Suppose that, while lying on a beach near the equator watching the Sun set over

Question

Suppose that, while lying on a beach near the equator watching the Sun set over a calm ocean, you start a stopwatch just as the top of the Sun disappears. You then stand, elevating your eyes by a height H = 1.85 m, and stop the watch when the top of the Sun again disappears. If the elapsed time is t = 13.2 s, what is the radius r of Earth?

Hint:

The Sun disappears when it lies along a tangent to the Earth's surface and through your eyes. Draw a tangent for each position of your eyes. For the second sunset, draw a right triangle involving Earth's radius and your eye height in the elevated position. The triangle involves the angle through which Earth rotates in the given time interval.

Explanation / Answer

In the lying position draw a tangent to the equatorial circle at the point A . When the earth has rotated through an angle , after a time 13.2 s , draw the radius of the circle (passing through your position on earth after 13.2 s) and extend it to point B by a distance 1.85 m where it touches the tangent. We get triangle with sides R , (R+1.85) and AB .

= 2*pi*13.2/24x3600 = 95.9931 * 10^-5 rad (1)

sin = = AB / (R + 1.85)
= {( R+ 1.85)^2 - R2} / (R + 1.85)..............(2)
= sqrt(2*1.85/R)

From (1) and (2) , equating the values of , squaring both sides.

sqrt(2*1.85/R) = 95.9931 * 10^-5
R = 4.0153 × 10^6 m

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