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# Aviation Math is Cool

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I use a simple math equation for figuring out how far planes will actually fly on a DME arc.  This is helpful for me at work because we sequence arrivals and it's good to know where the guys on the arc will end up in regards to other practice approaches.

I also find this useful in FSX, where I'm flying an arc, without any VNAV.  I can choose when to descend to keep my speed up on the approach.

Anyway, the official equation to solve for distance traveled on a DME arc is: 2 * Pi * r *(# degrees in arc / 360).  Pi being 3.14..., r is the radius (DME) of the arc, To find number of degrees just look at the approach plate, and do the basic math between the final approach course and the radial you start on.

To shorten this equation (rough math) just use 6 * DME * the % of a circle.  A lot of arcs are about 1/4 of a circle.  You after you solve 6 * the DME, just divide by 4 to see roughly how far you fly while on the arc.

For the VNAV math, if you descend at -1000 FPM on the arc, you can figure your ground speed in miles per minute.  Since you're descending at 1000 fpm its simple to decide when you need to descend!

Does anyone else have any helpful math to share?

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To descend at 300 ft. per NM, the required rate of descent is 5 x groundspeed.

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I use a simple math equation for figuring out how far planes will actually fly on a DME arc.  This is helpful for me at work because we sequence arrivals and it's good to know where the guys on the arc will end up in regards to other practice approaches.

I also find this useful in FSX, where I'm flying an arc, without any VNAV.  I can choose when to descend to keep my speed up on the approach.

Anyway, the official equation to solve for distance traveled on a DME arc is: 2 * Pi * r *(# degrees in arc / 360).  Pi being 3.14..., r is the radius (DME) of the arc, To find number of degrees just look at the approach plate, and do the basic math between the final approach course and the radial you start on.

To shorten this equation (rough math) just use 6 * DME * the % of a circle.  A lot of arcs are about 1/4 of a circle.  You after you solve 6 * the DME, just divide by 4 to see roughly how far you fly while on the arc.

For the VNAV math, if you descend at -1000 FPM on the arc, you can figure your ground speed in miles per minute.  Since you're descending at 1000 fpm its simple to decide when you need to descend!

Does anyone else have any helpful math to share?

Interesting Ryan (and I love stuff like this).

Now I have to put my shoes back on after using my toes to count.

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Hardly more than the good old 3:1 rule, or 10:3 rule for descents for me ...

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Easiest way to enter a DME arc, take your groundspeed lets say ~140kts (gs corrects for wind).  Drop the 0, divide by 2.  So 14/2=7.  Add 0.7 to your DME arc.   So for a 10DME arc, start a standard rate turn at 10.7 DME or 9.3 if comming from the VOR.  Turn 100 degrees, get onto the arc.  Fly in 10 degree segments.  Turn the CDI 10 degrees, when the CDI crosses center, turn the plane 10 degrees, turn the CDI 10 degrees...and so on until you reach your final approach course.  The HSI presents an easy 'picture' of where you are on the arc, but its good to practice with an old CDI for the extra challenge.

One advantage a traditional DME has over a GPS.   The DME gives you groundspeed relative to the station, so if you keep your DME groundspeed '0', you will know you are on the arc or how to quickly get back on if you drift.   If I recall from my checkride allowances, you have an allowance of 1nm either side of the arc, +100ft above -0ft below tolerance in altitude before you 'fail' the manuver.

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Deleted by KingGhidorah--originally put forth a rule of thumb, which on closer inspection, seems to be not very good.

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Aviation mathematics is no different from any math. It's simply an "Applied" mathematical concept in terms one can understand and prove .Its a proplem with teaching primary school math. No proof experimentation.

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Does anyone else have any helpful math to share?

Ed Williams' "Aviation Formulary"

http://williams.best.vwh.net

An enormous collection of formulae, algorithms and rules-of-thumb to solve almost any imaginable problem relating to navigation, wind corrections to track, heading and ground speed, aircraft performance -you name it, it's here.

Ed also developed a general algorithm to calculate magnetic variation at any point on the earth's surface, and that technique is presented with C source code.

There are general rules (like Ryan's example) that could be applied by pilots in flight using "in the head" math, or a simple calculator, as well as more complex formulae suitable for programmable calculators.

None of the math would be beyond anyone with a reasonable grasp of high school-level algebra and trigonometry.

Taken as a whole, Ed's formulary has all the algorithms that a programmer would need to write their own E6B app, or even the core of an FMS simulation.

He also has essays on the web site discussing the use of his formulae in practical navigation problem-solving, as well as covering numerous other aviation-related topics related to weather, aircraft performance etc.

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TJ I've got another one similar to that.

It's a rule of thumb for turns to final approach course. It's usually for radar vectoring but I use it all the time when self vectoring on FSX.

Assuming no wind and perpendicular course to final apch course. Divide GS by 100 (or drop a zero and put a decimal place between first two values). If you're flying 200 kts, at a point 2 mi from the final, you turn to an intercept heading.

We frequently vector fighters onto an ILS final, so we may turn then on 3-4 miles north or south of final (RY9/27).

Reference your arc rule of thumb: I like that better than what I was using.

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If that's how you atc guys do it, then that's how it is, but that sure seems early to me, Ryan, even if you allow for a much delayed response to your turn directive.  If I'm going 180 knots, I'm going to cover 3 nm in a minute.  It's going to take me a little over 30 seconds to turn 90 degrees.  I'll fly 1.5 nm in that time, but it is curving in to final approach course, not straight along my original, perpendicular, course, so we're only going to fly a portion of that distance towards the intercept, theoretically less than a mile.   I guess it makes sense though when one thinks about rolling in, rolling out, and whatever time it takes to respond to your directive.  Good thing to know.

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To determine the leadpoint to an DME/ARC: use 1% of ground-speed & 25° bank.

ex. 12 dme/arc, GS 250kts, leadpoint 2,5 nm. Turn 90° at 14,5 dme with 25° bank.

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To determine the leadpoint to an DME/ARC: use 1% of ground-speed & 25° bank.

ex. 12 dme/arc, GS 250kts, leadpoint 2,5 nm. Turn 90° at 14,5 dme with 25° bank.

So you're saying take 1%, while others above are basically saying .5%.  I propose a new rule of thumb by which we split the distance and use .75% of the ground speed   All in fun!

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