Maths Coursework – Artic Research
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Maths Coursework - Artic Research Speed of wind - 50 km/h blowing from the west Speed of aeroplane - 200 km/h I will treat the aircraft as a particle, instant and constant speed will be assumed, take of and landing times will be ignored. The aircrafts speed will be affected by the speed and direction on the wind. As we assume the aircraft speed, wind speed and direction will be constant; we can use vector diagrams to work out how long the journey will take. I will have my first base camp in the middle. I will then place the base camp at different areas in the circle, and see how this effects the journey time. I will then work out a general formula for any speed of aircraft, wind speed and direction, so the journey time could be worked for any variables of these 3. Circle 1 - radius = 50km. First observation site - north (0?). Resultant velocity will have to act from the base camp to the observation site. 50km h-1 Observation site 200km h-1 Resultant velocity ? Base camp Angle ? = Sine ? =50 / 200 ? =1/4 ? = Sine-1 (1/4) * =11.25? This angle 11.25? can be used to work out the bearing of the aircraft. Bearing = 360? - 11.25? = 348.75? So how long would the journey take? First we have to use Pythagoras' theorem to work out the speed of the aircraft.
The times would be variable because the weather is so unpredictable in the Arctic. The journey may only be, say 32 minutes, but the preparation time could be much longer. For a model of this simplicity, we don't take into account this extra time for take offs and landings; we just treat the aircraft as a particle, assuming instantaneous and constant speed. The speed of the wind is also unlikely to be constant in direction and speed. It is clear from the readings taken that a graph can be drawn that repeats itself after 180?. The time taken for 0? is the same as the time taken for 180?, and the total time for the site 90? is the same as the degree 180? later, 270?. I now need to derive a formula to calculate the time taken for any wind speed, aircraft speed and wind direction. W Observation site ? ?+90 ?-90 V R * ? Base camp R = resultant velocity W = wind speed V = aircraft speed R =? Using sin rule Sin A / a = sin B / b Sin (90+?) / V = sin ? / W Therefore, sin-1 (V sin 90 + ? / V) = ? Sin 90 + ? = cos ? Therefore, sin-1 (W cos? / V) = ? There are 180? in a triangle, therefore 180 = (90 + ?)
Therefore, e2 = d2 + r2 - 2 ? r ? d cos? This formula can be used to calculate the length of the resultant velocity from any given point. Conclusion - the formula to find time taken to travel to the observation site: R / V sin [[90 - ? - sin-1 (w cos ? / V)] / cos ?] To calculate the time taken to return to base camp; R / V sin [[90 - (? +180) - sin-1 (w cos (? + 180) / V)] / cos (? + 180)] Where R = radius of the circle, V = velocity of the aircraft, W = wind speed. So total time to and from base camp: [R / V sin [[90 - ? - sin-1 (w cos ? / V)] / cos ?]] + [R / V sin [[90 - (? +180) - sin-1 (w cos (? + 180) / V)] / cos (? + 180)] Formula to calculate the speed of the resultant vector; C = 2b cos ? ? ?4b2 cos2? - 4(b2 + a2) Where b = wind speed, a = aircraft speed, C = resultant velocity, ? = angle of resultant relative to the wind. Formula to calculate the length of the resultant velocity from any given point; e2 = d2 + r2 - 2 ? r ? d cos? Where d = distance from centre of circle to observation site, e = distance to observation site from base camp, r = distance to observation from centre of circle (the radius), ? = observation site bearing minus base bearing (? = ?2 - ?1).
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