PLEASE ONLY ANSWER PROBLEMS E-G 3) Longitude–DeadReckoning a. [5] Given that the

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PLEASE ONLY ANSWER PROBLEMS E-G

3) Longitude–DeadReckoning

a. [5] Given that the earth has a circumference of 40,075 km at the equator, how long is one degree of longitude in kilometers at the equator?

b. [5] The following is the equation to calculate the length of one degree of longitude (L) at a given latitude (lat) based on the length of one degree of longitude at the equator (Leq) : L = Leq * cos(lat)

Find the length of one degree of longitude in kilometers at a latitude of 28°

c. [5] On a 50 meter long ship sailing due west from the Canary Islands (latitude = 28° N) a piece of wood is dropped in the water at the bow (front of the ship) and timed as it floats down the side of the ship until it reaches the stern (back of the ship). Over the course of many such observations, it takes at a minimum of 18 seconds for the ship to pass the piece of wood and a maximum of 20 seconds. What is the minimum and maximum speed of the ship in kilometers per hour?

d. [10] If the ship started in the Canary Islands at the coordinates 28° 00’ 00” N and 18° 00’ 00” W and sailed due west, what is the minimum and maximum distance in kilometers it could have traveled in 120 hours?

e. [5] What is the new longitude for the ship (give minimum and maximum possible longitude)

f. [5] What is the distance between the minimum and maximum possible longitude in kilometers?

g. [5] Name and describe at least three possible sources of error in calculating a position this way.

Answers for Problems A-D :

(a) At equator, 1° longitude = (40075/360) = 111.319 km.

(b) L= 111.319 * 0.8829 = 98.2835 km

(c) Maximum speed of the ship = (1474.2525 -9) = 1465.2525 km per hour

(d) Resultant minimum speed is 9km/hr. So in 120 hours, it will go about (9*120) = 1080 km.

Resultant maximum speed is 10km/hr. So in 120 hours, it will go about (10*120) = 1200 km.

Please answer E-G

Explanation / Answer

e) Minimum longitude for the ship = Longitude + (Minimum distance travelled / length of one degree of longitude at latitude of 28o)

= 28o00'00'' + (1080/98.2835) = 28o00'00'' + 10.9886 degrees = 28o00'00'' + 10o59'19'' = 38o59'19''

Similarly Maximum = 28o00'00' + ( 1200/98.2835) = 28o00'00' + 12.2096

= 28o00'00' + 12o12'34'' = 40o12'34''

f) Find the difference between two longitudes

ie. 40o12'34'' - 38o59'19'' = 1o53'15''

Converting this to seconds = 3600 secs + 3180 secs + 15 secs = 6795 secs

Given 1o = 98.2835

therefore, 3600 secs = 98.2835 km

So, 1 sec = 0.02730 km

So for 6795 secs = 0.02730 X 6795 = 185.50 km.

g) Dead reckoning is simply an excercise in vector addition. You represent your current location as (X,Y) point ie. longitude and latitude and your velocity as an X,Y vector. Multiply the vector by time and add to the initial point.

The probem is made more complex by the fact that you are travelling across the surface of a sphere. This means your velocity vector is actually a curve rather than a straight line.

Dead reckoning is subject to cumulative errors. Ofcourse it can give best avilable information on position. But it is subject to significant errors due to many factors as both speed and direction must be accurately known at all instants for positions to be determined accurately.

Hope this helps

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