel camera picture.    Of course it is a lot less than that, as our eyes saccad and all that, but if you wanted to fool an individual and not model their eye saccades, if you where to project a 4 megapixel image in front of them, they presumably would have trouble telling the difference from reality.  and our visual response time is on the order of 20 frames a second, since TV is 24 and we can't see the difference.  As for the complexity, I know that using the 1 million color setting on my monitor doesn't really present any issues for me.  So log_2 1 million is 20 bits per pixel.  

So 20 bits * 4 10^6 pixels * 24 Hz is   2 10^9 bit hertz

Let's say we take the extreme example of not taking advantage of individual overlaps and the like, and precompute an entire life for everyone on earth.  That bound is 2 * 10^9 bit hertz  * 60 year lifetime gives me something like an exabyte of information.

So, it would take an exabyte of information per human to placate them.  Granted you sleep for about half your life and a lot of the information is redudant, but there you go.

How's that for a start?

[Alemi]

Some more.

3)  Google search reveals that it is 100 tons of copper.  WolframAlpha says that this is worth $650,000 or 6.5*10^7 pennies, if you sold the copper.  If you made a pre-1857 penny, which seems to have been (as it is currently) precisely 2.5 g, you'd get $363,000, a losing proposition, if you could get your hands on all the zinc you'd need, you could make $14 billion worth of pennies.  Huzzah!   Fun followup:  you'd need $12 million in zinc to pull this off.

4)  As a start the escape velocity of the moon is 2 km/s....  Will return to this later.

5) Well, my phone has a 5.55 Whr battery and with typical use lasts ~12 hr use on a charge (I'm a data hog with crappy reception).  So on average my phone is operating at 1/2 Watt.  That sounds high, but what are you going to do.  7 billion people, we're talking 3 *10^9 Watts excess.  Stefan boltzmann radiation for the earth is 2 * 10^17 W.  Let's consider small variations

hardly any difference

6) Interesting, will have to think.  Well, a couple things.  One, the earths rotation must be slowing by angular momentum conservation.  If I did the calculation right, and I'm not at all sure, this means that our day gets longer by a part in 10^-15 each year.  

You might also be able to estimate the time it takes for the moon to escape, or the age of the moon from that number.  

[Alemi]

Damn extra plusses.  Ignore almost all the plusses in the formulas above.

[Yariv Yanay]

On Sep 9, 7:40 pm, corky <rsw...@gmail.com> wrote:
> 1.)  How big are water droplets in clouds?  Alternative:  Why don't clouds
> fall out of the sky?

They do; I believe the phenomenon is called "rain".

> 5.)  Give everyone on earth a cell phone, how much does the mean global
> temperature increase?  Just curious.

Do you mean from the devices operating or from a carbon-footprint/
global warming perspective? (this is the only serious line in this
email)

> 6.)  The moon is moving away from the earth at 22 mm/yr.  What implications
> does this have?  On, like, anything.

As it's been explained to me, if you take this back about 65 million*
years into the past, you find that the moon was at a height of 3
meters. This explains the extinction of the dinosaurs, or at the very
least the taller ones.



* actual math gives 10 billion years but that's not conducive to my
dinosaur argument.