Helium-filled meteorological balloons are regularly used to provide measurements of pressure, temperature, humidity, wind speed and other useful properties of the atmosphere at different altitudes. Consider such a balloon occupying a constant volume V (i.e. not expanding nor contracting). For rough purposes, assume that the air density ρ is a linear function of altitude h, decreasing from a maximum value at ground level to 20% of that value at h = 10 km.
a) Determine the maximum total mass m_max of the balloon and instrumentation for the balloon to start rising from the ground when released.
b) Assume that the mass m of the balloon etc. is reduced to 0.8 m_max. Determine the altitude in km at which the balloon will stop rising.

Contributed by Stavros Tavoularis, Department of Mechanical Engineering, University of Ottawa, Ottawa, Canada. Image from Jet Propulsion Laboratory.