# 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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