is relative velocity formula wrong?

Question

I have a little but mind twisting problem.

let $\omega$ be the angular velocity of the earth. $A$ be the center of the earth and $B$ be a point on the surface of the earth. then join $A, B$ and extend up to $C$. assume that $BC$ is a pole. then by the relative velocity formula, we have

$$\begin{align}V_{BC}&=V_{BA}+V_{AC}\\ V_{BC}&=-\omega AB +\omega AC\\ V_{BC}&=\omega (AC-AB)\\ V_{BC}&=\omega BC\\ \end{align}$$

this means that there exists a velocity at the top of the pole relative to the base of the pole. and if something drops from the top of the poll it would not end up at the base but somewhere else which contradict the real world experience.can someone explain me where am I wrong? someone suggest that the formula for relative velocities is wrong. he says that we cannot add two velocities from two reference frames.

what I think is this is wrong because relative to the base $B$ of the pole there is no relative angular velocity at the top. so $V_{BC}=\omega BC=0$

Answer 1

You are not wrong. An object released from the top of a pole has slightly more angular momentum than the base of the pole so it will land to the east of the base of the pole by a very small distance. The reason we do not need to worry about this in real life (most of the time) is that the effect is very small indeed.

For a 5 meter high pole, the time taken to fall from the top to the bottom of the pole is approximately 1 second. The additional velocity at the top of the pole due to the Earth's rotation is $10\pi$ meters per day. This is about 0.36 mm per second. So the displacement away from the base of the pole is about 1/3 of a mm. You would have to carry out a very accurate experiment to be able to detect such a small effect.

However, effects due to the rotation of the Earth do become significant when calculating the trajectory of artillery shells, which can be in flight for 10s of seconds.

Answer 2

The formula is correct. As you have drawn it, both $B$ and $C$ are moving directly to the left and $C$ is moving faster because it is farther from the center. The two velocities you are adding are both in the same reference frame, the one centered on the earth and not rotating. In normal experience the poles we have are so small compared to the radius of the earth that you can't see that the object does not fall straight down. If you had a pole that wasn't small compared to the earth radius the object would follow an ellipse until it hit the earth.