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Okay what am I missing here. Isn't the correct answer New York?
No, because you are traveling West at a slightly higher latitude than your latitude when you are traveling East. So you will end up going a bit further West in terms of degrees longitude. The Empire State Building is in Manhattan so it's very close to the New Jersey border.
Doesn't this only hold if you're measuring the direction at each juncture rather than working from the NSEW coordinates of the Empire State Building?
Maybe that doesn't sound like the most intuitive way to think about it, but in my defense it's kinda similar to how bullseye navigation works.
(Also, since we're in an aircraft, pedantically we would need in theory need to account for the rotation of the Earth, which we can't do without knowing the airspeed.)
I don't understand. Let's suppose the helicopter has completed the first (northbound) leg of its journey and is about to turn West. To me, "west" would mean that the helicopter turns 90 degrees to the left. What direction would the helicopter go at that point if one were "working from the NSEW coordinates of the Empire State Building"?
If the helicopter makes 90 degree turns at each turn, it will return to the Empire State building, making a square with 300 miles to a side, right? Helicopters fly, so they don't need to respect the Earth's curvature - they can fly in a plane, at least until they exceed their operating altitude. So the 2D map view would be basically correct (if we don't worry about the Earth's rotation). This is the mental model I had in my head that told me we would return to New York (which now I feel a bit dumb about.)
But when after turning West, we turn back South, if we flew to the South pole from our location, we would collide with (intersect) with an aircraft flying due South from the Empire State Building (at the South pole). The lines aren't parallel; they intersect. So when we made our turn South, we will fly a different course if we turn "South" as in "South by compass" or if we turn "South" as in "parallel to a line extending due South from the Empire State Building." And if we fly South by compass, we won't be making a 90 degree turn, for the same reason that squares of latitude and longitude aren't perfect squares.
...I think that's all correct, but it's been a long time since I've thought about this, so thanks (it's good for me).
Ahh, I understand. Except that compass directions such as "north" are typically understood in respect of the Earth's curvature. So for example consider the following questions:
(1) A helicopter starts at the equator. How many miles due north does it need to fly before it reaches the north pole?
(2) A helicopter starts in NYC. How many miles due north can it fly before it is impossible to fly any further north?
Most people would reasonably understand these questions to have straightforward finite answers. But if "north" is understood to be on a plane which is tangent to the Earth at the point of departure, then answers are (1) it will fly forever without reaching the North Pole; and (2) it can fly North forever.
Yes, I think this is all right. The more erudite reason I can give for my initial interpretation is that in certain contexts (such as the military) navigation is done in reference to a fixed point (bullseye). But probably I was just having a grug-brained moment!
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Earth is a sphere, not a plane. Moving 100 miles west (on a line of latitude, not on a great circle) after you go north moves you through more degrees of longitude than moving 100 miles east does after you return to the latitude where you started.
At the extreme: Starting at the equator, if you go πr/2 north, 0 miles west, πr/2 south on a meridian of longitude 90 degrees east of the one where you started, and πr/2 west, then you will end up where you started (having traversed a triangle with three 90-degree angles).
Okay, but we're in an aircraft, which (if it wants to) can move in a plane relative to a fixed starting point, more or less.
...actually, I guess this means the answer to the question is unknowable with the information given, since without knowing the speed of the helicopter we can't ascertain if the rotation of the Earth impacts it at all.
ETA - but yes this otherwise makes sense given that we're modeling the directions based on the actual polar coordinates rather than based on the fixed starting point. But if you were modeling directions based on your fixed starting point rather than current position and didn't have to bother with the rotation of the Earth, you'd make a big square with 300 miles to a side.
Cardinal directions refer to the surface of the earth, not some abstract fixed plane.
Or at least to an abstract plane that is fixed to a point on the surface of the earth.
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