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

Question #126719

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

Expert's answer

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.

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