Some help with a few theory questions
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Some help with a few theory questions
Greetings fellow aviators, I'm a low timer(CPL) who is preparing for Airline entrance tests. Stumbled upon a few tricky ones( or perhaps I need some sleep 
!!) but was unable to get my head around the answers. So could anyone who would like to devote sometime on solving the following be kind enough to post the answers with the steps( or atleast the key steps)? Any help would be much appreciated. Thanks in advance
Q1) what is the rhumb line track from A (4500N 01000W) to B (4830N 01500W) ?
Q2) You are at A(Equator, 59 E) you travel on a bearing of 225 for 100 NM, find out the coordinates of the new location.
Q3) Calculate Density alt if Pressure alt at an aerodrome of elevation 4000' is 5000'. temperature on ad. id 27deg celscius.
!!) but was unable to get my head around the answers. So could anyone who would like to devote sometime on solving the following be kind enough to post the answers with the steps( or atleast the key steps)? Any help would be much appreciated. Thanks in advanceQ1) what is the rhumb line track from A (4500N 01000W) to B (4830N 01500W) ?
Q2) You are at A(Equator, 59 E) you travel on a bearing of 225 for 100 NM, find out the coordinates of the new location.
Q3) Calculate Density alt if Pressure alt at an aerodrome of elevation 4000' is 5000'. temperature on ad. id 27deg celscius.
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I was trying to hold myself, but I have to ask: how did you get your CPL if you can't solve these simple questions (the 3rd one specifically)? Anyway, I hope I can give adequate answers to your questions.
1. This one is very simple actually, you only need distances in the east-west and north-south directions and then use arctan trigonometric function to solve the problem. Given that at any given meridian, 1° = 60 NM, you can calculate the distance in the north-south direction by multiplying difference in latitudes by 60.
48°30' - 45° = 3°30' => 3,5° x 60 = 210 NM
The distance in east-west direction is a bit trickier, but not too difficult. Again, we get the difference between longitudes and multiply it by 60, but since 1° = 60NM is only valid at meridians (and equator), we have to multiply the result by cosine of average latitude.
(48°30' + 45°) / 2 = ~ 46,8°
15° - 10° = 5° => 5° x 60 x cos(46,8°) = 205,36 NM
So we have both distances, the only thing left is to calculate angle between catheti:
arctan(210/205,36) = 45,64°
Since we travel to the north and to the west, and I've considered the angle to be from parallel of latitude, we have initial track 270° (if we went only towards west) and we then add our calculated angle to get rhumb line track.
270° + 45,64° =~ 316°
2. This problem is inverse to the first, so it should be yet again very simple. We first calculate distances in north-south and east-west direction. We do that by calculating cosine and sine of our track distance.
North-south:
100 NM * cos(225° - 180°) = 100 NM * cos 45° = 70,71 NM
East-west:
100 NM * sin(225° - 180°) = 100 NM * sin 45° = 70,71 NM
I previously explained that on every meridian, you can take for granted that 1° = 60 NM. So this means, that we can easily convert north-east distance into change of latitude.
0°N/S - 70,71 NM / 60 = -1,1785° = 1,1785°S =~ 1°11' S
Once we have the new latitude, we can also calculate change of longitude:
(0 + 1,1785) / 2 = 0,5893°
59°E - 70,71 NM / (60 * cos(0,5893°)) = 57,8214°E =~ 57°49'E
Over all, new coordinates are 1°11'S 57°49'E.
3. This is the question that really sort of pissed me off: which examiner let you pass your CPL exam if you don't know how to calculate density altitude, which you need to calculate your aircraft's performance data ?!?!?!?
Anyhow, here it goes. The density altitude is important for everyday operations, since it gives you the altitude, which you use for performance calculations. The higher the density altitude, the less power you get, consequently you have smaller rate of climb and longer takeoff distances (and landing distances, but in SEP aircraft that I've flown the difference is only noticed if you compare performance at SL and 10.000ft DA). In order to get density altitude from indicated altitude you first need to convert indicated altitude into pressure altitude, which is basically the altitude your altimeter shows if you set it to QNE (1013,25 hPa). But in this case, you even have given pressure altitude, so you don't need to to this step. Don't get fooled with airfield elevation, aircraft (or its engine) doesn't know whether you're on grond and density altitude is 6000ft or you are at 10000ft true altitude and density altitude is again 6000ft. It only cares about air density, since available engine power is function of air density of air charge entering the cylinders and in standard (ISA) atmosphere density drops with increasing altitude, unless you use some sort of pressure charging (supercharger or turbocharger), but you don't have to worry about this here.
OK, now let's focus on the calculations. We have pressure altitude of 5000ft, and SAT of 27°C. We must first calculate ISA temperature at this pressure altitude, so we can see if actual air is hotter or cooler than ISA. Then we will apply rule of thumb: for every degree Celsius OAT is higher than ISA temperature, the density altitude increases by 120ft. Here it goes:
ISA temperature:
15 - 5 x 1,98 = 5,1°C
ISA deviation:
27°C - 5,1°C = +21,9°C
Density altitude
DA = 5000ft + 120 x 21,9 = 7628ft
This of course is calculation using rule of thumb, since sadly syllabus doesn't require you to know what's behind rules of thumb. However if you do want accurate calculation, let me know and I'll write it the long way.
The most important thing comes last: I would strongly advise you to study ATPL theory much more thorougly, otherwise I believe everybody will see right through you that you did pass the required exams, but you don't understand the subjects.
1. This one is very simple actually, you only need distances in the east-west and north-south directions and then use arctan trigonometric function to solve the problem. Given that at any given meridian, 1° = 60 NM, you can calculate the distance in the north-south direction by multiplying difference in latitudes by 60.
48°30' - 45° = 3°30' => 3,5° x 60 = 210 NM
The distance in east-west direction is a bit trickier, but not too difficult. Again, we get the difference between longitudes and multiply it by 60, but since 1° = 60NM is only valid at meridians (and equator), we have to multiply the result by cosine of average latitude.
(48°30' + 45°) / 2 = ~ 46,8°
15° - 10° = 5° => 5° x 60 x cos(46,8°) = 205,36 NM
So we have both distances, the only thing left is to calculate angle between catheti:
arctan(210/205,36) = 45,64°
Since we travel to the north and to the west, and I've considered the angle to be from parallel of latitude, we have initial track 270° (if we went only towards west) and we then add our calculated angle to get rhumb line track.
270° + 45,64° =~ 316°
2. This problem is inverse to the first, so it should be yet again very simple. We first calculate distances in north-south and east-west direction. We do that by calculating cosine and sine of our track distance.
North-south:
100 NM * cos(225° - 180°) = 100 NM * cos 45° = 70,71 NM
East-west:
100 NM * sin(225° - 180°) = 100 NM * sin 45° = 70,71 NM
I previously explained that on every meridian, you can take for granted that 1° = 60 NM. So this means, that we can easily convert north-east distance into change of latitude.
0°N/S - 70,71 NM / 60 = -1,1785° = 1,1785°S =~ 1°11' S
Once we have the new latitude, we can also calculate change of longitude:
(0 + 1,1785) / 2 = 0,5893°
59°E - 70,71 NM / (60 * cos(0,5893°)) = 57,8214°E =~ 57°49'E
Over all, new coordinates are 1°11'S 57°49'E.
3. This is the question that really sort of pissed me off: which examiner let you pass your CPL exam if you don't know how to calculate density altitude, which you need to calculate your aircraft's performance data ?!?!?!?
Anyhow, here it goes. The density altitude is important for everyday operations, since it gives you the altitude, which you use for performance calculations. The higher the density altitude, the less power you get, consequently you have smaller rate of climb and longer takeoff distances (and landing distances, but in SEP aircraft that I've flown the difference is only noticed if you compare performance at SL and 10.000ft DA). In order to get density altitude from indicated altitude you first need to convert indicated altitude into pressure altitude, which is basically the altitude your altimeter shows if you set it to QNE (1013,25 hPa). But in this case, you even have given pressure altitude, so you don't need to to this step. Don't get fooled with airfield elevation, aircraft (or its engine) doesn't know whether you're on grond and density altitude is 6000ft or you are at 10000ft true altitude and density altitude is again 6000ft. It only cares about air density, since available engine power is function of air density of air charge entering the cylinders and in standard (ISA) atmosphere density drops with increasing altitude, unless you use some sort of pressure charging (supercharger or turbocharger), but you don't have to worry about this here.
OK, now let's focus on the calculations. We have pressure altitude of 5000ft, and SAT of 27°C. We must first calculate ISA temperature at this pressure altitude, so we can see if actual air is hotter or cooler than ISA. Then we will apply rule of thumb: for every degree Celsius OAT is higher than ISA temperature, the density altitude increases by 120ft. Here it goes:
ISA temperature:
15 - 5 x 1,98 = 5,1°C
ISA deviation:
27°C - 5,1°C = +21,9°C
Density altitude
DA = 5000ft + 120 x 21,9 = 7628ft
This of course is calculation using rule of thumb, since sadly syllabus doesn't require you to know what's behind rules of thumb. However if you do want accurate calculation, let me know and I'll write it the long way.
The most important thing comes last: I would strongly advise you to study ATPL theory much more thorougly, otherwise I believe everybody will see right through you that you did pass the required exams, but you don't understand the subjects.
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its been a while since i got it & im rusty now, i did solve them but the answers in the options were varying by a digit or two, so needed an outside opinion,thanks for the help anyway
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i did solve them my answers are as follows;
1. 289 ..it should be 315( your method did the job, as I said I'm brushing up my theory after the sabbatical to try my luck again in aviation. Could you be kind enough to make me understand the concept why the diagonal line is a great circle & not a rhumb line ? secondly if u used arctan to fond the angle as 45,64°, can't trigonometry be used to find in terms of sine or cosine 45,64° to find the length of the hypotenuse ,but that gives a different answer !
if i understand why this hypotenuse is a Great circle,it would make sense why it has to be converted to rhumbline)
2.I got the answer as 1'10'S 57'49'E,but the correct option in the key is ..57'20'E of the given options of which mine is one.
3. Ur answer isn't their in the options, i got 7618,using 119deg/cel
I was randomly trying some sums. anyway u were kind enough to assist so thank you very much for the time and the steps.
1. 289 ..it should be 315( your method did the job, as I said I'm brushing up my theory after the sabbatical to try my luck again in aviation. Could you be kind enough to make me understand the concept why the diagonal line is a great circle & not a rhumb line ? secondly if u used arctan to fond the angle as 45,64°, can't trigonometry be used to find in terms of sine or cosine 45,64° to find the length of the hypotenuse ,but that gives a different answer !
if i understand why this hypotenuse is a Great circle,it would make sense why it has to be converted to rhumbline)
2.I got the answer as 1'10'S 57'49'E,but the correct option in the key is ..57'20'E of the given options of which mine is one.
3. Ur answer isn't their in the options, i got 7618,using 119deg/cel
I was randomly trying some sums. anyway u were kind enough to assist so thank you very much for the time and the steps.
