I read that in the 16th and 17th century, the question of whether the Earth rotates around its axis or all celestial bodies rotate around it was extensively debated. One of the anti-rotation arguments was that objects dropped from high places should move from the true vertical due to the ground having traveled meanwhile. Since the Earth does rotate, I'm trying to quantify the effect and whether it could be measured at the time. Would appreciate my argument being checked for basic sanity and correctness.
Suppose I climb a 100 meter tower and, standing near its edge, let go of a brick. The Earth rotates with the angular velocity $\omega = \frac{2\pi}{24\cdot 3600\ \mathrm{s}}$, and the speed of the base of the tower vs. the brick at the top before I let go of it are $R\omega$ vs. $(R+100)\omega$. This means that horizontally the brick is moving with the speed of $100\omega ~= 0.00727 \,\mathrm{m/sec}$ relative to the ground, while vertically it's dropping with the uniform acceleration $g=9.8\ \mathrm{m/s^2}$ and will hit the ground in $4.52$ seconds, having travelled $0.03\ \mathrm{m}$ horizontally. If I do the same from the Empire State Building ($381\ \mathrm{m}$), it comes out as about $22\,\mathrm{cm}$ (if the height grows by a factor of $X$, the distance travelled horizontally grows by $X^{3/2}$, the speed increase contributing $X$ and the time to drop $\sqrt{X}$).
So my questions:
Is this analysis basically sound? I realize I'm using the uniform vertical acceleration which is an approximation, and I'm using the circular motion of the Earth's surface when estimating velocities, but not when using them to find time and distance traveled. I guess the more exact calculation would be to treat the brick as a satellite on an elliptical orbit around the center of the Earth with the given initial position and velocity, find its orbit equation or simulate numerically, and find its intersection with the Earth's surface. It seems a bit daunting a task at the moment. Would that give substantially different results?
Is air resistance an important factor to take into consideration, for estimating the horizontal distance traveled? (if it is, I guess this still answers the same question for the Moon).
Assuming my estimates aren't too far-off, is that something that can be tested in a real experiment either now or in the 17th century? I think that'd depend on how exactly a true vertical we can guarantee the tower's wall to be?