Tuesday, May 25, 2010

Flying back

Those of us originating on the right side of the atlantic ocean are familiar with a little quirk of international flights: the flights home are shorter. Specifically, going from Tel Aviv to New York takes about one hour longer than going the other way around.

This is an oddity, and the very first explanation that comes to mind the rotation of the Earth. After all, our naive image of a plane going up in the air might be something a little like a rock being thrown up from a moving cart, and we would imagine the plane to pick up some relative speed by not rotating as fast as the Earth. Is this a factor in the plane's movement?

This gives us a perfect chance to use the Earth Units we introduced a few weeks ago. Specifically I'll use the Earth meter (equal to the radius of the Earth, which I'll dub e-m) and Earth second (one day, e-s).

First we want to figure out velocity of the airplane compared to the ground. When it is grounded, the plane and the Earth's surface both have an angular velocity of 2π 1/e-s; they do one revolution per day. This means the plane's linear velocity is 2π e-m/e-s, and its angular velocity once it's airborne is
\[2 \pi \cdot \frac{(1\; \rm{m_\oplus})}{(1\; \rm{m_\oplus}+A)} \;\rm{s_\oplus^{-1}},\]
where A is the altitude. That's the one number I'm going to pull out of thin air here; that being the thin air of the cabin where they always announce that we have attained a cruising altitude of 30,000 feet. In real people units, that's about 9,000 meters, or 9 kilometers - round it up to 10. Going back to the Earth day post 1 e-m is about 6380 km, so that the angular velocity of the airplane is about 0.9984 (2π) 1/e-s, and relative to the ground it is
\[0.0016 \cdot 2 \pi \;\rm{s_\oplus^{-1}}\]

So, over a journey of length of about 0.5 e-s, the overall distance traveled due to this effect would come to about
\[0.0008 \cdot 2\pi \;\rm{m_\oplus}.\]
Tel Aviv and New York are both in the mid-northern hemisphere and seven time zones apart, so a first-order estimate of the distance between them would be about
\[\frac{7}{24}\cdot 2\pi \;\rm{m_\oplus} \approx 0.29\cdot 2\pi \;\rm{m_\oplus}.\]

Overall, it looks like this effect is negligible. Indeed, anyone who gives the matter a second thought would notice that the planes should go faster when traveling westwards, as the Earth spins eastwards toward the rising sun. Anyone who looks even further into the matter finds that eastbound and westbound planes simply take different routes across the Atlantic, leaving us with a rather more mundane and less exciting explanation.

Still, I won't complain if it makes my flight any shorter. Now, if you'll excuse me, I have some beaches to catch up with.

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