calculating distance along a DME arc

[bluelake]

Quickly anyone, since I forgot.. whats the formula for measuring the distance travelled along an arc? I remember ONE time enroute question on the instrument written that requred using this.. and I just memorized the answer (1hr 20 minutes).. now I wish I knew the formula.

[desertdog71]

Originally Posted by bluelake

Quickly anyone, since I forgot.. whats the formula for measuring the distance travelled along an arc? I remember ONE time enroute question on the instrument written that requred using this.. and I just memorized the answer (1hr 20 minutes).. now I wish I knew the formula.

Funny how those things bite you later.....I wish I could help you, I had a short discussion with my Instructor on Approach plates, and he pointed out a DME arc. I will be A**hole deep in this stuff soon enough.

[SteveC]

Well the math isn't too hard, but there may be a shortcut that would give a "close enough" answer - someone else can add that part.

To solve the problem you need to know two things:

DME distance from the station (or radius if you will), and the number of degrees the arc encompasses. Call the first number "A" and the second number "B". "B" can be calculated by simply subtracting the initial radial from the final inbound course radial (or the other way around depending upon which is the greater number).

The distance around a circle is simply (PI x Diameter), thus the distance of a DME arc if you went all the way around (360 degrees) would be
(3.1416 x "A" x 2) miles. Since you only travel part way around the circle, we need to take the ratio of the arc compared to the whole circle, which would = "B"/360.

Final equation would be:
Distance (in miles) = (3.1416)("A")(2)("B")/360

I suppose that you could simplify it and say the length of the arc, in miles, is (0.017453)("A")("B").

Example:

DME distance from nav aid = 10 nautical miles ("A")
Length of the arc = 60 degrees ("B") <from the 130 degree radial to the 190 degree radial for example>.

The length traveled along the arc = (0.017453)(10)(60) = 10.47 nautical miles.




If you want to continue on from there and calculate the time required to travel the arc, simply divide the distance by the ground speed. If the plane in the example above was traveling at 120 knots, the time would be 10.47nM/120kts = 0.08725 hours. To change to minutes simply multiply by 60, giving 5.235 minutes, or 5 minutes 14 seconds.




I know that isn't the clearest explanation, so if you need further clarification let me know and I'll try to do a better job.

[TonyC]

Originally Posted by bluelake

Quickly anyone, since I forgot.. whats the formula for measuring the distance travelled along an arc? I remember ONE time enroute question on the instrument written that requred using this.. and I just memorized the answer (1hr 20 minutes).. now I wish I knew the formula.

If the question is distance, 1hr 20 minutes cannot be the answer.


For a rough answer, you can apply the 60-to-1 rule. As applied to this case, you might recall that at 60 miles from a given station, 1 degree of travel along the arc would amount to approximately 1 NM of travel. Flying around circle of 60NM radius would take approximately 360NM. For this distance, 1 radial = 1 NM

For a 20 DME arc, the distance traveled per radial is approximately 1/2 NM. Another method of describing it is 2 radials per NM.

A General formula that would cover both of these examples is there are ( 60 / DME ) radials per NM.

20 DME = 3 radials per NM
15 DME = 4 radials per NM
10 DME = 6 radials per NM
120 DME = 1/2 radial per NM

Let's say you want to know how many miles you travel when you fly 90 degrees along a given DME arc. In that case, you would take the the number of radials traveled and divide by the radials per NM (from the formula we just found above).

On a 60 DME arc: 90 / (1 radial/NM) = 90 NM
30DME: 90 / 2 = 45 NM
10DME: 90 / 6 = 15 NM

To combine the two, we'd use this formula

Distance traveled along an arc = Radials Travelled / (60 / DME)

Do a little algebraic juggling and you can get

Distance traveled along an arc = Radials * DME / 60

90 radials, 45 DME => 90 * 45 / 60 = 67.5
40 radials, 12 DME => 40 * 12 / 60 = 8
15 radials, 8 DME => 15 * 8 / 60 = 2


Does any of that help??






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[TonyC]

Originally Posted by SteveC

The length traveled along the arc = (0.017453)(10)(60)

Looks like we posted at the exact same time... well, minute, anyway - - you beat me to the Submit Reply button.



I used the approximation given by the 60-to-1 rule, and you used actual trig - - and we see how close they are to each other. Dividing by 60 is the equivalent of multiplying by 0.1666667. I think they're close enough to use.



The bigger lesson to learn here, of course, is that memorizing formulas, or answers, does one little good in the real world. It's far more important to understand the "why"s and "how"s of the world, and be able to figure out answers based on what we know.

Both of our posts offered good ways of analyzing the problem and working through to a usable solution. What a concept!






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[SteveC]

See.....I knew there would be a shortcut!

For fun, let's apply Tony's rough answer to my example of 10 DME, 60 degrees (or radials).

10DME = 6 radials per NM
60 radials total (190 - 130)

Therefore: (60 radials) / (6 radials per NM) = 10 NM

10 is pretty close to 10.47, and a very useful estimate I would say!

[aloft]

I use the '100 feet per degree per mile' rule-of-thumb. It's the same as Tony's, but I can remember "100 feet per degree per mile".

60 degrees x 10 miles x 100 ft = 60,000 ft = 10 nm

This is also useful in the vertical plane, btw.

[desertdog71]

Nice descriptions guys.

[RynoB]

There is a great book called Mental Math for Pilots by Ronald D. McElroy. It has all the shortcuts for finding distance along an arc, cloud bases, calculating VDPs, etc. You can get it through Sporty's or ASA's website.

[meritflyer]

Originally Posted by SteveC

Distance (in miles) = (3.1416)("A")(2)("B")/360

(0.017453)("A")("B").

Where did you come up with .017453?

[TonyC]

Originally Posted by meritflyer

Where did you come up with .017453?

(3.1416)(2)/360 = 0.0174533333






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[bluelake]

thanks for all the replies... now i know why I didnt remember it!!

and TonyC,
"If the question is distance, 1hr 20 minutes cannot be the answer."


thats because it was a Time Enroute question, like I said. Often those questions first require you to figure out distance as an intermediate step And I bet I aint the only pilot who ever memorized something without fully understanding it initially. "65 on short final.. 65 on short final.. my CFI says 65 on short final"


TOday I went hiking to clear my brain and I found a HUGE bone laying on the ground.. I think it is a femur of a bear or something like that! Now THAT was more fun to think about than this topic

[bluelake]

ok... I actually found an Instrument Gleim book and found the question. Here is what the FAA uses, and boy there are a LOT LESS digits and formula than the above (posters thank you still):

DISTANCE on an ARC = (CHANGE in DEGREES X DME ARC DISTANCE) / 60.

phew.. and I was also relieved to see that, in fact, the Time Enroute question that required this calc did have as its answer 1 hour 20 minutes..

[B767Driver]

Originally Posted by TonyC

For a 20 DME arc, the distance traveled per radial is approximately 1/2 NM. Another method of describing it is 2 radials per NM.


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Did you mean for a 30 DME arc?

[TonyC]

Originally Posted by bluelake

ok... I actually found an Instrument Gleim book and found the question. Here is what the FAA uses, and boy there are a LOT LESS digits and formula than the above (posters thank you still):

DISTANCE on an ARC = (CHANGE in DEGREES X DME ARC DISTANCE) / 60.

That's exactly what I said (and highlighted in dark orchid) in my first post, http://forums.jetcareers.com/showpos...24&postcount=4


Distance traveled along an arc = Radials * DME / 60






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[TonyC]

Originally Posted by B767Driver

Did you mean for a 30 DME arc?

Ummmm, yes, I did mean 30 DME arc. I'll correct the typo. Thank you!





Hmmmm. It's too late to edit! There goes my legacy.







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[SteveC]

Originally Posted by TonyC

Hmmmm. It's too late to edit! There goes my legacy.

Hmmmm. I think I'll bookmark that post. Got a feeling it might come in handy in the future.

[TonyC]

Hey, if they can change the title of a thread from "A post with 1000 threads" to "A thread with 1000 posts" I think I can survive this little misstep.



Ya ever get the feelin' that your work has gone unappreciated? Here, let me teach you how to do something. No thanks, just give me the formula out of the book - - I don't want to understand it, I just want the answer.

<sigh>










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[Chris_Ford]

Originally Posted by TonyC

Ya ever get the feelin' that your work has gone unappreciated? Here, let me teach you how to do something. No thanks, just give me the formula out of the book - - I don't want to understand it, I just want the answer.

You know what they say. Teach a man to fish and he'll eat for a lifetime. But GIVE a man a fish, and he'll come back the next day with a gun and demand all the fish you have.

It was tough growing up in Compton.

[ricecakecm]

I should probably know this, but why would anybody care what the distance they've gone around a DME arc is? I mean, is there any practical application for this?

[TonyC]

Originally Posted by ricecakecm

I should probably know this, but why would anybody care what the distance they've gone around a DME arc is? I mean, is there any practical application for this?

Take a look at this approach to Casper, WY (Natrona County International): http://204.108.4.16/d-tpp/0602/00072VD3.PDFYou begin the approach at HETTY (DDY 137/29).First question: Suppose your airplane has a turn radius of 2 miles. When should you begin your right turn to intercept the DDY 204 Radial inbound in order to roll out on centerline, no wind? (How many radials along the 29 DME arc constitutes 2 miles?)Second question: You want to be at 8400 feet when you begin your turn to intercept the DDY 204 Radial, but you have to be at or above 10,300 feet until you pass NUNTE. What descent rate should you use to arrive at 8,400 feet at the point where you plan to begin the turn inbound? (Again, how much distance between two radials along the arc?)Third question: If you failed to plan your descent properly, and you can't land because you were too high when you broke out of the weather, do you have enough gas to try it again? .
I don't know what's going on here, but I've tried to use nice paragraphs and all, but it keeps cramming it all together in one little nugget. My apologies for it being difficult to read....

[ricecakecm]

Originally Posted by TonyC

Take a look at this approach to Casper, WY (Natrona County International): http://204.108.4.16/d-tpp/0602/00072VD3.PDFYou begin the approach at HETTY (DDY 137/29).First question: Suppose your airplane has a turn radius of 2 miles. When should you begin your right turn to intercept the DDY 204 Radial inbound in order to roll out on centerline, no wind? (How many radials along the 29 DME arc constitutes 2 miles?)Second question: You want to be at 8400 feet when you begin your turn to intercept the DDY 204 Radial, but you have to be at or above 10,300 feet until you pass NUNTE. What descent rate should you use to arrive at 8,400 feet at the point where you plan to begin the turn inbound? (Again, how much distance between two radials along the arc?)Third question: If you failed to plan your descent properly, and you can't land because you were too high when you broke out of the weather, do you have enough gas to try it again? .
I don't know what's going on here, but I've tried to use nice paragraphs and all, but it keeps cramming it all together in one little nugget. My apologies for it being difficult to read....

Apology accepted

I just plug it all into the magic boxes and do what the little green things tell me to

[B767Driver]

Originally Posted by TonyC

Take a look at this approach to Casper, WY (Natrona County International): http://204.108.4.16/d-tpp/0602/00072VD3.PDFYou begin the approach at HETTY (DDY 137/29).First question: Suppose your airplane has a turn radius of 2 miles. When should you begin your right turn to intercept the DDY 204 Radial inbound in order to roll out on centerline, no wind? (How many radials along the 29 DME arc constitutes 2 miles?)Second question: You want to be at 8400 feet when you begin your turn to intercept the DDY 204 Radial, but you have to be at or above 10,300 feet until you pass NUNTE. What descent rate should you use to arrive at 8,400 feet at the point where you plan to begin the turn inbound? (Again, how much distance between two radials along the arc?)Third question: If you failed to plan your descent properly, and you can't land because you were too high when you broke out of the weather, do you have enough gas to try it again? .
I don't know what's going on here, but I've tried to use nice paragraphs and all, but it keeps cramming it all together in one little nugget. My apologies for it being difficult to read....

That's good advice TonyC is giving. I don't know what other guys do...but on an instrument arrival...I always look at the distance between stepdown fixes and plan the rate of descent I will need to comply with each one. Most of the time the distance between fixes are right on the 3:1 'typical' descent profile. Sometimes they are not.

FWIW, on a dme arc...a procedure i've not flown in ages by the way...I don't use the turn radius formula...but use a rule of thumb. Lead your turn on the arc to intercept the final approach course by a dme that is 10% of your groundspeed. If you're doing 180kts...lead the turn by 1.8 dme. Has always worked for me...and lets me keep as much of my brain power where it needs to be...on flying the gages. If you're like me...where brain power available is close to brain power required...this gets important!

[aloft]

Lead your turn on the arc to intercept the final approach course by a dme that is 10% of your groundspeed. If you're doing 180kts...lead the turn by 1.8 dme.

Of course, this is only useful for intercepting the final approach course if you're flying with GPS and there's a waypoint where the arc and the final approach course meet. For those of us without all the gadgetry, you need to guesstimate a lead radial at which to start your turn. Just how many degrees prior to the final approach course that lead radial needs to be is why you need to be able to calculate distance around an arc. If your turn radius at your approach speed is .75 nm (4500 ft) and you're on a 10 nm arc, using the 100' per degree per mile rule of thumb, you should start your turn to intercept the final approach course when you reach the radial 4.5 degrees prior to the radial defining the final approach course. (4500' turn radius divided by 100 divided by 10 = 4.5 degrees)

Most modern GPS systems like Ricecake's dual 530s will anticipate the turn and tell him when to begin it, which is why his brain has turned to mush.

[Alchemy]

Even in the RJ we still use the lead in radial when turning to the final approach course off an arc. Most of our Mexican plates have published lead ins about 10 degrees offset from the final approach course. They work pretty well.....the accuracy of Mexican VOR's on the other hand, well that's a different story.