Sponsor Message:
Aviation Technical / Operations Forum
My Starred Topics | Profile | New Topic | Forum Index | Help | Search 
Nice But Heavy Math Problem!  
User currently offlineCricri From France, joined Oct 1999, 581 posts, RR: 7
Posted (10 years 10 months 6 days 13 hours ago) and read 3447 times:

An imaginary aircraft is flying with constant speed at 360kts at an altitude of 30.000 feet (let's say 6nm). He spends a whole day airborne.

At sunrise, which occurs just in front of his nose on that June 21st day, the aircraft is at 45°N on greenwich.

During this day, the aircraft will keep the sun in front of him in his heading, turning right, so that at sunset, the sun keeps being on the longitudinal axe of the aircraft.

We will take the earth rotation into account

1 - tell what the flight of the aircraft will look like as if drawn on the ground

2 - give the flight duration, saying the flight starts at sunrise and stops at sunset.

Gentlemen, good luck, i failed and failed and failed....

23 replies: All unread, jump to last
 
User currently offlineCancidas From Poland, joined Jul 2003, 4112 posts, RR: 11
Reply 1, posted (10 years 10 months 6 days 13 hours ago) and read 3413 times:

i'm stumped. i'll be giving it to my physics professor. Big grin


"...cannot the kingdom of salvation take me home."
User currently offlineMjzair From United States of America, joined Nov 1999, 401 posts, RR: 0
Reply 2, posted (10 years 10 months 6 days 12 hours ago) and read 3383 times:

This seems like it is a trick question..
If the plane is at 45N on June 21st, the sun cannot be aligned directly on the longditudinal axis of the aircraft...
On June 21st the sun reaches it point furthest North at 23.5N.
That is the only latitude that the sun can be directly in front of the aircraft's longditudnal axis.
Plus, at 45N, the sun will always be south of you. Assuming you are following the sun, you are travelling West, so making turns to the right as indicated will put you off to the North, and cannot keep the sun directly in front of you.

The only thing that I can add if this is inface a genuine question is:
sunrise at 45N on June 21st is at 04:13 Local Mean Time
sunset at 45N on June 21st is at 19:52 Local Mean Time
The sun travels at 15.04 degrees per hour.
The circumfurence of the earth at 45N is 11347NM

Anyone else?


[Edited 2003-11-18 16:39:06]

User currently offlineMjzair From United States of America, joined Nov 1999, 401 posts, RR: 0
Reply 3, posted (10 years 10 months 6 days 12 hours ago) and read 3371 times:

I take back my previous post, I think I understand a little better, I will work on it, and get back to you...


User currently offlineB747Skipper From , joined Dec 1969, posts, RR:
Reply 4, posted (10 years 10 months 6 days 12 hours ago) and read 3375 times:

Salut Cricri -
xxx
Good one, I had a similar problem to solve some 30+ years ago, when PanAm had the strange idea to give pilots navigator licence courses. Now you catch me on a layover - and I dont have my good old HO.216 reduction tables, and my slide rule... but basically you are going to end up with a beautiful clockwise spiral. This because initially you are going towards the sun for the "local" morning, with your 360 KTAS... then follow it during what will be a very long afternoon and sunset...
xxx
Without my tables, cannot give you the accurate flight time, but it will be around some 18+ hours for sunrise to sunset, and if wou started like you said at 45.00N and 0.00W, you will probably land in Labrador, Canada... If you had given us much higher TAS, could end at the North Pole, and the flight would never end...
xxx
Oh la la, vous me donnez des cheveux gris, les copains du A.net...  Big grin
Happy contrails -
(s) Skipper


User currently offlineIkarus From United Kingdom, joined Jan 2001, 3524 posts, RR: 2
Reply 5, posted (10 years 10 months 6 days 11 hours ago) and read 3350 times:

Of course the sun can rise directly ahead of the aircraft. It means the pilot isn't necessarily flying at 90 degrees or 270 degrees, but a bit less.

So the plane starts out flying East (well, slightly south of due East), always keeps the sun ahead and ends up flying West in the end.

At 360 knots, the distance travelled during a day (if the 15.5 hours or so quoted by Mjzair are correct) will be just above 10,000 kilometres. That may well put the plane in a place where the sun sets at a different time, especially as it will spend most of the day flying southwards (with some east- or west- element in it). If the plane travels a quarter of the earth's cirumference in distance, and a lot of that southwards, it could end up south of the Equator. On the other hand, because it starts out flying East, it is speeding against the sun, so the day will be shortened (and in the evening, when it chases the sun, the day will be lengthened. If there is no wind - and you have to assume there is none - then the two should be exactly equal in effect and the day ends up at the same length. That would put the plane back on the exact same longitude as it started out as, just much further South (unless it actually crossed the Equator, and flew north for a while)) So if this is correct (and it may not be) you know at least one thing about the flight path: Start and End will be along the same longitude.

Now what happens in between? My initial guess was a sort of semi-ellipse, but I suspect I'm wrong, because the thing will be distorted by the location of the sun.

Hmmm. Let me think. Let's start with three lines: a vertical one (for the start & end location) and the 23.5th parallel (the path of the sun) and the 45th parallel (for the plane). Add two lines: One, 20,000 km to the East, and one 20,000 km to the West. Where these intersect the 45th parallel, the sun will rise and set. The sun travels at a constant speed. Maybe the easiest way to solve this would be with a computer program? Numerically? I know I'm ignoring the altitude of the plane and the Earth's curvature, but I think for a first estimate, this should be a feasible approach. Program it so that there's two points, at two coordinates. Let one travel on a constant path at constant speed, and let the other always head towards the first, at constant speed but naturally changing direction at every time step. Set the time step to ten minutes, and the total time to 12 hours. (err, why is the damn day longer than 12 hours? I hate 3D!!!!).

Ah, let's just say I failed miserably...

(Now wait for people more clever than myself to come in and say how wrong I am!  Wink/being sarcastic )

Good Luck

Regards

Ikarus


User currently offlineMjzair From United States of America, joined Nov 1999, 401 posts, RR: 0
Reply 6, posted (10 years 10 months 6 days 11 hours ago) and read 3344 times:

I respectfully disagree with the skipper. I dont think it is possible to end up in Labrador Canada..
Since the sun stays directly overhead 23.5 north and is aligned with the longditudinal axis of the aircraft.. Here is what I think will happen..
The plane is in the Southwest of France at sunrise.. and the plane is heading Southeast.. since sunrise is directly in front of him...
The aircraft will then follow a path southeast for the early morning hours, and will be in a constant but slow right turn as the sun is moving faster than the aircraft... the sun is moving 15.04 degrees per hour, and at 360 kts the aircraft is only moving 6 degrees per hour (of course, this number will fluctuate based on the latitude the aircraft is at.)
The aircraft will then make its final turn to a westerly heading on 23.5N somewhere over Africa at noon or just there after. Then, it will travel at 23.5N at 360kts for quite sometime as the sun races the plane accross the sky..
Well, that is all I can add to this.. the trig involved is too much for me..
Skipper, like I said I respectfully disagree with you, and I hate doing so.... I hope what I posted makes sense.


User currently offlineB747Skipper From , joined Dec 1969, posts, RR:
Reply 7, posted (10 years 10 months 6 days 11 hours ago) and read 3315 times:

Dear Mjzair -
A lot of your numbers are correct the 23.5, 15.04... 360 KTAS makes you move at 6 degrees per hour, but this in latitude, not longitude, except at the equator. Initially you would have a "sunrise heading" of about 120 degrees, this increasing to 180 degrees at local noon time, ending up heading 245 degrees as you chase the sun to its sunset... Would love to be at home and plot this on an old map... try do it per segments of 1 hour flight... looks like the end result is a perfect spiral to me... Fast enough, chasing the sun, we would end at the North Pole, and fly forever...
xxx
And... your are NOT disrespectful, my friend...  Smile
Happy contrails -
(s) Skipper


User currently offlineJetguy From , joined Dec 1969, posts, RR:
Reply 8, posted (10 years 10 months 6 days 11 hours ago) and read 3308 times:

Boy am I glad...
that there are plenty of folks here, much smarter than myself, who have problems with questions like this. Aren't you glad problems like this aren't on the FAA's ATP written test or used as oral questions? We'd all be looking for work.  Big thumbs up

Jetguy


User currently offlineCO2BGR From United States of America, joined Oct 2003, 558 posts, RR: 4
Reply 9, posted (10 years 10 months 6 days 8 hours ago) and read 3252 times:

Heres my input:

The flight path woul take it to northwest Egypt where it woud turn to slightly south of west and chase the sun. The plane would never make it to 23.5degN but quite close, and sunset would be over the middle of the atlantic at about 30degW. The path would look like a backward Cwith an extended flat bottom.

I think....



There are too many self indulgent weiners in this town with too much bloody money" Randal Raines- Gone in 60 Seconds
User currently offlineTimz From United States of America, joined Sep 1999, 6835 posts, RR: 6
Reply 10, posted (10 years 10 months 6 days 4 hours ago) and read 3154 times:

Nobody's going to come up with an actual answer, but still...

The guys that are saying the plane will start out headed southeast need to think a little more. If you're on the ground at 45 North at sunrise on 21 June, where does the sun rise? Well north of east, right? Actual azimuth would be somewhere around 55-56 degrees. The plane starts out headed that way; I suppose it ends up headed WNW, but could be wrong about that.


User currently offlineMjzair From United States of America, joined Nov 1999, 401 posts, RR: 0
Reply 11, posted (10 years 10 months 6 days 3 hours ago) and read 3131 times:

Timz...
I think maybe we are reading the problem a little differently.
It says that the plane is already in the air, and at sunrise the plane is directly facing the rising sun.
We established that the sun will rise at 23.5 N on June 21st, so from a position of 45N the sun must rise in the south east.


User currently offlineTimz From United States of America, joined Sep 1999, 6835 posts, RR: 6
Reply 12, posted (10 years 10 months 6 days 2 hours ago) and read 3122 times:

Like I said, it rises 55-56 degrees east of north at latitude 45 North on 21 June.

It rises north of east everywhere in the world on that day-- just as it rises south of east everywhere in the world on 21 December. On 21 March (or thereabouts) it rises in the east, everywhere.


User currently offlineCO2BGR From United States of America, joined Oct 2003, 558 posts, RR: 4
Reply 13, posted (10 years 10 months 6 days ago) and read 3095 times:

Timz:
If you are at a higher latitude than the sun it will rise south of east, lower latitude it will rise north of east. Bangor, ME is at about 45degN and it never rises north of east here even in 21Jun.



There are too many self indulgent weiners in this town with too much bloody money" Randal Raines- Gone in 60 Seconds
User currently offlineVikkyvik From United States of America, joined Jul 2003, 10027 posts, RR: 26
Reply 14, posted (10 years 10 months 6 days ago) and read 3095 times:
Support Airliners.net - become a First Class Member!

Timz,
How do you get the 55-56 deg E of N at lat 45N on June 21? I agree that it would rise north of east on that date...would the sun rise at 23.5 deg N of E at the equator on that date?
~Vik



"Two and a Half Men" was filmed in front of a live ostrich.
User currently offlineCO2BGR From United States of America, joined Oct 2003, 558 posts, RR: 4
Reply 15, posted (10 years 10 months 5 days 11 hours ago) and read 3000 times:

The sun would rise due east if one was at 23.5degN. If one is north of that it would rise South of East, If one was south of 23.5degN the sun would rise North of east. Therefore from that plane the sun would be rising south of east at about 105deg.


There are too many self indulgent weiners in this town with too much bloody money" Randal Raines- Gone in 60 Seconds
User currently offlineTimz From United States of America, joined Sep 1999, 6835 posts, RR: 6
Reply 16, posted (10 years 10 months 5 days 9 hours ago) and read 2970 times:

Okay, CO2BGR, try this. You agree that if you're at 70 degrees north latitude on 21 June the sun never sets, right? At "midnight" the sun will be pretty much north, right? And the same if you're at 67 degrees north: at "midnight" the sun will be just above the northern horizon.

So now move down to 66 degrees north latitude. If we ignore atmospheric refraction the sun will be just below the northern horizon at midnight-- then it will rise a few minutes later, just east of north.

Vikkyvik: the complication in calculating sun azimuth at sunrise/sunset is atmospheric refraction. If we were at sea level and the bottom of the sun were just touching the horizon, and for some reason the earth's atmosphere suddenly disappeared, the sun would disappear too. It would actually be just below the horizon, though previously the atmospheric refraction had made it appear to be above it.

But if we ignore that it's pretty simple. (If we also ignore the fact that the apparent horizon is slightly below a horizontal line from our eye.) Yes, at the equator on 21 June the center of the sun will rise 23.5 degrees north of east. If you're not at the equator-- well, I think there's a simpler formula than the following, but don't recall it offhand.

To specify the position in the heavens of the sun or any other celestial object we (in effect) give the position of the point on earth that's directly beneath it at that moment. If, at a given moment, the sun is directly above 23.5 degrees north, 60 degrees west, then the terminology is: the sun's declination is 23.5 degrees north and its Greenwich hour angle is 60 degrees. Declination is just like latitude, measured from 0 to 90 degrees north or south, but GHA goes from 0 to 360 degrees in a westerly direction.

Assume for the moment that you're on the Greenwich meridian at a latitude L, where north is positive. Then the sine of the sun's altitude (the angle between it and the horizon) is

(cosine GHA times cosine L times cosine declination)
plus (sine L times sine declination).

So if latitude is 45 degrees, GHA is 60 degrees and declination is 23.5 degrees, altitude comes out 37.3 degrees.

At sunset altitude is zero (we're assuming), so the above simplifies to

cosine GHA = -(tangent L times tangent declination)

so if we're still at latitude 45, the sun will set (not rise) when its GHA is 115.8 degrees.

Once we know its GHA, then the tangent of its azimuth is

sin GHA, divided by

[(cosine GHA times sine L) minus (cosine L times tangent declination)]

If you run through that you'll get azimuth -55.7 degrees-- i.e. 55.7 degrees west of north, at sunset.


User currently offlineTimz From United States of America, joined Sep 1999, 6835 posts, RR: 6
Reply 17, posted (10 years 10 months 5 days 9 hours ago) and read 2964 times:

"Bangor, ME is at about 45degN and it never rises north of east here even in 21Jun."

Take a look next year.


User currently offlineTimz From United States of America, joined Sep 1999, 6835 posts, RR: 6
Reply 18, posted (10 years 10 months 5 days 5 hours ago) and read 2914 times:

I knew I forgot something simple: for any object, seen from any latitude on the Greenwich meridian,

cosine altitude times sine azimuth = -cosine declination times sine GHA

So at sunrise/sunset, once you've calculated GHA you can use that to get the sun's azimuth.

And what if you're not on the Greenwich meridian? Well, longitude can start anywhere-- the formulas will remain the same. If the sun's GHA is 60 degrees at a given moment, and you're at 100 degrees west longitude, then the sun's "GHA with respect to you" (otherwise known as LHA, local hour angle) is 320 degrees.


User currently offlineCancidas From Poland, joined Jul 2003, 4112 posts, RR: 11
Reply 19, posted (10 years 10 months 5 days ago) and read 2873 times:

this one stumped my physics prof.


"...cannot the kingdom of salvation take me home."
User currently offlineRendezvous From New Zealand, joined May 2001, 516 posts, RR: 0
Reply 20, posted (10 years 10 months 4 days 22 hours ago) and read 2847 times:

The plane lands about 100 miles after the engines run out of kero to drink.

User currently offlineSkyguy11 From , joined Dec 1969, posts, RR:
Reply 21, posted (10 years 10 months 4 days 20 hours ago) and read 2832 times:

I think you'd end up in a graveyard spiral after burning your eyes out trying to stare at the sun all day long.

User currently offlineB747Skipper From , joined Dec 1969, posts, RR:
Reply 22, posted (10 years 10 months 4 days 12 hours ago) and read 2783 times:

OK, Friends -
xxx
Means we end up with a spiral...
And somewhere like Labrador, or Newfoundland... ?
That is what I volunteered to say in reply 4 above...
xxx
By the way I live at latitude 30 South.
Went out on the roof at sunrise last morning with a compass...
The azimuth of the sun was approximately 105/110 degrees.
We are one month away from our summer equinox here.
xxx
I showed this one to a retired navigator...
He describes this as a "constructive brainstorm"...
But a great one, Cricri...
xxx
Happy contrails  Smile
(s) Skipper



User currently offlineTimz From United States of America, joined Sep 1999, 6835 posts, RR: 6
Reply 23, posted (10 years 10 months 2 days 4 hours ago) and read 2534 times:

Couple of boneheaded mistakes in the calculation posted above (or maybe deleted by now). Try again:

Ignoring refraction, and assuming "sunrise/sunset" is when the sun is 90 degrees from the zenith, and assuming a spherical earth on which the aircraft covers 6 degrees per hour, sun's declination constant 23.44 degrees and its hour angle increasing at exactly 15 degrees per hour, then:

2.06 hours from start, position 48.67 N 17.20 E (farthest north)
5.22 hours 38.56 N 37.45 E (farthest east)
5.76 hours 35.42 N 36.42 E (start chasing sun instead of approaching it)
9.47 hours 29.16 N 12.66 E (farthest south)
11.31 hours 29.76 N 0.00 W
18.75 hours 43.26 N 51.49 W (sunset)



Top Of Page
Forum Index

Reply To This Topic Nice But Heavy Math Problem!
Username:
No username? Sign up now!
Password: 


Forgot Password? Be reminded.
Remember me on this computer (uses cookies)
  • Tech/Ops related posts only!
  • Not Tech/Ops related? Use the other forums
  • No adverts of any kind. This includes web pages.
  • No hostile language or criticizing of others.
  • Do not post copyright protected material.
  • Use relevant and describing topics.
  • Check if your post already been discussed.
  • Check your spelling!
  • DETAILED RULES
Add Images Add SmiliesPosting Help

Please check your spelling (press "Check Spelling" above)


Similar topics:More similar topics...
Heavy Touchdown Or Landing Gear Problem? posted Mon Sep 22 2003 05:22:13 by Sepang
Heavy Take-off With Tailwind posted Fri Oct 27 2006 20:02:52 by BA84
Very Heavy Traffic In Houston's IAH. posted Mon Aug 28 2006 22:47:57 by MexAir
737: GEN Fail With Flat Battery Problem posted Sat Aug 5 2006 07:14:40 by Mandala499
747 Brake System - How Heavy? posted Sat Jul 22 2006 20:36:23 by Tropicalsq744
Is This A Problem (SXM Landing Situation) posted Thu Jul 20 2006 23:06:33 by SuseJ772
Keeping A Heavy On The Taxiway posted Thu Jun 22 2006 04:03:21 by Evan767
Sita Problem Today posted Mon Jun 12 2006 18:25:56 by Zruda
Training Pilot About To Do Heavy Landing? Do What? posted Sun May 21 2006 19:36:33 by JulianUK
Embraer Heavy MX Programs posted Wed Feb 15 2006 09:08:23 by Ilikeyyc

Sponsor Message:
Printer friendly format