Finding radius of Earth through observation of Sun's motion The question I'm about to pose is from a physics book I had recently bought. Since I am very interested in physics I am quite keen in understanding how this question can be solved. Before I present this question, I would like to explain the context in which this question was posed. It is a question related to the chapter entitled - Measurement, dealing with the understanding of the basic quantities like mass, time, length etc. Now let me pose the question:Suppose that while laying down on a beach near the equator of the Earth, you watch the sun setting below the horizon. You start a stopwatch the moment the top of the sun disappears below the horizon. You then stand to a height of 1.70 metres, and stop the watch when the top of the sun disappears again. The time interval is exactly 11.1 seconds. What is the radius of the Earth?I must inform you that this is not a homework question or any such related assignment, I am simply asking a question whose solution I have been unable to find myself or online. I hope you'll be able to answer this question.
 A: You could use the distance to the horizon formula.
For $h=1.70$ m the horizon is $d=4.7$ km far away (roughly). When it sets again, the sun passed this distance. But we know that this must be equal to an angle of
$$ \gamma=\frac{11.1 \mbox s}{24 \mbox h}2\pi$$
but $\gamma = d/R$. So
$$ R = \frac{d}{\gamma}= \frac{24*60*60 \mbox s}{11.1 \mbox s}\frac{4.7 \mbox{km}}{2\pi}=5825 \mbox{km}$$
The answer should be 6350 km, but it's still a good result. Consider that a very small error on the time leads to big uncertainties on the result. A difference of one second would lead to the correct value.
Very nice question.
EDIT: I understood that the value Wikipedia reports for the distance of horizon uses the radius of Earth! This is not valid.
So we can use the original formula, $d=\sqrt{2Rh}$, then again $\gamma = d/R$ so that
$$ \sqrt{2Rh}=\gamma R$$
ignoring the $R=0$ solution
$$ 2h = \gamma^2 R $$
from $\gamma = 2\pi *11.1\mbox s/24\mbox h$ one obtains
$$R = 5223 \mbox{ km}$$
Notice that it is less precise: now we are using the experimental error twice: to estimate the horizon distance and to find $d$.
