chapter 16 ii earth as a sphere enrich

8/3/2019 Chapter 16 II Earth as a Sphere ENRICH

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CHAPTER 16: EARTH AS A SPHERE

Important Notes:

The distance between two points on the surface of the Earth is the length of the arc connecting

the two places. This distance (on the earth) is usually measured in nautical miles (n.m.)

The shortest distance from one point to another point on the surface of the earth is along the great

circle.

Distance DE = ( )60a nautical miles

Distance FG = ( )60b nautical miles

Distance JK = ( )60 cosc nautical miles

The distance along the longitude = the difference between two latitudes 60 n.m.

The distance along a parallel of latitude

= the difference between two the two longitudes 60 cos 0 n.m.

Example:

Earth As A Sphere 1

0 N

AB = ( )60 60 = 3600 n.m.

PQ = ( )130 60 = 7800 n.m.

AC = ( )20 60 cos 60 = 600 n.m.

CP A

600N

1000 E

S

N

O

Q

B

800

E

X

700S

00


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16.1 The Distance Along The Surface of the Earth (Along The Meridian)

1.

(a) Calculate the distance AB

b) Given the distance of AC is 9000 nm, find

the latitude of C.

2.

a) Given the distance PQ is 2700 n.m. and

P (0, 900W), find the latitude of Q

b) Given PQ = PR, find the latitude of R.

Earth As A Sphere 2

N

Q

R

O0

S

P

700N

500N

N

S

B

A

C


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3.

(a) Given POR = 1100, PQ = QR.

Find the distance of PQ.

b) State the latitude of P and R.

Latitude of P =

Latitude of R =

4. P(100 N, 800 E), Q and R are three points on

the earth surface. Q lies on to the north of Pand R lies on to the south of P.

a) The distance of PQ along the meridian is

2700 nm. Find the latitude of Q.

b) The distance of PR along the meridian is

1920 nm. Find the latitude of R.

Earth As A Sphere 3

P

R

O1100

Q00


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16.2 The Distance on the Surface of the Earth (Along The Equator)

1. Calculate the distance of PQ

(a)

The difference between two longitudes = 450 100

= 350

The distance PQ = 35 60= 2100 n.m

b) Find the distance PQ:

c) P (00, 700 W) , Q (00, 120 E)

d) P (00, 320 E) , Q (00, 400 W)

e) P (00, 70030 E) , Q (00, 290 30 W)

Earth As A Sphere 4

230

E

P

230 W

Q 00

QP

100 W450 W


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2.

Calculate the distance of PQ

The difference between two longitudes= 800 + 100

= 900

The distance of PQ = 90 60= 5400 n.m

Given PR = 3120 n.m

Find the longitude of R

The difference between longitudes P and R

=60

3120

= 520

Longitude of R = 100 + 520 = 620 E

3.

a) Given P (00, 100 E) and R (00, 100 W),

calculate the distance from P to R.

b) The distance of PQ along the equator is

2400 nautical miles. Find the longitude of

Q.

Earth As A Sphere 5

U

O

100 E

P

Q

R

S800 W

U

O

100 E

P

QR

S100 W


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4.

a) Calculate the distance PQ.

b) Given PR =3

1PQ, find the longitude of R.

5.

The distance of PQ along the equator is 900

nautical miles. Find the longitude of Q.

b) Given the distance of PR = 2PQ. Find the

longitude of R.

Earth As A Sphere 6

U

O

1000 E

P QR

S100 E

00

U

O

P

Q

R

S250 W

00


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16.3 The Distance Along The Surface of the Earth (Along The Common Parallel Of Latitude)

1.

a) Find the distance of PQ.

b) Given PR = 1200 nautical miles, find the

longitude of R.

2.

a) Find the distance of PQ.

b) Given PR = 2PQ, find the longitude of R.

Earth As A Sphere 7

200 W

450 W

N

Q 700N

P R

S

N

100 W

Q450 SP R

S

500E


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3.

a) Given PQ = 1234 nautical miles. Find the

longitude of Q.

b) Given the longitude of R is 1000 W.

Find the distance of QR.

4. Given P (750N, 420 E), Q (750N, 420 W)

and R lies in between PQ with PR = QR

a) Find the position of R.

a) Find the distance of PR measured along thecommon parallel latitude.

Earth As A Sphere 8

N

Q

00

PR

S

200

300W

O


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16.4 The shortest distance (is always the distance along the great circles)

Find the shortest distance betweenP and Q.

1.

POQ = 1800 (500 +500)

= 800

Distance = 80 60= 4800 n.m

The shortest distance between P and Q

= 4800 n.m

2.

3. 4.

Earth As A Sphere 9

1700 E100 W

N

Q500N

P

S

O

1700

E

100 W

N

Q

700NP

S

500N

00

300 E 400 S

N

Q

400N

P

S

N

P Q

S

250 W


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Find the shortest distance betweenP and Q. (You are advised to sketch a diagram)

5. P (400 S, 1000 W), Q (400 S, 800 E) 6. P (100 S, 700 W), Q (350 S, 1100 E)

P (00, 1100 W), Q (00, 100 E) 8. P (400N, 500 W), Q (400 S, 1300 E)

Earth As A Sphere 10


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Questions based on the examination format (Paper 2)

1. F (400 S, 720 E), G (400 S, 100 W), H and J are four points on the surface of the earth.

FH is the diameter of the earth and J is located at a distance of 3780 nautical miles due

north of F.

(a) State the position of H [1marks](b) Calculate the latitude of J [2 marks]

(c) Calculate the distance, in nautical miles from G eastwards to F, measured along

the common parallel of latitude. [5 marks]

(d) An aeroplane took off from G and flew due east to F and then due north to J.

The average speed of the aeroplane for the whole flights is 500 knots, find

the time of flight. [4 marks]

2. P (00, 720 E), Q and R are three points on the surface of the earth. Q is due south of P

and QR is the diameter of the parallel of latitude 20 0 S.

(a) Mark the position of P on the diagram above. [1 mark]

(b) State the longitude of R [2 marks]

(c) Calculate the shortest distance, in nautical miles, from Q to R, measured along the

surface of the earth. [3 marks]

(d) An aeroplane took off from P and flew due south to Q and then due west to R.

The average speed of its flight is 560 knots. Calculate(i) the total distance, in nautical miles, travelled by the aeroplane,

(ii) the total time, in hours taken by the aeroplane for the whole flight

[6 marks]

3. P (500N, 800 E), Q (500N, 100 W) and R are three points on the surface of the earth.

(a) Calculate the shortest distance, in nautical miles, from P to the North Pole

measured along the surface of the earth. [3 marks

(b) Given that R is 3620 nautical miles due south of Q. Calculate the latitude of R[4 marks]


Equator

R

S

N

Q

O


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(c ) An aeroplane took off from P at 0630 and flew westwards to Q along the

common parallel of latitude. The average speed of the flight is 350 knots.

Calculate the time of the aeroplane landed at Q. [5 marks]

4. F (00, x0 W), G (00, 600 E), H (460S, 600 E) and J (460S, 850 E) are four points on the

surface of the earth.

(a) Given that distance from F eastwards to G measured along the equator is 7920

nautical miles. Find the value of x. [4 marks]

(b) An aeroplane took off from G and flew due south to H. Given that the whole

flight took 42

1hours, calculate the average speed, in knots, of the aeroplane.

[4 marks]

(c) Another aeroplane took off from J at 1300, flew eastwards to H with an average

speed of 760 knots. Calculate its time of arrival at H. [4 marks]:

5. A (400 S, 300 W), B (400 S, 500 E) and C are three points on the earths surface. AC is

the diameter of a parallel of latitude.

(a) State the longitude of C. [1 marks]

(b) Calculate

(i) the distance, in nautical miles, from A eastwards to B, measured along

the common parallel of latitude.

(ii) the shortest distance, nautical miles, from A to C via South Pole.

[6 marks]

(c) An aeroplane took off from B and flew due north with an average speed of 570

knots. Calculate its latitude after flying 10 hours. [5 marks]

6. P and Q are two points on the surface of the earth with latitudes 400N. Longitudes of

P and Q are 200 E and 1600 W respectively.

(a) Calculate the distance, nautical miles, from P to Q, measured along thecommon parallel of latitude. [3 marks]

(b) An aeroplane took off from P and flew to Q at an average speed of 500 knots

via the North Pole. Calculate

(i) the distance travelled by the aeroplane,

(ii) the time taken of the whole flight

[4 marks]

(c) Another aeroplane took off from P and flew due west to Q half an hour after the

first aeroplane took off. Given that both of the aeroplanes reach Q at the same

time, calculate the average speed, in knots, of the second aeroplane.

[5 marks]



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7.

In the diagram above, positions of A, B, C and D are ( )30 ,20 ,S W

( )30 ,40 ,S E ( )60 ,40N E and ( )60 ,20N W respectively.

(a) Calculate the shortest distance, in nautical miles,

(i) between A and B as measured along the common parallel of latitude,

(ii) between B and C as measured along the meridian.

[7 marks]

(b ) An aeroplane X took off from B and flew due east to A with an average

speed of 400 knots. At the same time, another aeroplane Y took off from D

and flew due south to A with an average speed of 600 knots. Find the

distance of aeroplane Y from A when aeroplane X reached A.

[5 marks]

8. P, Q and R are three points on the surface of the earth. PQ is the diameter of the 50 N

parallel of latitude and PR is the diameter of the earth. The longitude of R is 75 .E

a) Findi) the latitude of R

ii) the longitude of P

b). Find the distance, in nautical miles, from P due east to Q measured along the

common parallel of latitude.

b) An aeroplane flew from P towards Q passing through the North Pole. The

aeroplane started from P at 0900 hours and arrived at Q at 1530 hours on the

same day. Find the average speed, in knots, of the aeroplane.

[50oS, 105oW; 6942 nm; 738.5 knots]


00

S

N

B

O

CD

A


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9. Points ( )43 ,78A S W , ( )43 ,14B S E , CandD are four points on the surface of the

earth. Point C lies to the north ofA andAD is the diameter of the earth.State the location of D

b) Given that the distance between A and C, measured along the meridian, is 4020

nautical miles, find the latitude of C.

c) Find the distance, in nautical miles, from B due west to A, measured along the

common parallel of latitude.

d) An airplane took off from B at 0700 hours and flew due west to A along the

common parallel of latitude, then due north to C. If the airplane reached C at 1924

hours, find the average speed for the whole flight.

[(43oN, 102oE); 24oN; 4037 nm; 650 knots]

10. ( )50 ,63E S E , F, G and H are four points on the surface of the earth. E, F and G lie ofthe common parallel of latitude, such that EF is the diameter of that common parallel of

latitude. The longitude of G is 47oW and H lies to the north of G.d) Find the longitude of F.

e) An aeroplane took off from E and flew due west to G. Then, the aeroplane

flew due north to H which is 5100 nautical miles from G. The average speed of

the aeroplane from E to H is 680 knot.

Calculate

i) the latitude of H

ii) the distance, in nautical miles, from E to G

iii)the time, in hours, taken for the flight from E to H.

[117oW; 350N; 4242.48 nm; 13.74 hours]

11. K, L and M are three points on the surface of the earth on the parallel of latitude56 S .

The longitude of K is 60 E whereas the longitude of L is10 E . Given KM is the

diameter of the parallel of latitude 56 S .

Find

a) the longitude of M.

b) the shortest distance, in nautical miles, between K and M measured along the


c) the distance, in nautical miles, from K to L measured along the parallel of

latitude.

d) the duration of the flight from L to the North Pole, along the shortest distance byan aeroplane at an average speed of 700 knots.

[120oW; 4080 nm; 1678 nm; 12 hours 31 minutes]

12. ( )71 ,18A N E and B are two points on the earths surface such thatAB is a diameter of aparallel of latitude.

a) Find the longitude of B

b) AC is a diameter of the earth. On a diagram, mark the positions of A, B and C.

Hence, state the latitude and longitude of C.

c) Calculate the shortest distance, in nautical miles, from B to the North Pole.



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d) An aeroplane took off from A and flew due west along its parallel of latitude at

an average speed of 540 knots. The aeroplane took 6 hours to reach a point X.

Calculate

i. the distance, in nautical miles, from A to X,

ii. the longitude of X.

[162oN; C (71oS, 162oW); 1140 nm; 3240 nm, 147o52W]

13. ( )50 ,80K S E , L and M are three points on the earths surface. KL is a diameter of

the parallel of latitude 50 .S M is 4860 nautical miles due north of K.

a) State the longitude of L.

b) Find the latitude of M.

c) Calculate the distance, in nautical miles, from K to L measured along the parallel

of latitude.

d) An aeroplane flew from L to K using the shortest route measured along the

earths surface and then flew due north to M. Given that the average speed forthe whole flight is 630 knots, calculate the total time of flight.

[100oW; 31oN; 6942.1 nm; 15 hours 20 minutes]

14. ( )0 , 24A E and B are two points on the equator while C and D are two points on thcommon parallel of latitude. C and D lie due north of A and B respectively.

a) Given that the distance from A to C, measured along the meridian, is 3360

nautical miles, find the latitude of C.

b) Given that the longitude of D is 42oW, calculate the distance from C due west to

D, measured along the common parallel of latitude.

c) An aeroplane took off from C and flew along the shortest route to D, then, due

south to B. If the average speed of the aeroplane for the whole flight was 550

knot, calculate

i) the total distance covered

ii) the time taken for the whole flight

[56oN; 2214.4 nm; 7440 nm; 13 hours 32 minutes]

15. ( )30 , 40A S E , ( )30 ,80B S W and C are three points on the surface of the earth

and AC is the diameter of the common parallel of latitude.

a) (i) Find the longitude of C(ii) Find the difference, in nautical miles, between the distance fro A to C via

the North Pole and the distance from A to C via the South Pole.

(iii) Calculate the distance, in nautical miles, from A due west to B, measured

along the common parallel of latitude.

b) Calculate the latitude of a point, D, which lies 4110 nautical miles due

north of B.

[140oW; 7200 nm; 6235.4 nm; 38o30N]



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16. P, Q, R and V are four points on the Earths surface. The longitude of Q and R are

50oW and 110oW respectively. Q and R are due east of P on the latitude 58oN such

that PQ = QR. Point V which is on the latitude 76oN is due north of R.

a) Find the longitude of P.

b) Calculate, in nautical miles,

i) the distance, measured along their common latitude, from Q due east to R.ii) the shortest distance, measured along the earths surface, from R due

north to V

c) An aeroplane flying at an average speed of 600 knots, flew due east from Q to R

and then flew due north from R to V. Calculate to the nearest hour, the total time

taken for the whole journey.

[10oW; 1908 nm; 1080 nm; 5 hours]

17. Two aeroplanes took off from an airport at ( )40 , 20 A N E and flew to their

destinations at an average speed of 600 knots. The first aeroplane flew due west and

arrived at B after flying1

5of the circumference of the parallel of latitude 40oN. The

second aeroplane used the shortest route to arrive at ( )20 ,160C S W .a) Find the longitude of B.

b) D is another point on the earths surface such that CD is a diameter of the

earth. Find the latitude of D.

c) Calculate

i) the distance each aeroplane travelled in nautical miles.

ii) the difference in time the two aeroplanes took for their respective flights.[52oW; 20ON; 3309.3 nm and 9600 nm; 10 hours 29 minutes]



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Past Year SPM Questions

Paper 2

November 2003 (Paper 2, Q 16)P (610N, 100 E) and Q are two points on the surface of the earth such that PQ is the diameter

of a parallel of latitude.

(a) Find the longitude of Q [1 mark]

(b) PR is the diameter of the earth. On the diagram below, mark the positions of Q and R.

Hence, state the position of R.

[4 marks]

(c ) Calculate the shortest distance, in nautical miles, from Q to the North Pole. [2 marks]

(d) An aeroplane took off from P flew due west along its parallel of latitude with an average

speed of 500 knots. The aeroplane took 9 hours to reach a point M.

Calculate

(i) the distance in nautical miles, from P to M

(ii) the longitude of M [5 marks]

2. July 2004 (Paper 2, Q 16)

( )35 ,58P S W , ( )35 ,24Q S E , R and V are four points on the surface of the earth. PR is a

diameter of the earth and V is located at a distance of 3060 nautical miles due north of Q.

(a) State the longitude of R. [2 marks]

(b) Calculate the latitude of V. [3 marks]

(c) Calculate the distance, in nautical miles, from P eastwards to Q, measured along the

common parallel of latitude. [3 marks]

(d) An aeroplane took off from P at 0800 hours and flew due east to Q and then due north to

V. Given that its average speed for the whole flight is 600 knots, at what time did theaeroplane arrive at V?

[4 marks]Earth As A Sphere 17

N

Equator

P

S


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3. November 2004 (Paper 2, Q16)

P (600

S, 700

E), Q and R are three points on the surface of the earth. PQ is the diameter of theparallel of latitude 600 S. R lies 4 800 nautical miles due north of P.

(a) State the longitude of Q. [2 marks]

(b) Find the latitude of R. [3 marks]

(c) Calculate the distance, in nautical miles, from P to Q measured along the parallel latitude.

[3 marks]

(d) An aeroplane took off from Q and flew towards P using the shortest distance, as measured

along the surface of the earth, and then flew due north to R.

Given that its average speed for the whole flight was 560 knots, calculate the total timetaken for the flight.

[4 marks]

4. July 2005 (Paper 2, Q16)

( )0 ,50F W , G and H are three points on the surface of the earth. G is due north of F andGH is the diameter of the parallel of latitude 30 oN.

On Diagram 10 in the answer space, mark the position of F.

[1 mark]

State the position of H.

[2 marks]

c. Calculate the shortest distance, in nautical miles, from G to H measured along the


[3 marks]

d. An aeroplane took off from F and flew due north to G and then due east to H. The

average speed of the aeroplane is 500 knots.

Calculate

i) the total distance, in nautical miles, travelled by the aeroplanes

ii) the total time, in hours, taken by the aeroplane for the whole flight.[6 marks]


N

Equator

H

S

G

DIAGRAM 10


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5. November 2005 (Paper 2, Q 16)

The table below shows the latitudes and longitudes of four points J, K, L and M, on the


Point Latitude Longitude

JK

L

M

20 Nx S

20 S

30 S

25 E25 E

y W

y W

(a) P is a point on the surface of the earth such that JP is the diameter of the earth.

State the position of P. [2 marks]

(b) Calculate

(i) the value of x, if the distance from J to K measured along the meridian is 4200

nautical miles.(ii) the value of y, is the distance from J due west to L measured along the common

parallel of latitude is 3270 nautical miles. [7 marks]

(c) An aeroplane took off from J and flew due west to L along the common parallel of latitude

and then due south to M. If the average speed for the whole flight is 600 knots, calculate

the time taken for the whole flight.

[3 marks]

6. July 2006 (Paper 2, Q16)

Diagram 8 shows the point ( )47 ,12P N W and the point Q on the surface of the earth. Thepoint C is the centre of the common parallels of latitude of P and Q.

(a) State the position of Q [3

marks]

(b) R is a point on the surface of the earth. It is given that R is situated at a distanceof 2400 nautical miles due south of P, measured along the meridian.


N

S

C

P

Q50O

DIAGRAM 8


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Find the latitude of R. [4 marks]

(c) Calculate the distance, in nautical miles, from P due east to Q, measured along

the common parallels of latitude. [3 marks]

(d) An aeroplane took off from Q and flew due west to P along the common parallel

of latitude. Then, it flew due south, along the meridian, to R.

It is given that the total time taken for the flight is 172

hours.

Calculate the average speed, in knots, of the aeroplane for the whole flight.

[2 marks]

7. November 2006 (Paper 2, Q16)

Diagram 9 shows four points P, Q, R and X, on the surface of the earth. P lies on the longitude

of80 W . QR is the diameter of the parallel of latitude of 50 .N X lies 5820 nautical miles due

south of P.

(a) Find the position of R. [3 marks]

(b) Calculate the shortest distance, in nautical miles, from Q to R, measured along

the surface of the earth. [2 marks](c) Find the latitude of X. [3 marks]

(d) An aeroplane took off from P and flew due west to R along the parallel of latitude

with an average speed of 600 knots.

Calculate the time, in hours, taken for the flight. [4 marks]


N

S

RP

Q

O

X

35O

50O

DIAGRAM 9


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8 Julai 2007.

P(250N, 600E), Q and R are three points on the surface of the earth.

PR is the diameter of the earth.(a) State the longitude of R.

(b) PQ is the diameter of the parallel of latitude 250N.

(i) State the position of Q .

(ii) Calculate the shortest distance, in nautical mile, from P to Qmeasured along the surface of the earth.

(c) An aeroplane took off fromPand flew due west to Q along thecommon parallel of latitude and then flew due south to R.

Calculate

(i) the distance, in nautical miles, from P to Q measured alongthe common parallel of latitude.

(ii) the time taken, in hours, for the whole flight if the average speed

of the whole flight is 650 knots.


[2marks]

[4marks]

[6marks]


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9 November 2007.

P(650N, 400W), Q ( 650N, 600E), R and V are four points on the

surface of the earth.PR is the diameter of the parallel of latitude 650

N.

(a) (i) State the longitude ofR.

(ii) Calculate the shortest distance, in nautical mile, from P to Rmeasured along the surface of the earth.

(b) V lies south of Q and the distance VQ measured along the surfaceof the earth is 4500 nautical mile.

Calculate the latitude of V.

(c) An aeroplane took off fromPand flew due east to Q and then flew

due south to V. The average speed for the whole flight was 550 knots.

Calculate

(i) the distance, in nautical miles, taken by the aeroplane from P to Q measured along the common parallel of latitude,

(ii) the total time taken, in hours, taken for the whole flight .

10 Jun 2008.

P(250N, 1200E), Q , R , M and V are five points on the surface of theearth.PQ is the diameter of the earth.R lies 2100 nautical miles along thecommon parallel of latitude due west ofP.

(a) State the location of Q.

(b) Fine the longitude of R.

(c) PM is the diameter of the parallel latitude 50N.Calculate the shortest distance, in nautical mile, from P to M,Measured along the surface of the earth.

(d) An aeroplane took off fromP, flew due west toR , along the

common parallel of latitude. Then it flew due south to V whichlies due east of Q. It is given that the average speed of the whole

flight is 560 knots.

Calculate the total time taken for the whole flight.


[4marks]

[3marks]

[5marks]

[2 marks]

[4 marks]

[2 marks]

[4 marks]


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11 November 2008.

P(53N, 84W), Q (53N, 25W) ,R and V are four points on the

surface of the earth.PR is the diameter of the parallel of latitude of 53N.

(a) State the location ofR.

(b) Calculate the shortest distance, in nautical mile, fromP to Rmeasured along the surface of the earth.

(c) Calculate the distance, in nautical mile, from P due east tomeasured along the common parallel of latitude.

(d) An aeroplane took off from Q and flew due south to V . Theaverage speed of the flight was 420 knots and the time taken

was2

16 hours .

Calculate

(i) the distance, in nautical mile, from Q to V measured alongthe meridian ,

(ii) the latitude of V.


[3 marks]

[2 marks]

[3 marks]

[4 marks]


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ANSWERS

Past Year SPM Questions

1 Nov 2003

(a) 170 W (b) 61 S , 170 W (c) 1740 (d) (i) 4500 (ii) 147.7 W

2 Jul 2004

(a) 122o E (b) 16o N (c) 4030 n.m. (d) 1949 hours

3. Nov 2004

(a) 110 W (b) 20 N (c) 5400 (d) 15 hours

4 July 2005

(a) (b) 30o N 130o E (c) 7200 n.m. (d) 11153 n.m. 22.306 hours

5. Nov 2005

(a) P(20 S , 155 W) (b) x = 50 , y = 33 (c) 10.45 hours

6. July 2006

(a) 47o

N 38o

E (b) 7o

N (c) 2046 n.m. (d) 592.8 knots

7. Nov 2006

(a) R(50 N , 135 E) (b) 4800 (c) 47 S (d) 9.32 hours

8 Jun 2007

(a) (i) 120 W (b) (i) Q (25 N , 120 W) (ii) 7800 n.m

(c) (i) 9788 n.m (ii) 19.67 hours

9 Nov 2007

(a) (i) 140 E , (ii) 3000 n.m (b) 10 S (c) 2535.71n.m (d) 12.79 hours

10 Jun 2008

(a) Q(50 S , 60 W ) (b) 65.55 E (c) 4800 n.m (d) 14.46 hours

11 Nov 2008

(a) (53 N, 96 E ) (b) 4440 (c) 2130.4 n.m (d) (i) 2730 n.m (ii) 7.5 N



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Enrichment


chapter 16 ii earth as a sphere enrich

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