Features and Location of Places The Equator, Great Circle, Small Circles, Meridian, Latitudes and Longitudes

Definition of latitude and longitude

The Earth is not a perfect sphere, as it is slightly flatter at the north and south poles than at the The position of any point on carth is located by circles round the carth, as follows: The earth rotates about its axis, which stretches from the north tothe South Pole Earth which go through the poles are circles of Longitude or meridians.

Consider the following diagram x Sane of longitude iene ge [ as le of latitud <\__ circle of latitude t ae pee eee XN /

SA. U4

is ua Normally Latitude is defined relative to the equator, which is the circle of latitude round the middle of the Earth while Longitude is defined relative to the circle of longitude which passes through Greenwich in London (Greenwich meridian), ‘The latitude of a position tells us how far north or south of the equator itis while the longitude of a position shows us how far east or west ofthe Greenwich meridian iti.

Latitude; If we draw a fine from the centre of the Barth to any position P, then the angle betwaen this line and the plane ofthe equator is the latitude of P

A Ne

/ SA { _ Latitude be — b——] equator“ if \ vi Ne Longitude: This is the angle between the plane through the circle of any Longitude P and the plane of the Greenwich meridian

Ff \y GiBgnwich

( F hs {<‘erfaaat i) a if Latitude can be either North or South of the equator while Longitude can be either East or West of Greenwich as When locating the latitude and longitude of place we write the latitude First then longitude

Example 1

Dar ¢s Salaam has latitude 72S (i.e. 7° south of the equator) and longitude 39°E (ec. 39° east of the Greenwich meridian), So Dar es Salaam is at (7°S, 39°E). NB; Greenwich itself has latitude S1°N ( 1.¢. 51 north of the equator)and longitude 0° (by

definition), Johannesburg has latitude 26°S (ie. 26 south of the equator) and longitude 28°E(i.e

28%east of the Greenwich meridian), therefore Johannesburg is at (26°S, 28°F) The north pole has latitude 90°S but its longitude is not defined. ( Every circle of longitude goes through the north pole):The south pole has latitude 90°s. Its longitude is not defined. So all points on the ‘equator (such as Nanyuki in Kenya) have latitude 0° Ranges; Latitude varies between 90°S (atthe south pole) to 90°N (al the north pole)

Ranges; Latitude varies between 90°S (at the south pole} to 90°N (at the north pole) Longitude varies between 180°F and 180°W. These are the longitudes on the opposite side of the Earth from Greenwich.

GREAT AND SMALL CIRCLES: There is an essential difference between latitude and longitude. Circles of longitude all have equal circumference. Circles of latitude get smaller as they approach the poles. The centre of a circle of longitude is at the centre of the earth. They are called great circles. For circles of latitude, only the equator itself is a great circle. Circles of latitude are called small circles,

Example 2

Find the latitudes and longitudes of A and B on the diagram below; es / cf rae / if / \ \ i le eae Kor} _) Se le t iss a i Y ea Oe PZ

Solution;

So the point A is at (60°N, 20°R) The point B is 10° below the equator, and on the Greenwich meridian, So the point B is at (10°S, 0°). 30°E sd ls L >\20°N \ [ys Te

• Copy the diagram show on the figure above and mark these points:

ww a (10PN, 30°F) > Q0°N, 20°~W) © (0, 20°W)

• Obiain a globe, and on it identify the following places

a (40°S, 30°E) b. 50°S,20°~W) © (10°N,40°W)

d ADN, 30°F)

e — (R0°N,10°F)

Difference between angles of latitude or longitude

Suppose two places have the same longitude but different latitudes. Then they are north and south of each other In finding the difference between the latitudes take account of whether they are on the same side of the equator or not.

• Ifboth points are south of the equator subtract the latitudes
• Ifboth points are the north ofthe equator subtract the latitudes
• fone point is south of the equator and the other north then add the latitudes

Similarly, suppose two places have the same latitudes but different longitudes:

• If both points are east of Greenwich subtract the Longitudes
• Ifboth points are west of Greenwich subtract the Longitudes
• fone point is east of Greenwich and the other west then add the longitudes

Suppose places 4 and Bare on the same longitude, then the difference in latitude is thoangle subtended by 4B at the centre of the earth, FTN f ' \B tea \ Lt “a Sele Then the difference in longitude is the angle subtended by AB on the earth's axis, [ST Sar / \ pf Locating a Place on the Earth’s Surface Locate a place on the Earth's surface sass Tes plcss on longitude 3°E ae Alexandvia (in Egypt at (31°N, 30°), Kigal (ia Rwanda) at

(2°5, 30) and Pitermaitabur in Sout Aiea) at (3075, 30°F) Find the eiffrance olan Between 2 Kigav andPitemaritaurg it

Solution

Therefore the difference is 28” The difference is 33° A plane starts at Chilcka airport (in Malawi) which is at (16°S, 35°E). It flies west for 50°. What

Solution

• In the diagram shown in the following figure find,

a The difference m longitude between A and B tow Ae cow OY or, ae

BA- ASS

[fp i Y

Eel =~ he Se Vitth stad K < \ Eas

• Find the difference in atinde between the fllowiog pars of places
• Iringa (8°S, 36°E) and Gendi (4°S, 36°E)
• Zanzibar ((6°S, 39°E) and Chiungutwa (11°S.39°E)
• Find the difference in longitude between the following pairs of places

a. Ibadan (Nigena), (7°N.4°E) and Makurd (Nigeria), (8°S,36°E) ‘'b. Ibadan and Kumasi (Ghana) (7°N, 2°W)

• The following is a list of places. Find pairs of places that have the same latitude or the

sama login: Fo each pret the sass lnginde, fd he tren a nde

A (30°S, 20°E) B(0°S, 50°E) C.(20°S, 40°W),

D (20°N, 40°W) E(O°N, SO°W) F (100°S, 50°E)

GQG0S, 40°) HQON. 10%) T@0"N, 40)

J (LON, 30°F) KGO°N, 40°W) L(10°S, 10°F)

• plane starts at (20°N, 27°) and Ges south for 42°, What nit new Inna and

engin?

• A plane starts at (51°N, 31°E) and flies west for 45°. What 1s its new latitude and

engine?

Distances along Great Circles

Distances along Great Circles

Cieaie dlances along great reer ‘ake paces de YW sme Io toni oe law in a thr we ki Suppose X dua north of, When aveing’noeh sea Y 1°X, you travel lon pat ofa cicle of longitude that is you travel along an arc of the circle.

1st far. \ f aie 7) Recall the formula forthe length of are A Ri ae ‘ < i Cy) A x R aor fan are subtends @ atthe Centre of the circle of radius R, then

aX ago " ee a ye r A € jo Ze, pa Tn the figure above R is the radius of the Earth and @ ib the difference between the latitudes of X and Y. Taking Rto be 6,400 km, the formula becomes

NB: Remember, to find the difference in latitudes, take account of whether the places are north or south of the equator. Ifthey are all found in south or north, then subtract the latitudes. If one is south and the other North then add the latitudes.

Nautical miles Distances are also measured in nautical miles. One nautical mile (nm) corresponds to ene minute of latitude. Let the difference in latitudes between two places bed The aumber of minutes in 8 is 606, since 1 degree has 60 minutes.

Hence the distance betweea the places, along the arc of longitude, is 608 am.

Lknotis speed of 1 nautical mile per hour. Navigation Related Problems. Solve navigation related problems

Example S

Find the distance between Alexandria (31°N, 30°E) and Kigali (2°S, 30°E)

Solution

‘Note that both places are on the same longitude The difference in lasitude is 33° ‘Use the formula Therefore the distance is 3,690 km.

This isthe distance in nautical miles. The distance is 1,980 am

Example 6

A plane starts at (20°S, 30°E), and flies north for 4000 km. Find its new latitude and longitude. ‘The plane flies north, hence its longitude is unchanged. The plane starts south of the equator, and ‘lying north. It may ccoss the equator, and so end up north of the equator. In this cate the latitude south of the equator willbe negative ‘Suppose the plane has flown along x° of latitude. Then using the formula for arc lenath

yn 10008860 gc g ak

A negative latitude south is equivalent toe latitude north, Hence the new latitude is 15.8°.N therefore rounding the answer tothe nearest degree the new position is (16°N, 30°F)

Example 7

A plane ilies north from (10°S, 30°E) to (27°N, 30°E) taking atime of 3 hours Find its speed, giving your answer both knots and kilometers per hour. Sotution ‘The plane is flying along a line of longitude.

‘The sumber of minutes i9 60 x 372.220", so st has flown 2.220 nautical miles. ‘To find the speed in knots, divide 2.220 nautical miles by 3, the speed is 740 knots. Recall that 1 nautical mile 15 1.862 km. So 1 knot (1 nm per hour) is equal to 1.862 km/hr.

Multiplying 740 by 1.862 gives the speed, so the speed is 1,378 km/hr Consider the following Questions, 1 Find the distance in kms between the following places. (@) Bangalore (13°N, 78°E) and Agra (27°N, 78°E)

• Asmara (15°N, 39°) and Mombasa (4°S, 39°E)

2, A plane starts as ((30°N, 40°W) and flies 300 kkm north. What is its new position?

• A ship stars at (0°N, 150°W) and sails south for 4.700 km_ what is its new position?
• A plane starts at (35°N, 90°W) and flies south for 5,590 km. What is its new position?
• A plane flies at 600km/h. How long does it take to go from (32°N, 51°E) to (5°S,51°E)

6 A ship sails at 20km/h. How long does it take to go from (31°N. 150°W) to @2°S,150°W)?

• Find the distances in nautical miles between the places in question 1
• A plane starts at (10°N, 18°E) and flies north for 1,200 nm. Find its new position
• A ship sails south at 10.4 knots. if it starts at (15°N, 150°W), find its position after 40 bas.

10, A Plane flies south from (12°N, 36°B) to (7°S, 36°E), taking 2¥ hours. Find its speed giving your answer in both knots and kms per hr

• A ship sailed north fiom (45°S, 28°W) to (39°S, 28°W). taking 30brs, What was its

speed? Give your answer in both knots and kms per he

Distances along Small Circles

Distance along Small Circles

Cateuate distance along small circles Suppose P and Q are places west or east of each other, Le they lie on the same circle of latitude. Then when you travel due east or west from P to Q you travel along an arc of the circle of latitude, The situation here is slightly different from that of the previous section. While circles of longitude all have the same length, cireles of latitude get smaller as they get nearer the poles.

Consider the circle of latitude $0°S. Let its radius be rkm. = aes / pesssceal 2 sor ] \ ‘pe / ‘Then taking a point P on the curcle, and letting R be radius of the earth:

In general, the circle oFlatirude a has radius Reos ct Thie is true for latimades both north and south of the equator. Now the distance along a circle of latitude can be found Suppose the difference between longitudes is @ Then the length of arc going from P to Qis

Nautical miles Generally if wo points on the same laimde cr have difference in longitude, then in nautical miles the distance along the circle of latitude is 608¢0s « nm

Example 8

Find the distance in km and nm along a eiscle of latitude between (20°N, 30°E) and (20°N, 40eW)

Solution

Both places are on latitude 20°N. The difference in longitude is 70°. Use the formula for distance.

The distance is 3,950 mm. Because it sails due west, the latitude remains unchanged. Suppose it has sailed through x°

kms per hr. ‘The difference in longitude between the two points is 8° Hence the distance, in nautical The speed ts 12.2 knots. To obtain the speed in kms per hr, multiply by 1.862.

• Find the distance in km between these places.

a. Johannesburg (26°S, 28°W) and Maputo (26°S,33°W) 'b, Washington (38°N.77°W) and San Francisco (38°N.122°W)

• A plane flies due east from (30°S,13°E) for 1,000km. What is its new position?
• A plane flies due east from (45°N, 5°W) for 1,200 km. what is its new position?
• A plane flies due east ftom (23°S.12°W) for 1,650km. What is its new position?
• A ship sails due west from (30°S,12°E) for 2,800km, what is its new position?
• Fins the distances in nautical miles between the places of question 1
• A plane starts at (40°S,30°E) and flies cast for 600nm. Find its new position.

8 A plane starts at (20°N, 10°W) and flies east for 720nm. Find its new position

• A ship starts at (40°S, 140°W), It sails north for 12 hrs, then east for 20brs. If ts speed

‘was constant 20knots, find the latitude and longitude of its final position.

• A ship starts at (30°N, 45°W), and sails north to (34°N,45°W). It then sails east to

(G4°NA1°W). if the joumey took 30 hrs, find the speed, given that it was constant.

• A ship can sail at 24lam/hr. it stasts at (20°S, 60°E). It sails due south for § hrs, then due

cast 2 hr. find the latitude and longitude of its final position. Navigation Suppose a ship is sailing in a sea current, or that a plane is flying in a wind. Then the course set the ship or plane is not the direction that it will move in. the actual direction and speed ean be Sound either by scale or by the use of Pythagoras’s theorem and trigonometry rave the line representing the motion of the ship relative to the water. At the end of this line

draw a line represeming the current. Draw the third side of the triangle. This side, shown with a double — headed arrow, is the actual course of ship.

Example 11

AA ship sets course due east. In still water the ship can sail at 1Skm/hr. There is a current following due south of 4kkmn/ir. use a scale drawing to find The speed of the ship b The bearing of the sip

Solution

In one hour the ship sails 1Skm east relative to the water. Draw a horizontal line of length 15em. In one hour the current pulls the ship 4km south. At the end of the horizontal line, draw a vertical line of length 4em.

15km ae km a (@) Measure the third side of triangle as 15.Sem. The speed of the ship 1s 15.5 km/hr. () Measuce the angle between the course of the ship and east as 15°, Add this to 90° The ship is sling along a bearing of 105° Note: These results can also be found by Pythagoras’ theorem and trigonometry.

Example 12

The ship of example 10 needs to travel due east. Calculate te following. a What course should be set? >. How long will the ship take to cover 120km?

Solution

‘The ship needs to set a course slightly north oF east, consider the following diagram iskm a 4k “A 6 (@) By trigonometry

A course of 074.5 “should be set.

• Using Pythagoras’ theorem in tangle.

Now divide 120by this speed. The ship will ake 83 hrs Note: Wino iret, the jurniby wild take Bh: The journey tai igh loge when there isa curren Suppose a ship or a plane doesnot directly reach a position, We can stl find how lowe the ship ox plane ist the position bs /6: Pe ra A In the Singrata aove,th tip 8 (x ple) israeli oa tagline AB, ‘The shortest distance to point P is when SP is perpendicular to AB.

inipte 13 A sil island is 200k say ona bearing Of 075°, A Ship sails on bearing of O70>Find the closest thatthe ship i to the island Cari oe snare bo NA S > 200 km I Thi btn cn ots id ed apt otis via ei ni li a id ti st i pa ok Then dic ai

• Find thedistance in nautical miles between the places in question

9.A plane flies on a bearing of 342°.There is a control tower 100 km from the plane on a bearing