TOPIC 7: THE EARTH AS THE SPHERE ~ MATHEMATICS FORM 3
Features and Location of Places
The Equator, Great Circle, Small Circles, Meridian, Latitudes and Longitudes
Describe 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 equator. But for most purposes we assume that it is a sphere. The position of any point on earth is located by circles round the earth, as follows:
The earth rotates about its axis, which stretches from the north to the South Pole.
Circles round the Earth perpendicular to the axis are circles of Latitude and Circles round the Earth which go through the poles are circles of Longitude or meridians.
Consider the following diagram
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
it is while the longitude of a position shows us how far east or west of
the Greenwich meridian it is.
latitude of a position tells us how far north or south of the equator
it is while the longitude of a position shows us how far east or west of
the Greenwich meridian it is.
Latitude; If we draw a line from the centre of the Earth to any position P , then the angle between this line and the plane of the equator is the latitude of P.
Longitude: This is the angle between the plane through the circle of any Longitude P and the plane of the Greenwich meridian .
Latitude can be either North or South of the equator while Longitude can be either East or West of Greenwich.
When locating the latitude and longitude of a place we write the latitude first then longitude.
Example 1
Dar
es Salaam has latitude 7°S (i.e. 7° south of the equator) and longitude
39°E (i.e. 39° east of the Greenwich meridian). So Dar es Salaam is at
(7°S, 39°E).
es Salaam has latitude 7°S (i.e. 7° south of the equator) and longitude
39°E (i.e. 39° east of the Greenwich meridian). So Dar es Salaam is at
(7°S, 39°E).
NB;
Greenwich itself has latitude 51°N ( i.e. 51° north of the equator)and
longitude 0° (by definition). Johannesburg has latitude 26°S (i.e. 26
south of the equator) and longitude 28°E(i.e. 28°east of the Greenwich
meridian), therefore Johannesburg is at (26°S, 28°E).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°
Greenwich itself has latitude 51°N ( i.e. 51° north of the equator)and
longitude 0° (by definition). Johannesburg has latitude 26°S (i.e. 26
south of the equator) and longitude 28°E(i.e. 28°east of the Greenwich
meridian), therefore Johannesburg is at (26°S, 28°E).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 (at the south pole) to 90°N (at the north pole).
Ranges; Latitude varies between 90°S (at the south pole) to 90°N (at the north pole). Longitude varies between 180°E 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.
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;
Solution;
The point A is 60° above equator, and 20° east of Greenwich.
So the point A is at (60°N, 20°E)
The point B is 10° below the equator, and on the Greenwich meridian.
So the point B is at (10°S, 0°).
Exercise 1
1. Write down the latitude and longitude of the places shown on figures below:
2. Copy the diagram show on the figure above and mark these points:
- (10°N, 30°E)
- (20°N, 20°W)
- (0°, 20°W)
3. Obtain a globe, and on it identify the following places.
- (40°S, 30°E)
- (50°S,20°W)
- (10°N,40°W)
- (40°N, 30°E)
- (80°N,10°E)
- (0°,0°)
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.
If both points are south of the equator subtract the latitudes
If both points are the north of the equator subtract the latitudes
If one 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
If both points are west of Greenwich subtract the Longitudes
If one point is east of Greenwich and the other west then add the longitudes
Suppose places A and B are on the same longitude, then the difference in latitude is theangle subtended by AB at the centre of the earth.
Suppose places A and B are on the same latitude.
Then the difference in longitude is the angle subtended by AB on the earth’s axis.
Locating a Place on the Earth’s Surface
Locate a place on the Earth’s surface
Example 3
Three
places on longitude 30°E are Alexandria (in Egypt) at (31°N, 30°E),
Kigali (in Rwanda) at (2°S, 30°) and Pietermaritzburg (in South Africa)
at (30°S, 30°E).
places on longitude 30°E are Alexandria (in Egypt) at (31°N, 30°E),
Kigali (in Rwanda) at (2°S, 30°) and Pietermaritzburg (in South Africa)
at (30°S, 30°E).
Find the difference in latitude between
- Kigali and Pietermaritzburg
- Kigali and Alexandria
Solution
(a)Both Towns are south of the equator. So subtract the latitudes. 30 – 2 = 28
Therefore the difference is 280
(b) Kigali is south of the equator, and Alexandria is north, so add the latitudes
31 + 2 = 33
The difference is 330
Example 4
A
plane starts at Chileka airport (in Malawi) which is at (16°S, 35°E).
It flies west for 50°. What is its new latitude and longitude?
plane starts at Chileka airport (in Malawi) which is at (16°S, 35°E).
It flies west for 50°. What is its new latitude and longitude?
Solution
Since it flies west, then subtract 35° from 50°. This gives 15°
The new longitude is now west of Greenwich, hence the plane is at (16°S, 15°W).
Exercise 2
1. In the diagram shown in the following figure find,
- The difference in longitude between A and B
- The difference in longitude between D and E
Distances along Great Circles
Distances along Great Circles
Calculate distances along great circles
Take
two places X and Y on the same line of longitude, i.e. one place is due
north of the other. Suppose X is due north of Y. When travelling north
from Y to X, you travel along part of a circle of longitude that is you
travel along an arc of the circle.
two places X and Y on the same line of longitude, i.e. one place is due
north of the other. Suppose X is due north of Y. When travelling north
from Y to X, you travel along part of a circle of longitude that is you
travel along an arc of the circle.
The diagram below shows two points A and B on the same circle of longitude.
In the figure above, the sector containing the arc AB subtending∅, is shown.
Recall the formula for the length of arc.
NB:
Remember, to find the difference in latitudes, take account of whether
the places are north or south of the equator. If they are all found in
south or north, then subtract the latitudes. If one is south and the
other North then add the latitudes.
Remember, to find the difference in latitudes, take account of whether
the places are north or south of the equator. If they 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
Navigation Related Problems
Solve navigation related problems
Example 5
Find the distance between Alexandria (31°N, 30°E) and Kigali (2°S, 30°E)
Solution
Example 6
A plane starts at (200S, 300E), and flies north for 4000 km. Find its new latitude and longitude.
Example 7
A plane flies north from (10°S, 30°E) to ((27°N, 30°E) taking a time of 3 hours.
Find its speed, giving your answer I both knots and kilometers per hour.
Exercise 3
Consider the following Questions.
Distances along Small Circles
Distance along Small Circles
Calculate distance along small circles
Suppose
P and Q are places west or east of each other, i.e 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.
P and Q are places west or east of each other, i.e 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, circles of
latitude get smaller as they get nearer the poles.
situation here is slightly different from that of the previous section.
While circles of longitude all have the same length, circles of
latitude get smaller as they get nearer the poles.
Consider the circle of latitude 50°S. Let its radius be r km.
Example 8
Find the distance in km and nm along a circle of latitude between (20°N, 30°E) and (20°N, 40°W).
Solution:
Both places are on latitude 20°N. The difference in longitude is 70°. Use the formula for distance.
Distance = 111.7 cos20° x 70°. Hence the distance in nautical miles is 60 x 70 x cos20°
The distance is 3,950 nm.
Example 9
A ship starts at (40°S, 30°W) and sails due west for 1,000 km. Find its new latitude and longitude.
Example 10
A ship sails west from (20°S, 15°E) to (20°S, 23°E), taking 37 hours. Find speed, in knots and in kms per hr.
Exercise 4
Consider the following Questions.
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 can be found either by
scale or by the use of Pythagoras’s theorem and trigonometry.
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 can be found either by
scale or by the use of Pythagoras’s theorem and trigonometry.
Draw
the line representing the motion of the ship relative to the water. At
the end of this line draw a line representing the current. Draw the
third side of the triangle. This side, shown with a double – headed
arrow, is the actual course of ship.
the line representing the motion of the ship relative to the water. At
the end of this line draw a line representing 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
A
ship sets course due east. In still water the ship can sail at 15km/hr.
There is a current following due south of 4kkm/hr. use a scale drawing
to find.
ship sets course due east. In still water the ship can sail at 15km/hr.
There is a current following due south of 4kkm/hr. use a scale drawing
to find.
- The speed of the ship
- The bearing of the sip.
Solution:
In
one hour the ship sails 15km east relative to the water. Draw a
horizontal line of length 15cm. In one hour the current pulls the ship
4km south. At the end of the horizontal line, draw a vertical line of
length 4cm.
one hour the ship sails 15km east relative to the water. Draw a
horizontal line of length 15cm. In one hour the current pulls the ship
4km south. At the end of the horizontal line, draw a vertical line of
length 4cm.
Example 12
The ship of example 10 needs to travel due east. Calculate the following.
- 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.
Note:
With no current, the journey would take 8hrs. The journey takes
slightly longer when there is a current.Suppose a ship or a plane does
not directly reach a position. We can still find how close the ship or
plane is to the position.
With no current, the journey would take 8hrs. The journey takes
slightly longer when there is a current.Suppose a ship or a plane does
not directly reach a position. We can still find how close the ship or
plane is to the position.
Example 13
A
small island is 200km away on a bearing of 075°. A ship sails on a
bearing of 070°.Find the closest that the ship is to the island.
small island is 200km away on a bearing of 075°. A ship sails on a
bearing of 070°.Find the closest that the ship is to the island.
Exercise 5
1. Find the difference in longitude between Cape Town (34°S, 18°E) and Buenos Aires (34°S, 58°W)
2. A ship startsat (15°N, 30°W) and sails south for 2,500km. Where does it end up?
3. Find thedistance in km along circle of latitude between cape Town and Buenos aires (see question 2)
4. A plane starts at (37°S, 23°W) and flies east for 1,500 km. where does it end up?
5. Find thedistance in nau
Great looking website. Assume yyou did a great deal of your very own coding.
Pretty! This has been an extremely wonderful post.Thank you
for supplying this info.
I’m very happly to discover this page. I need to to thank you for ones time just for this fantastic read!
I definitely really liked every little bit of it and
I have you bookmarked to check out new things in your blog.
Verry good post! We will bbe linking to this great content on our website.Keep up the great writing!
Way cool! Some very valid points! I appreciate you penning this post and
the rest of the website is also really good.
Would love to perpetually get updated outstanding web blog!
Hello! I wish to say that this post is awesome, great written and come with approximately all important infos.
I’d like to look extra posts like this!
🙂
I got this site from my pal who shaed with me concerning
this website and now this time I aam browsing thbis website and
reading very informative articles or reviews at this place.
I do not even know how I ended up here,
but I thought this poost was great. Idon’t know who you aree but certainly you
arre going tto a famous blogger if you arre not already 🙂
Cheers!
Way cool! Some very valid points! I appreeciate you penning this post annd the rest oof the website is also really good.
Great delivery. Great arguments. Keep up the amazing spirit.
Hello everyone, it’s my first visit at this website, and piece of
writing is genuinel fruitful designed for me, keewp up posting such articles or reviews.
It’s not my first time to go to see this web page, i am visiting
this web site very often and take good facts from here.
I’m very happy to discover this page. I need to to thank you for ones
time just forr this fantastic read! I definitely really liked every little bit of it and I have you bookmarked to check
out new things in your blog.
I am pleased that I detected this web blog, just the right
information that I was searching for!
Pretty! This has been an extremely wonderful post.Thank
you for supplying this info.
You need to take part inn a contest for one of the most useful websites
online. I will recommend thiss site!
Thhank you for this very good posts. I was wanting to know whether you were planning of publishkng
similar posts to this.Keep up writing superb content articles!
Howdy! I could have sworn I’ve been to this blog before buut after browsing through some of thhe post I realized it’s
new too me. Nonetheless, I’m definitely delighted I found it and I’ll be bookmarking and checking back frequently!
Fabulous, what a web site it is! This web ste provides
useful data to us, keep it up.
Excellent post. I wwas checking continuously this blog and
I am impressed! Extremely useful information. I care
forr such information a lot. I wass looking forr this certain information for a
very long time.Thank you annd good luck.
Excellent post. I was checking continuously this blog and I am
impressed! Extremely useful information. I care ffor
such information a lot. I was looking for this
certain information for a very long time.Thwnk you and good luck.
What’s up, I wish for to subscribe for this blog to take most
recent updates, sso where can i do it please heelp out.
Hi there to every body, it’s my first go to see of this webpage; this webpage contains ajazing and really excellent data in support of readers.
I’m very happy to discover this page. I neeed to to thank you for ones time just ffor
this fantastic read! I definitely really liked every little bit of it and I have you bookmarked to
check out new things in your blog.
I’m very happy to discover this page. I need to tto thank you for ones time just for this fantastic read!
I definitely really liked every little bit of it and I have you
bookmarked too check out new things in your blog.
Ohh, its fastjdious discussion about this article here at this web site, I haave read all that, sso now me also commenting
at this place.
Hello! Do you use Twitter? I’d like too follow you if that would be okay.
I’m definitely enjoying your blog and look forward too new articles.
Thank you for this very good posts. I was wanting to
know whether you were planning of publishing similar
posts to this. Keep up writing superb content articles!
I enjoy what you guys are usually up too. This type of clever work
and reporting! Keep up the very good works guys I’ve added you guys
to my own video.
I’m very happy to discover this page. I needd to to thank you for
onews time just for this fantastic read! I definitely
really liked every little bit of it and I haave you bookmarked to
check out new things in your blog.
This is the pdrfect blog for anybody who hopes to find
out about this topic. Youu definitely put a brand new spin onn a tooic which has been discussed for decades.Wonderful
stuff, just excellent!
Your mode of describing all in this piece of writing is in fact pleasant, all be capable of simply be
awwre of it, Thanks a lot.
Great post! We will be linking to this great post on our website.
Keep up the great writing.
Hi there to every body, it’s my first go to seee of this webpage; this webpage contains amazing and
really excellent data inn support of readers.
Wow, this article is nice, my sister is analyzing such
things, so I am going to inform her.
Fabulous, what a web site it is! This web site provides useful data to us, keep it
up.
Wow, this article iss nice, my sister is analyzing such things, so I amm going tto inform her.
I need tto to thank you for this fantastic read!! I definitely enjoyed every bit of it.I hae got you book-marked to
look at new things you post…
Hello everyone, it’s my first visit at thjs
website, and piece of writing is enuinely fruitful designed for me, keepp up posting
such articles or reviews.
This is the perfect blog for anybody who hopes to find out about this topic.
You definitely put a brand new spin on a topic whicch has been discussed
for decades.Wonderful stuff, just excellent!
Hello! I wish to ssay that this post is awesome, great written and come with approximately all important
infos. I’d like tto look extra posts likke this!
🙂
I simply want too input that you have ? goold website ?nd I enjoy the desxign and als
artcles ?n it!
Hi there to every body, it’s myy forst go too seee of this webpage; this webpage contaains amazing and really excellent data in support of readers.
What’s up, after reading this awesome post i am as well delighted
to share myy experience here with friends.