Maths Question arising from "1984"

[Dekan]

I'd be grateful for advice and guidance on this question please. Given - that in 1984, London ("chief city of Airstrip One") is being subjected to a sustained bombardment by rocket-bombs, and:

 

1. The rocket-bombs are falling at the rate of 20 to 30 a week (say, 4 each day);

2. Each bomb destroys buildings within a 50-metre radius of the impact-point;

3. No rebuilding is undertaken;

4. London has an area of 40 x 40 kilometres, ie 1,600 square kilometres;

 

Then - is it possible to calculate how long it would take for London to be completely destroyed?

The difficulties in such a calculation seem to be as follows:

 

A. As the bombardment goes on, there's an increasing probability that incoming bombs will land in the already destroyed area. Eg, when 50% of London has been destroyed, 50% of newly-arriving bombs will land in this destroyed area, and so do no more than "convulse the rubble".

 

B. There's an infinite degree of "overlap" between the impact-points of successive bombs. Ie, a new bomb might land right on the dead centre of a previous detonation, or 1 metre from it, or 5 metres from it, or 10 metres.... and so on.

 

I raised this question many years ago, on another science forum, and was told that because of factors A and B above, no exact mathematical calculation can be made. Granted that this is so, is it at least possible to make a kind of "statistical"prediction - on the lines of "after x number of years, there's a 99% probability that London will be 99% destroyed "?

 

This question has nagged at my mind ever since first reading the book (which was before the actual year 1984!) Any advice will be very much appreciated - thanks!

Edited February 17, 2014 by Dekan

[Bignose]

There probably isn't a nice closed functional form that would describe the situation, but a Monte Carlo simulation of the scenario could be run many 1000s of times and give a pretty decent idea of the average devastation v. time curve. Run it a bunch more and you can probably even build decent estimates of the confidence intervals, too.

[mathematic]

If you are looking for an average after n bombs, there is a straightforward approach.

Let x be the fraction of the area affected by one bomb. The first bomb destroys x, the second bomb destroys (on average) x(1-x), the nth bomb destroys (on average) x(1-x)^n-1. Add all this up to get 1-(1-x)^n destroyed (on average) after n bombs.