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## The Cambridge Positioning System

A cache by mjouk Message this owner
Hidden : 02/26/2014
Difficulty:
Terrain:

Size:  (small)

#### Watch

How Geocaching Works

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### Geocache Description:

A puzzle inspired by the mathematics used in the GPS.

## Caveat Cacher

The coordinates above are convenient for parking, but there's nothing to see there, except a handy car park.

## Introduction

The GPS is almost miraculous. It allows us to work out where we are by measuring the time it takes for a radio signal to travel from a network of satellites to our receiver. Signals from nearer satellites will arrive more quickly, so if we know where the satellites are we can work out where we are. As a bonus, we can also work out what the time is.

Happily our GPSr does all the maths for us, but to find this cache, you'll have to do a similar calculation yourself.

## The Puzzle

Rather than satellites orbiting the Earth, imagine a cyclist travelling east along an East-West path. When he starts cycling he briefly rings his (very loud) bell; when he reaches the middle of his he rings it again; when he gets to the end he rings it a final time.

Our cyclist takes three minutes to cover the 1.2km track and so rings:

• At 11:58:30.00 when he's at TL 38839 53670;

• at noon when he's at TL 39439 53670;

• at 12:01:30.00 when he's at TL 40039 53670.

These coordinates are close to the One-Mile Radio Telescope at Lord's Bridge whose track is almost perfectly East-West (to within 1cm in 1 mile).

If you were sitting on the cache armed only with a stopwatch you could measure the intervals between the rings: 89.20s and 89.40s respectively.

Your task is to work out where you are, and exactly when you heard the middle ring.

You should:

• Do all your calculations using OSGB36 National Grid coordinates.

• Assume that the speed of sound is 300ms-1.

• Assume that the cache is north of the track.

The puzzle can be solved with nothing more than pen and paper, but you'll probably find a calculator makes it easier.

## Ordnance Survey (OSGB) Coordinates

The Earth's surface is curved which makes it difficult to do calculations. Things are even more complicated if we specify our position with angles of latitude and longitude. Happily the Ordnance Survey invented a set of coordinates which make things easier.

If you use six-figure references for the Easting and Northing, each unit corresponds to 1m on the ground. It might be useful to know that TL eeeee nnnnn corresponds to 5eeeee 2nnnnn.

You can use the normal Cartesian formula to calculate (approximate) distances.

This approximation isn't perfect: for example it means that a line of constant Northings isn't a line of constant latitude. Our cyclist's track is the former; the One-Mile's track the latter.

Converting between OSGB coordinates, and latitude and longitude is fiddly and some online converters give different answers. To avoid such problems I suggest you use the OS's own converter.

## Acknowledgements

Thank you to Cambridge Past, Present & Future for their help in setting up this cache.

## Congratulations!

...to crb11 for being first to find it.

Frr gur trbpurpxre.

Decryption Key

A|B|C|D|E|F|G|H|I|J|K|L|M
-------------------------
N|O|P|Q|R|S|T|U|V|W|X|Y|Z

(letter above equals below, and vice versa)