The Trip to the Sun demonstrates that, due to its curvature,
the Earth falls away below you on your journey. The
curvature of the Earth can be noticed by watching a ship
sail beyond the horizon. The ship sinks below until only
the highest parts of the ship can be seen. From the top of
a tall building you can see much further. Clint Brookhart,
in his book called Go Figure, Using Math to Answer
Everyday Imponderables (1998, Contemporary Books)
gives an approximate formula for determining the
distance to the horizon for different heights above the
ground. The formula is said to be accurate to about 0.5%,
and to work for altitudes of up to 180,000 feet. The
formula is given as:
Distance D (in miles) =1.225 (h)1/2 where h is the height
above the ground in feet.
For example, a person at the seashore whose eyes are 4
feet above the ground can see an ocean horizon about
two and a half miles away. From the top floor of the
John Hancock Building in Chicago, which is 1,440 feet
above the local ground, one can see 46.5 miles. And for
an airplane flying at 40,000 feet, the horizon would be
245 miles in each direction. Notice that the formula
makes sense as it predicts the horizon to be zero distance
away for a height of zero feet. That means that an ant
with its eye to the ground has a horizon that is very near,
and about zero.
The formula above doesn't take into a small optical
effect caused by the fact that air gets less dense with
altitude and can bend the light slightly. This means that
the different layers of air you see through act like a lens
and bend the light from beyond the horizon over the
horizon into your view. So you can actually see a bit
further than the formula above would predict. A revised
formula more accurately describes the distance to the
horizon, taking into account the refraction or bending of light. It is attributed
to noted geographer Arthur Strahler and means you can see about 8% farther than
the
geometry would indicate: D = 1.317 (h)1/2
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