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# Maths Coursework &amp;#150; Artic Research

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Introduction

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

θ

Base camp

Angle θ= Sine θ=50 / 200

θ        =1/4

θ= 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.  Pythagoras’ theorem is “the sum of the squares of the opposite and adjacent = hypotenuse” or a2 + b2  = x2

2002 = V2 + 502

2002 – 502 = v2

37500 = v2

v = 37500

Resultant speed of aircraft = 193.6km h-1 (3s.f)

Middle

Wind speed 50 kmh-1

Aircraft speed = resultant velocity = 200 – 50 = 150 kmh-1

T = 50 / 150

T = 0.333 hours

T = 20 minutes

Total time taken to travel to observation site and return  = 20 minutes + 12 minutes = 32 minutes

The time taken to reach the observation site west (270°) of the base camp will be the same as the time taken to reach the observation site east of the base camp. This is because the vector diagram travelling from the base camp to the east observation site will be the same the vector diagram from the west observation site to the base camp, and the vector diagram from the base camp site to the east observation site will be the same as the vector diagram travelling from the west observation site to the base camp. So total time taken to reach went observation site and return to base camp = 32 minutes.

The results taken would not be very realistic, as the journey times are quite short take off and landing times will be significant. In a place like the arctic take off and landing times would be very significant, as it would take variable and mostly long periods of time to get the aircraft prepared. 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.

Conclusion

Base camp: measured in bearing from the centre of the circle, with the distance in km.

Base camp

e

Observation site

d

r

θ1

θ

θ2

Centre of circle

Where d = distance, e = distance to observation site from base camp, r = distance to observation from centre of circle (the radius), θ = observation site bearing minus base bearing (θ = θ2 - θ1).

Using cosine rule – a2 = b2 + c2 – 2bc cos A, where a = e, b = d, c = r, A =θ.

Therefore, e2 = d2 + r2 – 2 × r × d cosθ

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 - θ - sin-1 (w cos θ / V)] / cos θ]

R / V sin [[90 – (θ +180) - sin-1 (w cos (θ + 180) / V)] / cos (θ + 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 - θ - sin-1 (w cos θ / V)] / cos θ]] + [R / V sin [[90 – (θ +180) - sin-1 (w cos (θ + 180) / V)] / cos (θ + 180)]

Formula to calculate the speed of the resultant vector;

C = 2b cos θ±4b2 cos2θ - 4(b2 + a2)

Where b = wind speed, a = aircraft speed, C = resultant velocity, θ = 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 cosθ

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), θ = observation site bearing minus base bearing (θ = θ2 - θ1).

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