Ground/Air Nautical Miles

David CL Francis wrote in message ...
On Wed, 3 Nov 2004 at 12:33:39 in message
.net, John T Lowry
wrote:
A trip of 100 nm over the ground, in an hour, if into a 10 knot direct
headwind, would be a trip of 110 nm relative to still air.

Are you sure about that?
Aircraft flying at 200k effective speed over the ground 190 knots
agreed?
Time taken = 100/190 = 0.5263157 hours
Effective distance at 200k TAS is 200*0.5263157 = 105 .26 nm

I'm no expert, but I think you misread and overanalyzed his statement
grin.

He said the 100nm trip was actually made in 1 hour while into a 10kt
headwind. Therefore the effective speed was 110kts, and the effective
distance (which is the point of ANM) was 110nm in that 1 hour.

ANM is sometimes used to compute the theoretical range of an aircraft.
For example, you fly a prototype jumbo jet from NYC to Paris, and it
took X thousand pounds of fuel. But after you add in the headwind x
time, then you can figure out the total ANM that it it went, and thus
compute its range or fuel usage/nm.

Kev

On Fri, 05 Nov 2004 22:14:57 GMT, David CL Francis
wrote:

On Fri, 5 Nov 2004 at 11:16:17 in message
, Todd Pattist
wrote:

David CL Francis wrote:
That does not answer the fact that the original statement by John
appears to me to be wrong.

It's not wrong.

Nice to meet you here again Todd. I agree I was wrong, in that the
problem he postulated was different from what I assumed. I think my
calculations were right though.

I apologise to those concerned

However John's actual statement now I read it more carefully seems to
imply that given the wind speed you must find the TAS at which you
must fly to get there in an hour!

Is this the calculation that is intended? Even more trivial than my
calculations!

John wrote:
"A trip of 100 nm over the ground, in an hour, if into a 10
knot direct headwind, would be a trip of 110 nm relative to
still air."

Thus we have an unknown cruise TAS cruise speed
Let that be V

We have a 100 nm distance and a head wind of 10k
We have a time of flight of exactly one hour

Therefore 100/(V-10)= 1

and V -10 = 100

it follows that V = 110k

So at a TAS of 110k you travel a ground distance of 100nm against a wind
of 10k and surprise, surprise you than fly 110 'air' nm

More or less self evident so I am unclear what that achieves?

You're wrong, here's why:

you're still wrong and that is why you're not achieving anything.
in the real world your cruise speed remains constant so what happens
is that the 110 nautical miles that the wind makes the 100 miles seem
like, takes longer to fly. engine running for longer equals more fuel
burn from the fixed tankage in the aircraft hence the need to pre calc
the usage and plan for it.

if you are actually a pilot you are an accident waiting to happen.
Stealth Pilot

On Fri, 5 Nov 2004 at 09:44:49 in message
.com, Raul Ruiz
wrote:
Brush up on your math...
http://www.math.utah.edu/~alfeld/math/0by0.html

I may be out on limb here with modern high school math but I don't
really agree with everything there! Perhaps it is my great age! I try to
learn form what people tell me but it gets harder and harder. :-(

(Infinity + infinity) = infinity so I agree you cannot just subtract one
of them each side and say infinity = 0 !!!

What is (1 + 1/n)^n as n tends to infinity? (Clue: it's a very
important number - assuming I have not screwed up that expression at my
great age.)

As I understand it only under very special circumstances can infinities
can be cancelled out. But it is done in some esoteric equations.

What is tan(Pi/2)? Better still plot tan(theta).

Of course infinity cannot be treated as an ordinary number but it still
'exists' and you can compute larger and larger numbers as long as you
like.

What I did was to plot three points on a function which at one point
tends or goes to infinity.

Do you agree that 20/b gets larger and larger as b gets smaller and
smaller and that there is no point at which you can say that ends? I
can say with confidence that 20/0 is infinity and I can go on from that
to say that 20/0 = 40/0. What I cannot do is to infer from that that 20
= 40. But that is the nature of infinity. I know that dividing both
sides of equation by zero cannot be done with impunity either. I agree
that you cannot place a value on (infinity * 0) but I didn't want to do
that.

Let a = b
multiply both sides by b a*b = b^2
subtract a^2 from both sides a*b - a^2 = b^2 - a^2
therefore a*(b - a) = (b + a)*(b - a)
cancel (b - a) then a = b + a
but a = b therefore 1 = 2

But I expect you all know that one!

It's all good fun!
--
David CL Francis

Of course infinity cannot be treated as an ordinary number but it still
'exists' and you can compute larger and larger numbers as long as you
like.

Infinity is not a number. Because of this, when you get infinity as an answer, final or intermediate, you need to look closer. That said...

There is more than one size of infinity. Consider for example the positive integers and the odd positive integers. Obviously the set that is missing all the even integers has to be smaller. But it ain't so. You can pair each member of one set
with a member of the other set, and you'll never run out.

1..2
2..4
3..6
etc.

Ok, so much for that. What about fractions? There have to be more fractions than integers, because the fractions include the integers (6/3 is just another way of spelling 2). Let's arrange a grid however:
(note - at this point you'll probably want a fixed spacing font to see the grid)

* 1 2 3 4
1 1/1 1/2 1/3 1/4 ...
2 2/1 2/2 2/3 2/4 ...
3 3/1 3/2 3/3 3/4 ...
4 4/1 4/2 4/3 4/4 ...
....

This chart has to have all the fractions in it. And all the integers are listed in the column to the left, going down. Each row has an infinite number of fractions in it, so there just =have=to= be more fractions than integers. Well, no. (and it
has nothing to do with the fact that 2/1 is the same as 4/2)
Consider a path that starts with 1/1, and goes diagonally up right as far as it can (which isn't far at all!) then goes to the next diagonal down, and the next diagnoal up, zigzagging along, until it reaches the end (which is, of course, never). It
would cover (and I've paired them with integers starting with 1, below)
1/1 1/2 2/1 3/1 2/2 1/3 1/4 2/3 3/2 4/1 ...
1 2 3 4 5 6 7 8 9 10 ...

I'll never run out, and I'll never miss a beat. There must be the =same= number of fractions as there are integers.

Ok, we get three strikes in baseball, I get three ups here. Lets look at all the real numbers beween 0 and 1 and try to list them. They are listed as infinite decimals,
though some of them may end in lots of zeros - i.e. .5 is the same as .500000.... (and also .4999999..., which I won't get into here)

Here's my list. Itegers on the left, decimals on the right (in no particular order):

1 .348791037984....
2 .500000000000....
3 .000023416898....
4 .142857142857....
5 .141592653589....
6 .414213562373....
.... ...

No matter what I do, I can't list all the decimals on the right, and not because I can't afford the paper. However the list is created (and I have not put them in an order for several reasons), there is always at least one number that's not on the
list. Create it thusly: Write a decimal point and then take the first decimal place of the first number on the list, and write it down. Take the second decimal digit of the second number and write it down... take the nth decimal digit of the nth
number on the list.. and write it down... (see below)

..300893...

Now, just below it write a number whose digits differ in every place.

..411904

That number IS NOT ON THE LIST! ("Sure it is... it's the 52342th one, you must have missed it." "nope, the 52342nd digit is different." "oh yeah... oh wait, here it is, it's the 230498103984th one on the list." "nope... "

So, there are more real numbers between zero and one than there are integers! There are at least two sizes of infinity... the "original size" and the "giant size".

(actually, there are an infinite number of sizes of infinities, but that step is less mind boggling than the first one)

So, what does this have to do with aviation? Well, it will give you something to ponder on those long cross countries, it will explain why your fuel calculations were a bit off in the headwind (Oh, it must have been a big infinity in the
calculations instead of a little one), and it will give you something to impress the girls with when a pilot certificate doesn't do the trick.

Ok, maybe not. But it's still interesting.

Jose
--
Freedom. It seemed like a good idea at the time.
for Email, make the obvious change in the address.

Jose wrote:
Of course infinity cannot be treated as an ordinary number but it still
'exists' and you can compute larger and larger numbers as long as you
like.

Infinity is not a number. Because of this, when you get infinity as an

Actually, in the world of digital computers, infinity *is* a finite
number.

(*evil laugh*)

On Sat, 6 Nov 2004 at 22:00:32 in message
, Stealth Pilot
wrote:

So at a TAS of 110k you travel a ground distance of 100nm against a wind
of 10k and surprise, surprise you than fly 110 'air' nm

More or less self evident so I am unclear what that achieves?

You're wrong, here's why:

you're still wrong and that is why you're not achieving anything.
in the real world your cruise speed remains constant so what happens
is that the 110 nautical miles that the wind makes the 100 miles seem
like, takes longer to fly. engine running for longer equals more fuel
burn from the fixed tankage in the aircraft hence the need to pre calc
the usage and plan for it.


No one disputes that statement of the obvious I would guess. Anyway as
far as I know you slightly adjust your cruise speed according to the
situation. That wasn't what anyone was discussing as far as I can see.
Perhaps you could tell me what I was trying to achieve as you seem to
know better than I do? On second thoughts don't bother.

if you are actually a pilot you are an accident waiting to happen.
Stealth Pilot

I'm not, just an elderly aeronautical engineer, so that's all right
then!
--
David CL Francis

On Fri, 5 Nov 2004 at 14:40:51 in message
, Kevin Darling
wrote:
David CL Francis wrote in message
...
On Wed, 3 Nov 2004 at 12:33:39 in message
.net, John T Lowry
wrote:
A trip of 100 nm over the ground, in an hour, if into a 10 knot direct
headwind, would be a trip of 110 nm relative to still air.

Are you sure about that?
Aircraft flying at 200k effective speed over the ground 190 knots
agreed?
Time taken = 100/190 = 0.5263157 hours
Effective distance at 200k TAS is 200*0.5263157 = 105 .26 nm

I'm no expert, but I think you misread and overanalyzed his statement
grin.

Your right - I did. See other posts.

He said the 100nm trip was actually made in 1 hour while into a 10kt
headwind. Therefore the effective speed was 110kts, and the effective
distance (which is the point of ANM) was 110nm in that 1 hour.

ANM is sometimes used to compute the theoretical range of an aircraft.
For example, you fly a prototype jumbo jet from NYC to Paris, and it
took X thousand pounds of fuel. But after you add in the headwind x
time, then you can figure out the total ANM that it it went, and thus
compute its range or fuel usage/nm.

That makes sense.
--
David CL Francis

On Sat, 6 Nov 2004 at 18:38:22 in message
, Jose
wrote:
That number IS NOT ON THE LIST! ("Sure it is... it's the 52342th one, you must have missed it." "nope, the 52342nd digit is different." "oh yeah...
oh wait, here it is, it's the 230498103984th one on the list." "nope... "

So, there are more real numbers between zero and one than there are integers! There are at least two sizes of infinity... the "original size" and the
"giant size".

(actually, there are an infinite number of sizes of infinities, but that step is less mind boggling than the first one)

That was fascinating and I enjoyed it but I do have trouble with the
concept of different 'sizes' of infinity although I feel sure they
exist. 0r do they? What do I mean by 'exist' anyway? :-)

I feel a bit like the definition of the size of space in the
'Hitchhiker's Guide to the Galaxy'. "Space is big!' If you think its a
long way to the corner shop..."

I can't remember it properly!

Thanks
--
David CL Francis

On Sat, 6 Nov 2004 at 13:49:45 in message
, Blanche
wrote:
Jose wrote:
Of course infinity cannot be treated as an ordinary number but it still
'exists' and you can compute larger and larger numbers as long as you
like.

Infinity is not a number. Because of this, when you get infinity as an

Actually, in the world of digital computers, infinity *is* a finite
number.

No, it just needs a computer memory of infinite size to display it and I
guess an infinite time to move the bits into this very large register.

(*evil laugh*)

(*Hysterical laugh*)
--
David CL Francis

That was fascinating and I enjoyed it but I do have trouble with the concept of different 'sizes' of infinity although I feel sure they exist. 0r do they? What do I mean by 'exist' anyway? :-)

Infinities (it works for finites too) are compared by attempting a one to one comparison between their elements. If one can achieve this, then they are the same "size". If it is impossible to achieve this, then one of them is "bigger".

The number of errors that can be made is bigger than the number of people.
The number of ways to skin a cat is bigger than the number of cats.

There are actually an infinite number of sizes of infinity. The best way to think of an infinity is that it's the size of a set that contains all the elements. For example, the "size" (quantity of elements) that a set containing all the integers is
infinite. It is however smaller than the set of real numbers between zero and one, as shown in my original post. More generally, the quantity of subsets of an infinite set is bigger than the set being subsetted (to coin a word). (a subset is a set
which contains "some" of the elements of the original set, and no other elements, where "some" could be "all" or could be "none". Sets are designated (sometimes) by listing their elements inside curly braces; a few (i'll show three) subsets of the
days of the week are {sunday} and {tuesday, thursday, friday} and {} (that last one being the empty, or "null" set).

So, the set of positive integers ( {1,2,3,4,...} ) is not as large as the set of subsets of the positive integers ( {}, {1}, {2}, {1,2}, {1,2,3,4}, {8, 9, 423}, {500}, ... )
Note that it is perfectly fine for a set to contain sets as elements. Don't confuse a set with an element however: 1 is different from {1}. "Monday" is different from "the SET of days between Sunday and Tuesday". A car (something you can drive)
is different from "car" (the word describing something you can drive).

Ceci n'est pas une pipe.

Jose
--
Freedom. It seemed like a good idea at the time.
for Email, make the obvious change in the address.