MATHS COURSEWORK

“ARCTIC RESEARCH”

Introduction:

In my investigation, a group of researchers are to set up a series of observation sites at equal distances from each other in the Arctic on the circumference of a circle of radius 50 km. The aim of my investigation is to figure out how long the researchers should be prepared to travel for on each journey. I am also to find where in the circle that the base camp should be located in order to minimize these journey times as short as possible. Using some calculations I will determine all of this taking into account that there is a strong westerly wind present constantly at all times in the area. I found out, from research, that average wind velocities in the Arctic are from 25 – 35 km/h. I will use this range in my investigation. The bearing at which the planes must depart is affected by this westerly wind therefore, this angle must be calculated.

If this investigation was to be physically carried out there would be some factors that needs to be considered. In the arctic there are many mountains and obstacles a plane would have to take into account during flight. To take into account these factors in my calculations would be impractical therefore, I must establish variables that could be controlled and make some assumptions in order to make it practical.

These are the variables that I must establish;

- Strength of the westerly wind.
- Speed that the researcher’s plane would be travelling at. I will investigate into how both of these affect the overall flight time by varying the velocities for both the plane and the wind.

These are the assumptions that I must establish;

- I will be treating the people in the plane and the plane itself as particles. This will mean that they have no weight at all and therefore, this will not affect the flight of the plane. On the other hand, if it was considered the flight of the plane would be affected.
- The flight velocity of the plane is constant and the speed at which the plane travels will be achieved instantaneously. This will prevent complications concerned with takeoff and landing times.
- Wind velocity is also constant and always blowing in the exact, same direction of the west, as the flight times for a constantly changing wind velocity would be almost impossible to calculate.
- Mountains and obstacles in the arctic that a plane would have to take into account during flight causes complications that are unnecessary for this investigation, so I will assume that the path of the flight is perfectly straight.
- The plane will fly parallel to the ground, therefore, keeping the distance that the plane travels equal to the horizontal distance from base camp to observation site.
- It takes no time for the plane to take of or land.
- Time is flowing forward at a constant rate.
- There is no air resistance or friction except for the wind. In real life there are other forces that could affect the flight of the plane, however, to make this investigation practical I will not be considering any of these.
- The height at which the plane will be during flight will not be considered.

Modelling:

To make the modelling of my investigation clear, I will refer to my observation sites around the circumference of my circle, using letters in the alphabet. This will range from A – H as there are going to be 8 observation sites. I will start with my base camp in the middle. This is a good position to start as all the observation sites are at equal distances of 50 km (which is the radius) from the base camp. It will also enable me see how the journey times are affected in this position taking into consideration the westerly wind, and allow me alter the position of the base camp afterwards in order to find the best location within the circle where the journey times to all 8 of the observation sites are limited.