How far to the east will the object land? Express your answer in terms of h, g, and the angular speed ω of th?
Consider a tall building located on the Earth's equator. As the Earth rotates, a person on the top floor of the building moves faster than someone on the ground with respect to an inertial reference frame because the person on the ground is closer to the Earth's axis. Consequently, if an object is dropped from the top floor to the ground a distance h below, it lands east of the point vertically below where it was dropped.
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Let R be the earth's radius (this will cancel out in the end). Then, as the earth rotates, the bottom of the building is traveling eastward at speed ωR, while the top of the building is traveling eastward at a rate of ω(R+h). This means a "stationary" ball at the top is traveling at a relative eastward speed of: ω(R+h) − ωR = ωh, relative to the ground. It maintains that excess horizontal speed as it falls vertically. It takes t = √(2h/g) seconds to fall, and in that time it covers a relative horizontal distance of: x = vt = (ωh)√(2h/g) = ω√(2h³/g)