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FMSP Year 10 Team Mathematics Competition 2012 Introduction to Distances on the Earth

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Introduction to Distances on the Earth The main ideas are: Great circles Small circles Position on the Earth (Latitude/Longitude) Nautical miles Shortest distance between two points on a great circle Distance between two points on a small circle Before you continue, download the template and instructions to make a model of the Earth.model of the Earth

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Introduction to Distances on the Earth The shape of the Earth closely resembles a flattened sphere (a spheroid) with an equatorial radius of 6,378 km, whilst the distance from the centre of the spheroid to each pole is 6357 km. We are going to model the earth as a perfect sphere.

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Introduction to Distances on the Earth So the equator and lines of longitude all lie on great circles. These are circles with a radius equal to the radius of the Earth (approx. 6400 km). This also means that the plane of the circle intersects with the centre of the Earth.

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Introduction to Distances on the Earth A nautical mile (NM) is defined as the distance travelled on a great circle when moving through 1 minute of arc. There are 360 o in a circle and 60 minutes in a degree. So there are 360 x 60 = 21600' in a circle. So the distance around the equator is 21600 NM.

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Introduction to Distances on the Earth So to calculate the shortest distance from 60 o W to 10 o E along the equator (a great circle): Smallest angle along the equator between the two positions is 60 + 10 =70 o Shortest distance = 70 x 60 = 4200 NM Remember to convert degrees into minutes.

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Introduction to Distances on the Earth Calculate the shortest distance between (0 o N, 42 o 15' W) and (0 o N, 127 o 32’ W) Smallest angle along the equator between the two positions is 85 o 17'. Shortest distance = 85 x 60 + 17 = 5117 NM Remember to convert degrees into minutes.

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Introduction to Distances on the Earth All circles which pass through the North and South poles are great circles. Lines of longitude lie on great circles. Directly opposite meridians form great circles. So the Greenwich (Prime) meridian 0 o W and the International Date Line 180 o W form a great circle.

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Introduction to Distances on the Earth The longitudes 90 o W and 90 o E form a great circle as they are opposite each other. Calculate the shortest distance between (70 o N, 90 o W) and (50 o N, 90 o E): Smallest angle along the great circle is 20 + 40 = 60 o. Shortest distance = 50 x 60 = 3000 NM. Smallest angle passes over the North Pole

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Introduction to Distances on the Earth Small circles are parallel to the equator. Lines of latitude are small circles. The latitude of the equator is 0 o. The latitude of the North pole is 90 o N. The latitude of the South Pole is 90 o S. All other latitudes lie between these two angles.

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Introduction to Distances on the Earth As you travel around a line of latitude (small circle) the distance travelled is shorter than the distance covered on the Equator (great circle) for a given angle. Radius of the small circle (r) Radius of the Equator (R) Latitude angle Then r = R cos So the small circle smaller than the great circle by a factor of cos . Equator Latitude 41 o N N S 41 o R r N r R

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Introduction to Distances on the Earth Calculate the distance along the line of latitude from (41 o N, 36 o W) to (41 o N, 155 o E). Smallest angle is not 191 o, but 169 o. Converting this angle into minutes: Angle = 169 x 60 = 10140'. Since the measurement is along a small circle the distance is reduced by a factor of cos 41 o. Distance in nautical miles = 10140 x cos 41 o = 7653 NM Equator Latitude 41 o N N S 41 o R r 0o0o N 36 o W 155 o E This is not the shortest distance! Remember: The shortest distance is on a great circle passing through these two points, centred at the Earth’s centre.

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Introduction to Distances on the Earth From (20 o S, 15 o E) I travel West 3000 NM. What is my new position? Since we are moving along a small circle, the angle will be larger than 3000'. Angle = 3000 ÷ cos 20 o = 3193' = 53 o 13' Moving West from our original position will take us past the Greenwich meridian to a westerly point at an angle of (53 o 13' - 15 o ). New position is (20 o S, 38 o 13' W) Use the scale factor to convert from the small circle to the great circle, as if we are moving along the Equator. Latitude 20 o S Equator N S 0o0o ? o W 15 o E

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Introduction to Distances on the Earth For further reading and questions, you can copy and paste this link: http://books.google.co.uk/books?id=vk0Nt1cN_9MC&pg=PA146&dq=Earth+as+a+Sphere+mathematics&hl=en&sa=X&ei=k8rrTu_0B cOSiQfC6eWzBw&sqi=2&ved=0CDwQ6AEwAQ#v=onepage&q=Earth%20as%20a%20Sphere%20mathematics&f=false

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