while lying on the beach......?

Question

a man (lying down) wachtes the sun set. he starts a stop watch as soon as the sun disapperas. he then stands (height of his eyes from ground is 1.70m) and stops the watch when the sun disapperas again. the time elapsed on the watch is 11.1s. what is the radius of earth?

Answer 1

h = 1.7 m
t = 11.1 s
R = ?

Time taken for 1 rotation by Earth = 24 hrs = 86400 s.

The angle through which the Earth rotates for the sun to disappear again = q = (11.1 / 86400) * 2π (360 degress = 2π rads)

Now cos q = R/(R+h)

R (1-cos q) = h cos q

R= (h cos q) / (1- cos q)

Now you could substitute the values of h and cos q to evaluate R. But you can also evaluate the expression using limits.

Now, if you observe q is very very small. i.e: q -> 0.

(h cos q) / (1- cos q) -> h (cos q + cos²q) / (sin²q).

Now sin q -> q as q -> 0 . Also cos q -> 1 as q -> 0.

R = 2h / q²
R = 5217.95 x 10³ m
R ~ 5218 km

Answer 2

We know the correct answer is R = 6378 km. Let's see if we can come close to that.

This is similar to the problem of determining distance to the horizon. See link below. In the no-refraction diagram in that link, h = 1.7 meter. R + h is the hypotenuse of triangle OCG. The side adjacent to angle dg is side CG = R. The Earth rotates 15 deg/hour, so the angle dg is 15*11.1/3600;
and cos(dg) = 0.999999674
1-cos(dg) = .000000326

R = (R+h)cos(dg)
1 = (1 + h/R)cos(dg)
1- cos(dg) = (h/R)cos(dg)
(1-cos(dg)) = h/R
R = h/(1-cos(dg)) = 1.7/.000000326 = 5,215 km. Not quite the accuracy we should hope for, but not unreasonable considering all the likely sources of error. Two observers at the top and bottom of a tower could get a much better result.

Maybe you will get a better result using the refraction-considered diagram.

Answer 3

Using h = 0 for the height of his eyes when lying down,
θ = 360(11.1/86,400)
R = (R + h)cosθ
R = h/(1 - cosθ)
R = (0.0017 km)/(1 - cos(360(11.1/86,400)))
R ≈ 5,218 km

This does not agree well with the known radii of 6,356.750 km (polar) -- 6,378.135 km (equatorial)
It could, however, represent the E-W radius at about 55° n or S latitude.

Answer 4

well sun cant be disappear twice sun only ones a day set not twice

Answer 5

Half of the diameter. which is roughly egual to 5300km

Answer 6

The answer can't be solved. There is an elephant in the way.

Answer 7

((3 600 * 24) / 1.1) * 1.7 = 133 527.273