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Why if the earth is spinning at 1000mph a person falls from 10 stories his body didn't land about 1.4 miles away but right down as if the earth didn't spin at all

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I trust your calculation and he does end up $1.4$ miles away. But while he falls, the earth spins and the building from which he falls moves $1.4$ miles also. So, he ends up next to the building!

Why not use a ball next time? It's much less bloody.

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It's essentially because right before you enter free fall, you are stationary with respect to the earth, so you are moving with the earth, and when you fall, you still maintain that same tangential speed (tangent to the surface of the Earth) that you had before you entered free fall.

Why do you maintain that tangential speed that you had right before free fall?

Well, it follows from Newton's second law. Specifically, to good approximation, there are no external forces acting on you in the tangential direction, so your acceleration in that direction is zero, and you thus keep moving in that direction with the same speed.

Some mathematical detail

The earth spins with a period of about one day. Let's assume that you're at the equator and that the spin axis goes through the poles, then when you're standing on the building, you and the earth are both moving with a tangential speed of about $$ v = R_E\Omega = (6378\,\mathrm{km})\left(\frac{2\pi}{1\,\mathrm{day}}\right) \approx 464\,\mathrm{m}/\mathrm{s}. $$ If we assume that 10 stories is about 100 feet, which corresponds to falling for about $2.5\,\mathrm s$, and given the tangential speed just computed, this means that you will travel about $1160\,\mathrm m$ in the tangential direction. This is such a small amount compared to the circumference of the Earth's equatorial circle, which is on the order of $2\pi R_E \approx 40,000\,\mathrm{km}$ that essential what happens is that you, and the point right below you before you jump, tangentially move exactly in sync until you meet up; the curvature of the Earth is essentially negligible.

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  • $\begingroup$ That is a great answer but if that is true why does a plane from east to west take the same time as west to east one plane should take very little time while the other moving slower than the rotation of earth should never reach its location I.e a car going 50mph will never catch the car going 60mph the planes have no attachment to the physical earth $\endgroup$
    – Terrance Riggs
    Commented Mar 15, 2013 at 20:39
  • $\begingroup$ Like the person falling from the building, planes do initially (just before takeoff) travel with the earth. $\endgroup$
    – joshphysics
    Commented Mar 15, 2013 at 20:51
  • $\begingroup$ And the air moves nearly with the earth, too. But not exactly, see Trade wind. $\endgroup$
    – erik
    Commented Mar 15, 2013 at 20:55
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    $\begingroup$ There's a tiny eastward deviation from Coriolis force effects, with maximum at the equator. $\endgroup$
    – Michael Luciuk
    Commented Mar 15, 2013 at 22:28

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