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calculating distance along a DME arc

Discussion in 'Technical Talk' started by bluelake, Mar 16, 2006.

  1. bluelake

    bluelake Well-Known Member

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    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.
  2. desertdog71

    desertdog71 Girthy Member

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    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. :)
  3. SteveC

    SteveC Really? Staff Member

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    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.


    :)
  4. TonyC

    TonyC New Member

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    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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  5. TonyC

    TonyC New Member

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    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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  6. SteveC

    SteveC Really? Staff Member

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    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!
  7. aloft

    aloft Suspended

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    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.
  8. desertdog71

    desertdog71 Girthy Member

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    Nice descriptions guys.
  9. RynoB

    RynoB That One Guy

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    Good Book

    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.
  10. meritflyer

    meritflyer Well-Known Member

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    Where did you come up with .017453?
  11. TonyC

    TonyC New Member

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    (3.1416)(2)/360 = 0.0174533333






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  12. bluelake

    bluelake Well-Known Member

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    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 :)
  13. bluelake

    bluelake Well-Known Member

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    this thread can now RIP

    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..

    :):):)
  14. B767Driver

    B767Driver New Member

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    Did you mean for a 30 DME arc?
  15. TonyC

    TonyC New Member

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    That's exactly what I said (and highlighted in dark orchid) in my first post, http://forums.jetcareers.com/showpost.php?p=336524&postcount=4


    Distance traveled along an arc = Radials * DME / 60






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  16. TonyC

    TonyC New Member

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


    :)


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


    ;)




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  17. SteveC

    SteveC Really? Staff Member

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



    ;)
  18. TonyC

    TonyC New Member

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    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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  19. Chris_Ford

    Chris_Ford Suspended

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    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.
  20. ricecakecm

    ricecakecm Well-Known Member

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    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?

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