Answer to Question #126719 in Mechanics | Relativity for stephen smith

Question

hi i live in hawkesbay new zealand and get to see the sun come up over the sea most days , it takes about 7 minutes and i have been wondering with the spin of the earth how far have i traveled in those 7 minute , cheers

Answer 1

The latitude of Hawke's Bay is approximately $39^\circ25'S$ . Therefore, the linear velocity of rotation of Earth is the equatorial velocity multiplied by the cosine of the latitude.

We know that the Earth makes a rotation in approximately 24 hours, so the linear equatorial velocity is

$v_e = \dfrac{2\pi R_{\oplus}}{T}= \dfrac{2\pi\cdot6378\,\mathrm{km}}{86164\,\mathrm{s}} = 465\,\mathrm{m/s}$ .

On the latitude of the Hawke's Bay the velocity is $v = v_e\cos\varphi = 359 \,\mathrm{m/s}$ . So in 7 minutes you travel

$7\cdot60\cdot0.359 = 151$ km due to the rotation of the Earth.

Worth noting, the orbital velocity of the Earth is 29.8 km/s, so in 7 minutes you travelled $7\cdot60\cdot29.8 = 12516$ km due to the orbital motion of Earth.