A bucket of water is pulled up in humid air

[Karol]

Homework Statement

A bucket with surface A, full of water, is pulled upwards through humid air, humidity ρ[mass/unit volume], at constant speed v₀. the cloud is at rest and the drops instantly stop at hitting the water.

Homework Equations

Newton's second law: $F=\frac{d(ma)}{dt}=\dot m v+m\dot v$

The Attempt at a Solution

The bucket that collects water isn't an isolated system, so:
$$F=\dot m v_0+\dot v m+\Delta P=\dot m v_0+0+\rho A v_0^2=2\rho A v_0^2$$

 

[insightful]

Can a bucket that is full of water accumulate any more water?

 

[jbriggs444]

The problem as posed leaves much to the imagination. Among the things left unspecified is the question that is to be answered.

At a guess, we are expected to imagine the air as a cloud of water droplets. The air has zero viscosity and is massless. The suspended water droplets merge with the bucket as the bucket's top surface is pulled through their location. The question is what force (beyond that required to support the bucket against gravity) is required due to the interaction with the water droplets.

The "Relevant equation" listed in of the OP includes a typo.
The "attempt at a solution" contains an equation that does not follow from the relevant equation.