Start with local horizon coordinates, i.e., a rotating Cartesian coordinate system with the x, y, and z axes aligned as follows:
x = East; y = North; and z = Zenith.
Let r, v, and a be the position, velocity, and acceleration vectors.
Thus v = dr/dt and a = dv/dt.
The equations of motion are
a = - g - 2 omega cross v,
where the gravitational acceleration is g = -|g|(0,0,1),
the Coriolis acceleration is - 2 omega cross v,
and "cross" is the vector cross product.
The angular velocity of the earth is omega = |omega| (0,c,s), where c and s are respectively the sin and cos of north latitude. Both c and s will be positive for a point in the Northern hemisphere.
The magnitude of omega is approximately |omega| = 2 pi / (24 hours).
The initial position is r = (x,y,z) = (0,0,H), where H is the height of the tower. We can assume H < 1 km, which is a tiny fraction of the Earth radius. The initial velocity is v = (0,0,0).
Without actually solving the equations of motion, we can see that the solution will be like this: The downward acceleration will cause a buildup of downward velocity, such that the Coriolis force will cause an Eastward acceleration, which will cause a buildup of Eastward velocity, such that the Coriolis force will impart a Southward acceleration. Thus the trajectory will veer to the Southeast.
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