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

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.

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.

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.

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 360o 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.

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

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

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 0o W and the International Date Line 180o W form a great circle.

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

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 0o. The latitude of the North pole is 90o N. The latitude of the South Pole is 90o S. All other latitudes lie between these two angles.

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 41o N N S 41o R r N r R

Introduction to Distances on the Earth Equator Latitude 41o N N S 41o R r Calculate the distance along the line of latitude from (41o N, 36o W) to (41o N, 155o E) . Smallest angle is not 191o, but 169o. 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 41o. Distance in nautical miles = 10140 x cos 41o = 7653 NM 0o N 36o W 155o 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.

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

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_0BcOSiQfC6eWzBw&sqi=2&ved=0CDwQ6AEwAQ#v=onepage&q=Earth%20as%20a%20Sphere%20mathematics&f=false