Q:

Helium balloon energy gain

I answered a question about a "vacuum balloon" and came up with a problem I feel should be simple, but I cannot find the answer. Imagine a zero-drag balloon of a density $a$. It floats up to a height $h$ against gravity of constant acceleration $g$ (we assume small $h$ relative to Earth's diameter). The atmospheric density is given by a function $\rho(h)$ (for simplicity, deal with all densities as multiples of sea level density). At this point, what velocity has the balloon gained?
My first idea is to assume a cubical balloon and find the loss of potential energy of the entire column of air above it between its start and stop positions dropping by its side length, but then all of the column will be slightly underpressured, causing more air to drop, and I cannot tell if that is going to make a significant contribution to the result.

A:

To get a velocity, you need an acceleration, and that requires a mass. So, find the buoyant force as a function of altitude and integrate (including gravity) to find the loss of potential.  Use that to find the gain in kinetic energy. With a real balloon, drag is significant and it quickly reaches a terminal velocity.

