[Guest]

A pilot once flew north 500km, after which he changes his direction to fly west for another 500km. Again he changes his direction and this time south covering 500km, and as you might expect, he finally flies east for another 500km.

Is he at the same place from where he started?

Edited April 27, 2008 by esjohnson

[akaslickster]

those are not directions you gave , they are names of the airlines. good one.

no because it states once north.

Edited April 27, 2008 by akaslickster

[Guest]

those are not directions you gave , they are names of the airlines. good one.

no because it states once north.

No Akaslickster, they are directions, and by 'once' I meant past tense.

Edited April 27, 2008 by esjohnson

[bonanova]

Yes - if he began the trip 250 miles S of the equator.

Otherwise not.

If he started North of that latitude, he ends up somewhat west of his starting point.

That's because his western leg covers more longitude than his final eastern leg.

By the same reasoning, if he starts out south of that latitude, it covers less longitude

and he ends up east of his starting point.

Only if his west leg is 250 mi N and his east leg is 250 mi south of the equator

will he end up exactly where he started.

Edit - change miles to km.

Sorry to be so US-centric.

[Guest]

Yes - if he began the trip 250 miles S of the equator.

Otherwise not.

If he started North of that latitude, he ends up somewhat west of his starting point.

That's because his western leg covers more longitude than his final eastern leg.

By the same reasoning, if he starts out south of that latitude, it covers less longitude

and he ends up east of his starting point.

Only if his west leg is 250 mi N and his east leg is 250 mi south of the equator

will he end up exactly where he started.

Edit - change miles to km.

Sorry to be so US-centric.

Excellent Bonanova, you are exceptionally great.

If its your reply, surely its gonna be correct !!!

No doubts at all.

[bonanova]

[spoiler=And there are other solutions... ]There are infinitely more possible starting points...

[1] 500km south of a latitude where a 500 km west leg will put him exactly north of where a 500 km west leg would have put him if he'd done that leg first. It would involve covering all the longitudes once plus that amount. I could do the math if I were at a desk, but I'm typing this from my bed... but you can visualize this if you think about it. It's maybe about 600 km S of the North pole.

[2] a little north of [1] and covering all the longitudes twice, plus the first-leg-west amount.

[3] a little north of [2] and covering all the longitudes three times ... etc.

[abhisk]

Excellent Bonanova, you are exceptionally great.

If its your reply, surely its gonna be correct !!!

No doubts at all.

Totally agree with you there.

[bonanova]

I've calculated the other solutions, approximating the near polar region as a flat map.

Referring to the diagram, where L = 500 km:

The pilot starts out L+r km from the North Pole, then:

• flies north L km - to a distance r km from NP.
  
• flies west L km - in doing so, he circles the pole n times plus a degrees longitude.
  
• flies south L km.
  
• flies east L km - returning to starting point.
  

West leg:

[1] L = r[2npi + a]

East leg:

[2] L = a[r + L]

Define f = r/L. then from [2]

[3] a = 1/(1+f)

Substitute [3] into [1] and simplify:

[4] f² + f - 1/(2npi) = 0 which solves to

[5] f = [-1 + SQRT(1 + 2/npi)]/2

The starting distances from NP are given by 500 (1+f) km.

Here are the values for n = 1, ..., 10

569.83

537.04

525.25

519.16

515.44

512.93

511.12

509.76

508.69

507.83 km from NP.

[Guest]

Great job bn.