spherical geometry

**mat** · Nov 11th 2008, 04:55 PM

calculate to the nearest mile the distance between
P(26 degrees north,70 degrees west), Q(72 degrees S, 70 degrees W)

and

R(o degrees,50 degrees East), S(o degrees, 32 East) along the equator.

and

A ship leavers rio at midday and sails south at a steady 15 knots.
What are the co-ordinates od the ship at 2000 hours.
If it continues on the same bearing at the same speed what time should it reach its co-ordinates (28degrees south, 43 degrees West)

discuss your solutions the mathematical models that enable these calculations to be made and the crediibilty you can give their accuracy.

thanks.

**earboth** · Nov 12th 2008, 01:10 AM

Originally Posted by mat

calculate to the nearest mile the distance between
P(26 degrees north,70 degrees west), Q(72 degrees S, 70 degrees W)

and

R(o degrees,50 degrees East), S(o degrees, 32 East) along the equator.

...

Since P and Q are located on the same longitudinal line (=great circle) you only have to calculate the difference of the latitudes:

$d_{lat} = |26^\circ - (-72^\circ)|=98^\circ$

Let R denote the radius of the earth. Then the distance between P and Q is calculated by:

$d(PQ)=\dfrac{98^\circ}{360^\circ}\cdot 2 \pi \cdot R$

The second example has to be calculated similarly because the points are located on a great circle (=equator) too.

**earboth** · Nov 12th 2008, 01:19 AM

Originally Posted by mat

...

A ship leavers rio at midday and sails south at a steady 15 knots.
What are the co-ordinates od the ship at 2000 hours.
If it continues on the same bearing at the same speed what time should it reach its co-ordinates (28degrees south, 43 degrees West)

...

Do you mean Rio de Janeiro?
Do you mean 20:00 o'clock?

Keep in mind that one nautical mile covers exactly one minute on a great circle.

If the ship travelled for 8 hours on a longitudinal line, then the position is

$8\ h \cdot 15 \frac{nm}h =120 \ nm$ This distance correspond to 120' = 2°.

So the ship is 2° south of Rio d.J.

**mat** · Nov 13th 2008, 02:22 PM

hello ear.
for the first question I understand the equation

for kilometres

but not for nautical miles.
in my notes its AB=60 theta
or degrees dont know.

**earboth** · Nov 14th 2008, 04:41 AM

Originally Posted by mat

hello ear.
for the first question I understand the equation

for kilometres

but not for nautical miles.
in my notes its AB=60 theta
or degrees dont know.

The difference is

$\theta = 98^\circ$

Since each degree contains 60 nautical miles the distance in nautical miles is actually

$d = \theta \cdot 60\ \dfrac{nm}{^\circ}$

**mat** · Nov 19th 2008, 06:09 PM

Originally Posted by earboth

The difference is

$\theta = 98^\circ$

Since each degree contains 60 nautical miles the distance in nautical miles is actually

$d = \theta \cdot 60\ \dfrac{nm}{^\circ}$

hello
ear for the first answer pq=10946km.Then I found out that 1 nautical mile is equal to 1.852km.
So the answer for pq is 5900nm to 2 significant figures.
rs =14521km so 7800 nautical miles to 2 significat figures.
is this correct?

I still dont understand the other question .let me rephase it.
A ship leaves rio de Janerio with coordinates (23 south,43 west).
at midday and sails due south at a steady 15 knots.This equals to 15 nautical miles I was told.
What are the coordinates of the ship at 20.00 hours (8pm)?
If it continues on the same bearing and at the same speed, what time should it reach its coordinates (28s, 43W)?
how do I draw this?

another question.

what is a small circle and a great circle of the earth?
What is the prime meridan coordinates?
a plane flies 3600 nautical miles in seven and half hours.Find its average speed in knots (nautical miles)?

**earboth** · Nov 19th 2008, 10:48 PM

Originally Posted by mat

hello
ear for the first answer pq=10946km.Then I found out that 1 nautical mile is equal to 1.852km.
So the answer for pq is 5900nm to 2 significant figures.
rs =14521km so 7800 nautical miles to 2 significat figures.
is this correct? ...... yes

I still dont understand the other question .let me rephase it.
A ship leaves rio de Janerio with coordinates (23 south,43 west).
at midday and sails due south at a steady 15 knots.This equals to 15 nautical miles per hour I was told.
[1]What are the coordinates of the ship at 20.00 hours (8pm)?
[2] If it continues on the same bearing and at the same speed, what time should it reach its coordinates (28s, 43W)?
how do I draw this?

another question.

[3]what is a small circle and a great circle of the earth?
...
[4]a plane flies 3600 nautical miles in seven and half hours.Find its average speed in knots (nautical miles)?

[1]. Speed is defined as

$speed=\dfrac{distance}{time}~\implies~distance=spe ed \cdot time$

The ship sails during 8 hours at a constant speed of 15 knots = 15nm/h. The total distance is:

$d = 8\ h \cdot 15\frac{nm}h=120\ nm$

Since 1 nm correspond to 1 min on a great circle, the ship has covered $120' = 2^\circ$ to South. The new position is $(25^\circ S, 43^\circ W)$

[2]. The ship has to go the distance $28^\circ - 23^\circ = 5^\circ = 300'$

300 min correspond to 300 nm. Since the ship is sailing at 15 nm/h it needs $t=\dfrac{300\ nm}{15 \frac{nm}{h}} = 20\ h$

The ship started at noon; therefore the ETA is 8:00 am the next day.

[3]. All circles on the surface of the earth which have the same radius as the earth are great circles (for instance all longitudinal circles and the equator) all other circles are small circles (all latitudinal circles except the equator).
If you consider the plane geometry to be a special case of the spherical geometry then the great circles correspond to straight lines and the small circles to curves in the plane.

[4]. Use the definiton of speed given above:

$v=\dfrac{3500\ nm}{7.5\ h} = 466.7 \dfrac{nm}h = 466.7 \ kts$

**mat** · Nov 20th 2008, 07:13 PM

thankyou for a quick reply ear.
one more question
new york (40n,74w)
rio de janerio (23s, 43w)
Find the time diffrence between these 2 cities to the nearest hour.
if it midnight in new york what time is it in rio de janerio?

is the time diffrence is 2 hours to nearest hour?

rio is 2:00am?

**earboth** · Nov 20th 2008, 10:58 PM

Originally Posted by mat

thankyou for a quick reply ear.
one more question
new york (40n,74w)
rio de janerio (23s, 43w)
Find the time diffrence between these 2 cities to the nearest hour.
if it midnight in new york what time is it in rio de janerio?

is the time diffrence is 2 hours to nearest hour? ........ Yes

rio is 2:00am? ........ Yes

Your answers are correct.

Do yourself and do us a favour and start a new thread if you have a new question.

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