Math IA - curve of a flight math PDF

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Math IA - curve of a flight math...

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Introduction:  Since my dad was an expatriate when I was young, we would have to fly long-haul internationalflights frequently, and everytimebefore takeoff I would stare at the map and wonder why the planes fly in the route they do and specifically why the paths are curved and not straight.Therefore my aim for this exploration is to find the distance and time of the shortest flight path from NewYork to Manila in addition to investigating whether it is better to fly East or West from New York. By learning about spherical trigonometry and great circles on my own since it is beyond this course, I will use different formulas and spherical triangles to find the distance between two destinations.  From this new found information, since I may want to go to college in the East Coast, this could impact which flight I would pick, whether I would fly over the West Coast  and back over the Pacific Ocean, or fly over the Atlantic Ocean and Europe. In order  to investigate the fastest and shortest flight distance from New York to Manila, I used two  different methods. Firstly, I used the latitudes and longitudes of these two cities to find the central angle formed between these two cities if drawn to the core of the Earth. From this, the  length of the great circle path the flight takes can be determined and the approximate time as well. The second method I will use is by drawing spherical triangles on the lenart sphere. By constructing a triangle that goes through the two cities, the angles of the triangle can be found. Through the spherical law of cosines, these angles can beused in order to determine the angle formedagain at the core of the Earth. The second aim I will be investigating using the first method mentioned above, is which flight path from NewYork to Manila that exists in reality has the fastest travel time. Since there are no direct flights to Manila, I will be investigating one with a stopover in Vancouver and one with a stopover in Hong Kong. Since these two are different distances, this could possibly affect the flight times as one of the stopovers is approximately in the middle of the journey from New Yorkto Manilawhereas the other stopover is significantly closer to Manila. This could possibly affect the outcome as the Philippine Airlines flight has two legs more similar in length while the Cathay Pacific flight has one long leg and one short leg.  Through  this exploration, I hope to learn about flight paths and the great circles that they fly in, in addition to being able to determine whether it is better to fly heading West, or fly heading East.     1

Optimum Flight Path Method 1:  In order to find the optimum flight path from New York to Manila, spherical geometry needs to be used. Spherical geometry is commonly used for navigation since spherical geometry is thegeometry of two-dimensional surfaces on a sphere. Therefore, in order to findtheoptimum flight path, the great circle connecting these two points needs to be found since flying along a great circle is the shortest distance between two points on a sphere.  During  this  exploration, I learned about great circles and spherical geometry on my own since this was necessary for what I wanted to do but is not taught in the course. I read about spherical geometry on various websites in addition to using the Lenárt Sphere to teach myself about this new type of trigonometry. The Lenárt Sphere is a tool used to learn spherical geometry through hands on experimenting with spherical surfaces. With the  Lenárt  sphere  you  can  accurately measure and draw shapes on a spherical surface using it for different applications of spherical geometry. The Lenárt Sphere is used below in my second method for finding the optimum flight path. For this first method, by finding the central angle at the core of the Earth between two points, the arc length of the flight path on the great circle can be found through the formula shown below using the longitudes and latitudes of the different cities.  Figure 1: Diagram Showing Center Angle

 2

I  constructed  the  diagram  on the previous page on Google Earth to help visualize how I am going to find this flightpath.Ilearnedhow to use Google Earth and how to plot points accurately in order toobtainanaccurate representation of the places the flight path passes through. By using the path tool, I connected the two coordinates of New York and Manila  to  show the great circle path a pilot would take to fly between these two points. Then  by picking a third point and connecting paths between this third point a spherical triangle can be formed. From this, I learned that the optimum flight path from New York to ManilawouldpassthroughtheArctic and eastern Russia which is contrary to what one might think by looking at a map. By finding the center angle in the middle of the Earth, Angle O as shown in the diagram above, since arc length AB is the arc across from this angle,  the  formula that relates the arc length of a circle to the center angle can be used. The diagram below shows how a pyramid is constructed and the angle in the center of the Earth is used to find the arc length or great circle distance.  Figure 2: Diagram Showing Center Angle in a Sphere

1



  I will be using the equation below to find the angle made in the center of the Earth between New York andManilaif a spherical triangle is drawn between the three points as shown in Figure 1. The equation is derived from the haversine formula which is an important  equation used innavigation, giving great circle distances between two points on a sphere using their longitudes and latitudes. By substituting the latitudes and longitudes as shown below into the equation, the center angle can be found to later on be used to find the distance on a great circle between New York and Manila.   1

Spherical Triangle. Digital image. Spherical Triangle. WolframMathWorld, n.d. Web. 8 Apr. 2016.

3

When attempting to usethisequation, I wondered how this equation came about so I decided to attempt to derive it. In order to derive this equation, we first start with the Haversine Formula where for any two points on a sphere the central angle between them is given through:  2 hav ( rd ) = hav (ϕ2 − ϕ1 ) + cos (ϕ1 ) cos (ϕ2 ) hav (λ2 − λ1 )  

Where hav is the haversine function and θ = rd 

hav (Θ) = sin2 ( Θ2 ) = 1−cos(θ) 2

sin2 ( Θ2 ) =

1−cos(θ) 2



due to the trigonometric identites of double angles.

On the left side of the equation,

( dr ) is the central angle in the Earth, the angle

necessary to  find to calculate the shortest flight path. In order to derive the equation for the central angle, we must solve for

( dr ) .



hav (Θ) = hav ( rd ) = sin2 ( Θ2 )  hav ( rd ) = h  h = sin2 ( Θ2 )  

Since we want Θ alone, we must take the square root of both sides to get: 

√h = sin( Θ2 )  arcsin√h =

Θ 2





Since Θ =

d r

, you can substitute this back into the equation to get:



d  arsin√h = 2r 2arcsin√h = dr  

Next, since hav ( d ) = h , this means that r 

h = hav (ϕ2 − ϕ1 ) + cos (ϕ1 ) cos (ϕ2 ) hav (λ2 − λ1 ) , and since

hav (Θ) = sin2 ( Θ2 ) , this means that  h = sin2

(

ϕ 2−ϕ 1 2

) + cos (ϕ ) cos (ϕ ) sin ( )  1

2

2

λ2 −λ1 2



Lastly, h can be substituted from 2arcsin√h = dr , to give the equation  2

"Haversine Formula." Wikipedia  . Wikimedia Foundation, n.d. Web. 26 Apr. 2016.

4

d r

= 2arcsin



√

sin 2 ( ϕ2−2 ϕ1 ) + c osϕ1cosϕ2sin 2 (λ2−λ1 ) 2

I Since this formula was foreign to me, I needed to find an online source that was able to guide me in the right direction for deriving this formula. However, this online source was  only a  guide  as  it did not show the full steps to deriving the formula for the central angle and I was able to derive this independently by learning more about the Haversine Formula. Through this, I was able to better understand where the formula came from and understand it well, giving me a greater depth of knowledge and understanding for my exploration.  Below, the formula that I derived is used to find the central angle needed to find the fastest flying distance between New York and Manila.  ϕ 1 = l atitude of 1st place in radians, λ1 = l ongitude of 1st place in radians  ϕ 2 = l atitude of 2nd place in radians, λ2 = l ongitude of 2nd place in radians   d r

= 2arcsin

√

sin 2 ( ϕ2−2 ϕ1 ) + c osϕ1cosϕ2sin 2 (λ2−λ1 ) 3 2

 New York: ϕ 1 = 40° 43′N , λ1 = 73° 45′W 4 Manila: ϕ 2 = 14° 35′N , λ2 = 120° 59′E 5   Since  the  angle needs to be expressed in radians in order to use the equation that relates  the arc length of a circle to the center angle, the longitudes and latitudes of each city must be converted into radians as well. The coordinates are written with the first two digits  as  the  degrees and the second two digits as minutes out of 60. Therefore for example the latitude of NewYork would be40 ° and43/60minutes. A sample calculation of converting these coordinates into radians is shown on the next page.

ϕ1 =

40+(43/60) 180

ϕ1 =

40+(0.716...) 180

×π ×π

3

Bourne, Murray. "3-D Earth Geometry." Intmathcom  RSS. N.p., n.d. Web. 27 Feb. 2016. "Latitude and Longitude Facts." Latitude And Longitude. N.p., n.d. Web. 27 Feb. 2016. 5 "Location of Philippines." Philippines  Latitude, Longitude, Absolute and Relative Locations. N.p., n.d. Web. 27 Feb. 2016.

4

5

ϕ1 =

40.716 180

×π

ϕ1 = 0.7106...  The same process of calculations was usedtofindthe other latitude and longitudes in radians of New York and Manila.  New York: ϕ1 = 0.7106..., λ1 = 1.2871...  Manila: ϕ2 = 0.2545..., λ2 = 2.1115...   For the center angle equation, the difference between the two latitudes and the two longitudes is necessary, this equation is shown below. 

ϕ2 − ϕ1 = 0.2545... − 0.7106... = − 0.6047...  λ2 − λ1 = 2.1115... − 1.2871... = 0.8243...  By using these different latitudes and longitudes and differences between the two, these valuescan be substituted into the center angle equation to find the angle in radians the two cities make. If x = central angle then the domain is 0 < x ≤ 180. This is because if you want to find the shortest flight path, you need the shortest arc length of the great circle and therefore it needs to be the smaller angle out of the two. Since on a circle there will always be two angles to get from one point to the other, x and (360-x), the smaller angle less than 180 will always be the one needed if you want to find the shortest distance. 

2arcsin

√

2arcsin

√sin

sin 2 2

(

ϕ2−ϕ1 2

) + cosϕ1cosϕ2sin

2

 ( λ2−λ1 2 )

( −0.456... ) ) + cos (0.7106...) cos (0.2545) sin 2 ( 0.8243... 2 2

= 0.942117161   With this center angle, the formula that relates arc length of a circle to the center angle in radians can be used. The approximate radius of the Earth is 6371 km6 therefore this is the radius that will be used in the equation. By finding this arc length, this is the same as finding the distance on a great circle the plane will travel from one destination to another. 6

"Earth Radius." Wikipedia  . Wikimedia Foundation, n.d. Web. 04 Apr. 2016.

6

 S = Let the arc length between the two cities 

S = rΘ  S = 6371 × 2.1994...  S = 14012.85836  This  distance  found is in km but in order to find the approximate time it takes to travel this flight path, the distance needs to be converted to nautical miles since this is the unit of measurement used for travel.  1 Nautical Mile (NM) = 0.539957 Kilometers

N M = 14012.8583 × 0.539957  N M = 7566.340961  With the distance in nautical miles of the flight path known, the approximate time in hours of the flight can be found. Here, the NM are divided by 500 because this is approximately how many nautical miles an airplane flies in an hour.7 

T ime = Distance (NM ) ÷ 500  T ime = 7566.3409... ÷ 500  T ime = 15.1326819 hours   Therefore, with this first method, it should take approximately 15.1 hours to fly fromNewYorktoManilaon a Great Circle path. This procedure is accurate to a moderate extent since it does not take into account other external forces such as winds and jet streams. In addition I think it is accurate to a moderate extent until the time is determined because it is very difficult to estimate how many nautical miles a plane flies per hour since it varies with each plane, flight path, and vectors. The result is a bit faster than I had expected but this is most likely due to the flight time not taking into account the wind and jet streams as mentioned above.  Optimum Flight Path Method 2: 

7

"Convert Knots to Miles Per Hour." Knots  to Miles Per Hour. N.p., n.d. Web. 25 Apr. 2016.

7

The second method I used to calculate the optimum flight path from New York to Manila was by using the Lenárt Sphere. I taught myself how to use the Lenárt Sphere from  the booklet provided, learning how to plot accurate coordinates and drawing great circles  using  the tools provided. Additionally, in order to use the Lenárt Sphere to calculate the optimum flight path, I had to teach myself spherical trigonometry and how it works.Spherical trigonometrywouldbe used in this situation, since by drawing spherical triangles, the distances between the two points can be found using trigonometry. These curved  distances  are on great paths therefore being the shortest distance between two points. Below is a picture of the Lenárt Sphere apparatus used.  Figure 3: Various Equipment for the Lenárt Sphere

  First, I plotted the coordinates of New York and Manila on the sphere using the protractor and compass provided. Once these coordinates were plotted, I used the spherical ruler to draw a great circle through these points. Next, in order to create a spherical triangle with lengths part of great circles, I picked a third point on the sphere and  drew  great  circles through those points as well. From this constructed spherical triangle, I found the angles using the protractor. With the three angles of the spherical triangle, it is possible to find the center angle again at the center of the earth by using the spherical AAA law of cosines. This center angle can then be used to find the arc length or distance from one point to another.  Figure 4: Spherical Triangle on the Lenárt Sphere

8

  Let c = the center angle in the Earth between the two cities 

( c = arccos ( c = arccos

cos(c) + cos(b)cos(a) sin(a)sin(b)

)  ) 8

cos(130) + cos(34)cos(63) sin(63)sin(34)

c = 122.3237154  In order  to  use  the formula that relates arc length of a circle to the center angle, the  angle  needs  to  be changed from degrees to radians since radians is used to calculate arc length. 

Radians =

Degrees 180

×π

Radians =

122.32... 180

×π

Radians = 2.13495188   From this information, the arc length can be found from New York to Manila. 

S = rΘ  S = 6371 × 2.13...  8

"Three Angles Given (Spherical AAA)." Wikipedia  . Wikimedia Foundation, n.d. Web. 27 Feb. 2016.

9

S = 13601.77656 km  Since S uses the radian of the Earth and the angle between the two cities, and because the angle between the two cities has to be less than or equal to π , then the domain for S is 0 < S ≤ 20015.0868   Again,  using  the same formula above, the approximate time it takes to travel this distance can be used after converting kilometers to nautical miles. This distance in nautical  miles  is  7,344.37  nautical miles. Using this formula we can approximate that it takes about 14.69 hours to travel from New York to Manila.  For both of these optimum flight paths, from New York to Manila, the spherical triangles showthat you should travel West over the Arctic and eastern Russia. Although the Philippines is south of New York and it does not seem to make sense to fly North as the shortest flight path, flying over the Arctic is the shortest flight path since New York, Manila, and apointintheArctic are all collinear locations in spherical geometry and lie on the same great circle thereforemakingthis theshortest flight path since great circle paths are the fastest. Through both of these methods, the flying times differ by less than an hour. Considering this, I believe that these methods are rather accurate. I think that the reason  these  values  differed slightly is solely due to human error as using the lenárt sphere  is  not 100% accurate since the points I plotted are not as precise as the latitudes and longitudes although I tried to make it as precise as possible. Therefore, due to this inaccuracy the angle measurements would differ affecting the solution as the equation used relies greatly on the angles formed.

 Real Flight Path Flying West:  In order to investigate whether it is better to fly East or West, I calculated the distances of two real flight paths to see which one is shorter. The first flight path I used is Philippine Airlines with a stopo...