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A maths problem for Spitfires (and other aircraft)

Postby Admin (Philstyle) » Wed Aug 08, 2018 8:37 am

Here's the problem:

At any specific altitude, how far away from the centre of your aircraft, if the land/ ground feature you can see immediately off the wingtip?

My question is inspired by a quote from Johnnie Johnsons's book "ing leader" where he talks about the huge land distances which were obscured yb the Spitfire's wings at altitude. I ahve also noticed online that many people tend to reference their location by what they can see off their wing tip, or behind their trailing edge. Often, it seems to me, people massively under-estimate how far away those land features actually are from their current position.

So, the problem is relatively basic trig.
In level flight (assuming alos level attitude) by drawing a triangle from the pilot's eye posiiton to the wingtip we have a hypotenuse (c) and and angle "B" relative to the aircraft's z-axis.
The distance from the Pilot's eyes down the a-axis to the Wing Datum line gives us out "a" side of the triangle.
The distance along "b" is simple half of the aircraft's wingspan.
See image:
Image

We can then simply extend the z-axis line down the full altitude of the aircraft to get the totla length for the "ground positon" "a" side of th etriangle. Then using trig we can work out how far away down the hypotenute (c) the landmarks are, at any given altitude.

I already know the following:
side b = half of 11.23m (11230mm) = 5615 mm

Sides c and a I cannot know until I can work out what the distance is (typically) between the pilot's eye level in the cockpit and the wing datum.
I have this document which has some measurements:
https://i.imgur.com/gtmjNU4.png

I think length "L" is a start, but I need to project tha up tot he eyeline of a pilot... say 15cm below the canopy roof . . .

Does anyone have access to a diagram or a best-guess of the distance between a pilot's eye level and the "wing datum" of a spit IX?
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Re: A maths problem for Spitfires (and other aircraft)

Postby Admin (Philstyle) » Wed Aug 08, 2018 8:50 am

OK, based on using a ruler in the image (knowing what "L" is, I just assumed a consistent scale and measured the distance on the image with a rule....) I provided in the previous I am going to call side "a" to be 1350mm

So our triangle wiitnh the spitifre itself is:
a = 1350 mm
b = 5450 mm
c = 5615 mm

B = 76.0882 degrees
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Re: A maths problem for Spitfires (and other aircraft)

Postby Admin (Philstyle) » Wed Aug 08, 2018 9:02 am

Aaaaaaand, the results are in (needs checking.. Dietrich?)

For any given altitude (in level flight) this ready reckoner will tell you how far away a land-feature on your wingtip is from your aircraft's location on the ground, approximately:

Incidentally, as a quick rule, one can basically say:
0 to 10,000ft - wingtip features are 1km away for each thousand feet altitude
10,000 to 20,000ft - wingtip features are 1km away +3 for each thousand feet altitude
20,000 to 30,000ft - wingtip features are 1km away +5 for each thousand feet altitude

Image
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Re: A maths problem for Spitfires (and other aircraft)

Postby DD_Fenrir » Wed Aug 08, 2018 9:45 am

Don't forget to account for the dihedral Phil, IIRC it's 6°
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Re: A maths problem for Spitfires (and other aircraft)

Postby Admin (Philstyle) » Wed Aug 08, 2018 10:44 am

DD_Fenrir wrote:Don't forget to account for the dihedral Phil, IIRC it's 6°


This would chanfe the "B" angle to around 74 degrees. The overall difference in distance is about 10%.
I will update with a new table shortly.
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Re: A maths problem for Spitfires (and other aircraft)

Postby Admin (Philstyle) » Wed Aug 08, 2018 11:16 am

Using the 74.02 degree agle for B (assiming wingtips are 50cm higher due to the dihedral) we gett he following:

Boils down to a more simple rule too:
0 to 10,000ft = 1km distance per 1,000ft altitude
10,000 to 30,000ft = 1km distance per 1,000ft altitude +1km

Image

So at 8,000ft a feature on the wingtip is 8km away from your map location
At 23,000ft a feature on the wingtip is 24km away from your map location
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